Wireless Networks

, Volume 16, Issue 5, pp 1427–1446

Optimal aggregation factor and clustering under delay constraints in aggregate sequential group paging schemes

Authors

  • Hung Tuan Do
    • Department of Operations ManagementPurdue University
  • Yoshikuni Onozato
    • Department of Computer Science, Graduate School of EngineeringGunma University
    • Department of Computer Science, Graduate School of EngineeringGunma University
Article

DOI: 10.1007/s11276-009-0212-z

Cite this article as:
Do, H.T., Onozato, Y. & Yamamoto, U. Wireless Netw (2010) 16: 1427. doi:10.1007/s11276-009-0212-z
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Abstract

This paper considers several optimization problems of sequential paging with aggregation mechanism which has been shown to reduce significantly the paging cost of a wireless communication system. An important problem is to find the optimal aggregation factor subject to a constraint on the average paging delay. Another problem is, given a cost function that depends on both paging cost and paging delay, how to find the optimal aggregation factor to minimize that cost function. We have formulated and shown that these can be solved nicely due to the monotonicity and convexity of the average paging cost function and paging delay function. We demonstrate that the optimization problems of the aggregate factor and subnet clustering are not separable. This leads to joint optimization problems of aggregation factor and clustering that are investigated in this paper. The paper presents different algorithms to solve these joint optimization problems using the monotonicity in the aggregation factor and the number of clusters of the average paging cost and delay with the unconstrained optimal clustering and the structures of the constrained optimal clustering.

Keywords

IP pagingPaging costPaging delayJoint optimizationConvex optimization

1 Introduction

In wireless mobile networks, paging is a process to determine the exact location of a specific mobile terminal (MT) in Personal Communication Systems (PCSs) or a mobile node (MN) in Mobile IP [1] when this terminal or node is in stand-by state. Paging service is popularly deployed in wireless networks for two major benefits [2, 3]: to reduce location update cost and to save power consumption of mobile terminals or nodes.

In paging, a mobile node is not required to register its location per each inter-cell or inter-subnet move, but it needs to update its location if it moves out of a determined set of cells or subnets called a paging area (PA). In this paper, we use the term PA for location area (LA) in PCSs and registration area in Mobile IP networks. In each paging cycle, the network sends paging request to a predefined subnet of cells/subnets. If the mobile node is located, it updates its location with the network and the paging process is terminated successfully. Otherwise, the paging process will continue with another subnet of cells/subnets and so on. The paging latency (or delay) is defined as the number of paging cycles needed to locate the mobile node. Usually, there is a time constraint imposed on the paging latency.

It is well-known that paging cost and location update cost represent a trade-off in node paging. If the mobile node updates its location frequently, the network can know its location more precisely, and thereby the paging cost can be reduced. Nevertheless, the location update cost certainly increases in this case. On the other hand, if the mobile node performs location update not frequently, a larger PA must be paged to locate it when it is wanted. Thus, the paging cost incurred will be larger. Also, there exists a more obscure trade-off between paging cost and paging latency. Paging cost is proportional to the number of paging cycles and the number of subnets paged during each cycle. With the same paging algorithm, if the number of subnets being paged in each cycle is larger (i.e. the subset of subnets being paged is bigger) then the mobile node is more likely to be found faster, yet the paging cost may be higher. An important area of research on paging systems is to address these two main trade-offs. Considering the two basic trade-offs in paging, the crucial problems of research in paging are to minimize the total signaling cost of both paging and location update and to strike a balance between paging cost and paging latency, which is specified by various constraints.

It should be noted that in PCSs, paging is implemented at the link layer, while in Mobile IP networks, paging is employed at the network layer (hence, named IP paging). Some points of difference between layer 2 paging and layer 3 paging in implementation is investigated in Ref. [4]. Several IP paging protocols have been proposed in [2, 3, 5, 6]. References [7, 8] provide good surveys on mobility management in IP-based wireless systems. In [9], the authors proposed an IP architecture for systems with different infrastructure.

Some Individual IP Paging schemes in which PA construction is customized to each mobile node were first proposed in [10]. In PCS networks, construction of the registration areas adaptive to user mobility and call patterns is introduced in [11, 12]. The idea of Individual Paging is completely ramifying from the traditional paging, in which PA is fixed and common to all mobile hosts. Some merits of Individual IP paging are investigated in [13].

Most of research works in the literature are concerned with Static Aggregate Paging schemes [14] that are independent of MNs, i.e. the PAs are fixed and common to all MNs [2, 3, 15]. An important problem in this context is to construct an optimal PA to solve the trade-off between paging cost and location update cost. In [15, 16], this problem is addressed in various scenarios. For a comparative study between Individual Paging and Static Aggregate Paging schemes, please refer to [14].

However, for a given and fixed PA, a prominent problem is to solve the trade-off between paging cost and paging delay with multi-step paging. Multi-step (sequential) paging is proposed to reduce the paging cost at the expense of higher paging latency. In multi-step paging, all the subnets of a PA are partitioned into groups. Upon a paging request, all the subnets of the first group are paged at the same time in the first round then, if the user can not be located, all the subnets of the second group are paged, and so on. Usually, multi-step paging algorithms can work in both PCSs and wireless IP networks because they deal with clustering subnets and specifying the paging sequence.

The works most related to our work here are [1719]. The authors in [17] showed how location probabilities of cells can be used to cluster the cells of a LA in PCSs into groups so that the average paging cost is minimized subject to delay constraint. This is the first paper investigating sequential group paging in cellular networks. The optimization of clustering subject to a constraint on paging delay is formulated and solved using a continuous density function that approximates the discrete location probability distribution of cells. The problem is then formulated as a convex optimization problem. However, the clustering obtained is just an approximate solution because the problem of concern is discrete.

In [18], the optimization problems as presented in [17] are solved using the structures of the optimal solution and then dynamic programming is employed to solve the problem recursively in polynomial time.

We introduced paging mechanisms with aggregation in [19]. With an aggregation of paging requests, the system can aggregate k paging requests looking for k mobile users into one request packet to find all k users together, where k is a control variable. This mechanism can significantly reduce the paging cost, but at the same time may increase the paging delay as demonstrated in [19]. The aggregation of paging requests is especially useful when the rate of incoming messages is large and some level of service delay is acceptable at the beginning of a communication session. When the call arrival rate to dormant MNs in the PA (i.e. the paging request rate to the PA) is large, there will be a queue of paging requests built up at the Paging Agent, and hence, a batch processing of aggregate paging would be necessary.

Reference [19] also proposes effective paging schemes based on the balanced partitions of subnets. We can see that the sequential group paging scheme considered in [17, 18] is a special case of sequential paging with aggregation when the aggregation factor is equal to one. However, when aggregation mechanism is employed the average paging cost and delay functions become more complicated and the structure of the optimal clustering is changed as shown later in this paper.

For aggregate sequential group paging, there are several important questions left unanswered in the literature. What is the optimal aggregation factor subject to a maximum average paging delay? Given a utility function what is the value of the aggregation factor to minimize it under constraint on the average paging delay? More generally, what is the best paging scheme regarding the aggregation factor, the number of clusters, and clustering itself?

This paper concerns with these problems, more precisely optimization problems of sequential paging with aggregation mechanism. An important problem is to find the optimal aggregation factor k subject to a threshold of paging delay. Another problem is given a cost function that depends on both paging cost and paging delay, how to find the optimal k to minimize that cost function. The problem of joint optimization of both aggregation factor k and clustering is also a topic of this paper. The main contributions of this paper are as follows:
  1. 1.

    Derivation of the average paging cost and delay functions as functions of aggregation factor k and a clustering which is by ordering property represented by a partition vector v. Decreasing monotonicity of the average paging cost function and increasing monotonicity of the average paging latency with respect to k are proven. These properties are essential in finding the optimal aggregation factor k given a clustering in an effective way.

     
  2. 2.

    Proof of the convexity of the average paging cost function and the average paging delay function with respect to k. These findings will formulate a convex optimization problem if the utility function preserves the convexity. Moreover, since k is an integer, if the objective function is convex, we can use a very simple algorithm to find the optimal k.

     
  3. 3.

    Proof of the optimal stochastic ordering of subnets according to their location probabilities for a general k. With this feature, a clustering can be represented simply by a vector so called a partition vector, and thus, a sequential paging strategy is represented by this vector as well.

     
  4. 4.

    The unconstrained and constrained optimization of subnet clustering given an aggregation factor k. The monotonicity in the aggregation factor and the number of clusters of the average paging cost function and delay function for the unconstrained optimal clustering is demonstrated. This helps to solve unconstrained joint optimization problem with much less complexity.

     
  5. 5.

    The approaches to the joint optimization problems of aggregation factor and clustering. The two optimization problems are found interdependent instead of the first possible intuition that they seem independent. In this paper, the recursive structures of both unconstrained optimal clustering and constrained optimal clustering are presented. These structures can be seen as generalizations of the structures found in [18]. The constrained joint optimization problem then can be addressed using dynamic programming with polynomial time complexity. We can do better with the unconstrained optimization thanks to the monotonicity in the aggregation factor and the number of clusters mentioned above.

     

The rest of the paper is organized as follows. Section 2 describes the system model, assumptions and the formulation of the average paging cost and paging delay. Section 3 presents the main properties of sequential paging with aggregation that are essential in solving the optimization problems. The optimization problems and approaches to them are discussed in Sect. 4. We illustrate our results with some numerical examples in Sect. 5. In Sect. 6, we discuss relaxation of assumptions, extension of our analysis, and some other related problems. Finally, Sect. 7 concludes the paper.

2 System model and problem formulation

2.1 System model and assumptions

2.1.1 Network architecture

Our proposed model of sequential paging scheme with aggregation is presented in Fig. 1. There is a paging agent that initiates paging processes for each PA, preferably located at a higher level in the network hierarchy. This paging agent can be co-located with a router or even a Home Agent (HA). When there comes an IP session to an idle MN residing within the PA, the paging agent initiates a paging request message that contains the IP address of the wanted MN and sends it to the FAs of the subnets within the first group g1. Within each subnet, upon a paging request arrival, the Foreign Agent (FA) formulates and broadcasts a paging message including the IP addresses from the paging request message over the air to locate the MN within their subnets. If the MN is not found, the paging agent sends the request to g2, and so on until the MN is located.
https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig1_HTML.gif
Fig. 1

Proposed model of sequential paging scheme with aggregation

In aggregate sequential paging schemes, the paging agent can aggregate k paging requests looking for k mobile users into one aggregate paging request message to find all k users together. In our proposed schemes, this can be performed as follows: the paging agent first reads the address fields of the incoming packets and then generate a single message called aggregate paging request message including all the addresses of the MNs. As proposed in [6, 20], this implementation is readily accommodated since for efficiency, it is allowed to have multiple IP addresses in the address field. Then the aggregate paging request message is transmitted down to the subnets in the PA, cluster by cluster, until all k users are located or the entire PA is searched. An MN when getting the request destined for it will change to active mode and perform a location update to the system.

For example, there are two paging requests, which come close to each other in time, to 2 MNs: X residing in group g2 and Y dwelling in group g3 of the PA. When k = 1, the system will first page for X by paging g1 then g2 and find X in some subnet of g2. Next, the system pages all over again for Y, starting from g1 to g2 and finally it finds Y in g3. When k = 2 and suppose the paging agent aggregates these two paging request, then the system will page for both X and Y in one round. That is it pages g1, then g2 and finds X there, then it finally pages g3 and finds Y there. Intuitively, the paging cost in the first case will be significantly larger than that in the second case.

The parameters used in our analysis are shown in Table 1. The location probability of a subnet si is the probability that an arbitrary MN residing in the PA will be found in si. This location probability can be obtained from collective data, and in this paper, we assume they are calculated and available. The location probability of paging group gj is the probability that an arbitrary MN residing in the PA will be found in gj, which can be calculated as the sum of the location probabilities of all its subnets. Note that λ also represents the paging request rate. If a paging cycle time is normalized to 1 and λ > 1, there will be a queue of paging requests at the Paging Agent and therefore, a paging scheme without aggregation could cause unacceptable paging delay.
Table 1

Parameters in our analysis

Symbol

Meaning

N

Total number of subnets/cells in the PA

si

Subnet i in the PA (1 ≤ i ≤ N)

m

The number of paging groups in the PA (1 ≤ m ≤ N)

gj

jth group in the paging sequence (1 ≤ j ≤ m)

PS(si)

Location probability of subnet si

PG(gj)

Location probability of paging group gj

CG(gj)

Paging cost incurred by paging group gj

λ

Average call arrival rate to dormant MNs in the PA

We propose an improved version of aggregate sequential paging as follows. As a paging process goes, the IP addresses of the MNs that are already located will be removed by the paging agent from the list of k wanted MNs’ IP addresses. Thus the sizes of paging requests sent from the paging agent to FAs and paging messages broadcasted by FAs are maximal for paging the first cluster, but then reduced as the paging process goes. This enhanced version definitely reduces the paging overhead cost mentioned later in this section and in Sect. 6.2, while offering the same average paging delay. Although an analysis of this enhanced version would be out of scope of this paper, the assumption of negligible overhead cost would be more reasonable if the paging agent uses it.

Note that in this paper we use the abbreviation PA for paging area, while the term paging agent is not abbreviated. We also use the terms cluster and group, partition and clustering pair-wise interchangeably.

2.1.2 Assumptions

Note that inter-PA mobility of MNs is handled by a handoff procedure, which is usually required to be fast. Moreover, inter-PA mobility, which is similar to inter-domain mobility, is relatively rare in practice. To focus on the optimization of PA clustering and paging order, we assume that the handoff procedure is fast enough to assume zero missing probability as in most of related works in the literature. In other words, we assume that all the MNs are residing in the PA during a paging process, i.e. any MN of interest will be surely found in the PA. This also means no inter-PA mobility occurs during a paging process. The assumption can be formally stated in the following condition:
$$ \sum\limits_{j = 1}^{m} {P_{G} (g_{j} )} = \sum\limits_{i = 1}^{N} {P_{S} (s_{i} )} = \rho = 1 $$
(1)

In fact, this assumption is not required throughout our formulation and analysis, and thus, can be easily relaxed without changing our analysis. When there is a positive paging probability, some MNs move out of the current PA while the paging agent is paging for them, these MNs will be required to make their registration with their new PAs and subsequently, they belong to these new PAs. The current paging agent will be updated with these address changes. From now on their location management is not associated with the current PA, and therefore, not relevant to our formulation. When this assumption is relaxed (i.e. 0 ≤ ρ < 1), the term “location probability distribution” should become “truncated location probability distribution”.

Arrival process of paging request packets is uniform or Poisson with the mean rate λ.

When aggregation of paging packets is employed, the paging agent needs to process the incoming paging packets by reading their MNs address fields and then creates an aggregate packet including all the addresses of the aggregated paging packets. This processing time can be assumed to be proportional to the number of the paging packets, i.e. the aggregation factor k. However, the processing time is assumed negligible in this paper.

By including multiple addresses of MNs in an aggregated paging message, we can increase paging packet size, thus incurring some overhead paging cost. However, as in the paging message structure proposed in [6, 20], the field “Mobile Nodes’ IP addresses” is readily designed to accommodate multiple addresses. Furthermore, usually this field may account for a small fraction of a paging message. Also, as we will show when k ≥ 2, the paging cost could be reduced multiple times. Therefore, for simplicity, it is reasonable to assume that the overhead incurred by sending the aggregated packet down to the paging groups of subnets is negligible. This assumption would be even more reasonable if the enhanced version of aggregate paging mentioned in Section Network Architecture is employed. We discuss a relaxation of this assumption in Sect. 6.2.

2.2 Average paging cost and paging delay functions

Let C(k) and L(k) be the expected paging cost function and expected paging delay function with respect to aggregation factor k, respectively. Let (g1, g2, …, gm) be an m-partition of the PA. As derived in [19], the expected paging cost per each MN being paged can be calculated as follows:
$$ C(k) = {\frac{1}{k}}\left( {\sum\limits_{i = 2}^{m} {\left( {\left( {\left( {\sum\limits_{t = 1}^{i} {P_{G} (g_{t} )} } \right)^{k} - \left( {\sum\limits_{t = 1}^{i - 1} {P_{G} (g_{t} )} } \right)^{k} } \right)\sum\limits_{t = 1}^{i} {C_{G} (g_{t} )} } \right)} + \left( {P_{G} (g_{1} )} \right)^{k} C_{G} (g_{1} )} \right) $$
(2)

Note that once an aggregate paging request packet has been generated and transmitted, it acts exactly as a non-aggregate paging request, from a mobile user perspective. Basically, in (2), C(k) is calculated from the probability of finding k users in the first i clusters and the corresponding paging cost of paging these i clusters. Note that i is the smallest number of first clusters that k wanted users can be found.

The average paging delay L(k) for an arbitrary user can be seen as the summation of a half of aggregation delay and the average paging delay without aggregation as follows:
$$ L(k) = {\frac{{L_{\text{aggr}} (k)}}{2}}\; + L_{s} $$
(3)
where Ls denotes the average paging delay due to sequential paging without aggregation, and Laggr(k) represents the average aggregation delay. Ls is calculated as follows [17, 18]:
$$ L_{s} = \sum\limits_{i = 1}^{m} {P_{G} (g_{i} )} i $$
(4)
For uniform or Poisson arrival process with rate λ, the average delay due to the aggregation Laggr(k) is (k − 1/λ. Note that we count the aggregation time since the time of the first arrival. So the average delay L(k) is calculated as follows:
$$ L(k) = {\frac{k - 1}{2\lambda }} + \sum\limits_{i = 1}^{m} {P_{G} (g_{i} )} i $$
(5)
For Poisson arrival process with rate λ, the aggregation time with factor k can be computed as the time to the (k − 1)th arrival of paging packets Xk, and this is given by a Gamma distribution with moment generating function as follows:
$$ f_{{X_{k - 1} }} (t) = \lambda e^{ - \lambda t} {\frac{{(\lambda t)^{k - 2} }}{(k - 2)!}}\quad \Uppsi_{{X_{k - 1} }} (s) = \left( {{\frac{\lambda }{\lambda - s}}} \right)^{k - 1} $$

So we have: \( L_{\text{aggr}} (k) = E\left[ {X_{k} } \right] = {\frac{{d\Uppsi_{{X_{k} }} (s)}}{ds}}\left| {_{s = 0} = {\frac{k - 1}{\lambda }}} \right. \). As a result, formula (5) is used for both uniform and Poisson arrival processes.

For simplicity of presentation, we introduce the following notations. For a given set of subnets and its partition (g1, g2, …, gl), define:
$$ \begin{aligned} &A_{0} = 0,A_{i} = \sum\nolimits_{t = 1}^{i} {P_{G} (g_{t} )} \hfill \\ &B_{i} = \sum\nolimits_{t = 1}^{i} {C_{G} (g_{t} )} \hfill \\ \end{aligned} $$
(6)

Without loss of generality, assume the paging cost per subnet is 1, then we have: \( B_{i} = \sum\nolimits_{t = 1}^{i} {C_{G} (g_{t} )} = \sum\nolimits_{t = 1}^{i} {\left| {g_{t} } \right|} , \) where |gt| denotes the number of subnets in group gt.

In the new notation we express C(k) and L(k) for the set of all subnets of the PA with a partition (g1,g2,…,gm) as follows:
$$ \begin{gathered} C(k) = {\frac{1}{k}}\sum\limits_{i = 1}^{m} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)B_{i} } \hfill \\ L(k) = \sum\limits_{i = 1}^{m} {P_{G} (g_{i} )i} + {\frac{{L_{\text{aggr}} (k)}}{2}} = \sum\limits_{i = 1}^{m} {\left( {A_{i} - A_{i - 1} } \right)i} + {\frac{{L_{\text{aggr}} (k)}}{2}} \hfill \\ \end{gathered} $$
(7)
where \( 0 = A_{0} \, < \,A_{ 1} \, < \, \cdots \, < \,A_{m} = \sum\nolimits_{j = 1}^{m} {P_{G} (g_{j} )} = \sum\nolimits_{i = 1}^{N} {P_{S} (s_{i} )} = \rho = 1 \) and B1 < B2 < … < Bm = N, Bj is a positive integer.

3 Analysis

We can see that the average paging cost and paging delay are functions of the location probability distribution of the subnets, the clustering of subnets and the aggregation factor. In this section, we consider the most essential properties and forms of optimality of sequential group paging with aggregation. Note that when there is no aggregation, i.e. k = 1, we have the sequential paging schemes as presented in [17, 18]. From these findings, we will state and address important optimization problems in Sect. 4.

3.1 Monotonicity of the paging cost and paging delay functions

Proposition 1

Given a fixed clustering, C(k) is a strictly decreasing function of k, while L(k) is a strictly increasing function of k.

Proof

We will show that it holds for any positive real number k. Then it also holds for any positive integer k.
$$ \begin{aligned} {\frac{{{\text{d}}C(k)}}{{{\text{d}}k}}} = {\frac{1}{{k^{2} }}}\left( {k\sum\limits_{i = 1}^{m} {\left( {A_{i}^{k} \log A_{i} - A_{i - 1}^{k} \log A_{i - 1} } \right)} B_{i} - \sum\limits_{i = 1}^{m} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)B_{i} } } \right) \\ = {\frac{1}{{k^{2} }}}\sum\limits_{i = 1}^{m} {\left( {\left( {1 - k\log A_{i - 1} } \right)A_{i - 1}^{k} - \left( {1 - k\log A_{i} } \right)A_{i}^{k} } \right)} B_{i} \\ \end{aligned} $$
Let’s investigate the function f(x) = (1 – k log x)xk, x ∈ (0, 1]. Take the first order derivative of this function:
$$ {\frac{{{\text{d}}f(x)}}{{{\text{d}}x}}} = kx^{k - 1} - kx^{k - 1} (k\,{ \log }\,x + 1) = kx^{k - 1} ( - k\,{ \log }\,x) > 0 $$

Thus f(x) is a strictly increasing function. Since Ai−1 < Ai, we can see that the derivative of C(k) is negative, and therefore, C(k) is strictly decreasing.

Recall that \( L(k) = L_{S} (k) + L_{\text{aggr}} (k)/2 \). For a fixed clustering of subnets, LS(k) is a constant, while \( L_{\text{aggr}} (k) = {\frac{(k - 1)}{\lambda }} \) is linearly increasing in k. Hence, L(k) is strictly increasing in k. □

3.2 Optimal paging sequence

When there is no aggregation employed, it is shown that the optimal paging order, which minimizes both average paging cost and average paging delay, is the descent order of the subnet location probabilities [17, 18]. The proof is fairly easy for k = 1. In this part, we will investigate this optimal paging order for the paging with a general aggregation factor k. To prove for a general k, we can argue that it suffices to show for the case of two adjacent groups because of bubble-sorting algorithm.

Proposition 2

For sequential paging with a general aggregation factor k, paging in descent order of the location probabilities of subnets minimizes both the average paging cost and paging delay.

Proof

Appendix.

From now on, we can restrict our discussion to ordered sequence of subnets. A partition (g1,…,gm) is therefore completely specified by a so-called partition vector v = (v1,…,vm) = (|g1|,…,|gm|). Note that, given the location probabilities of subnets, a clustering is also uniquely defined by a group location probability vector a = (A1,…,Am) or a group paging cost vector b = (B1,…,Bm), where Ai and Bj are defined in (6). Hence, any one of these three vectors may be used to represent a clustering.

Paging cost function and paging delay function now can be expressed as the functions of the aggregation factor k and the partition vector v: C(k, v) and L(k, v), respectively.

3.3 Convexity of paging cost and paging delay

The convexity of the average paging cost function and affinity of the average paging delay function are essential in formulating various optimizations as convex problems. The Slater condition is easily satisfied and thus we have no duality gap. As a result, the optimization problem can be solved by the duality approach. Furthermore, for k is an integer, a more effective iterative algorithm might be employed to solve the optimization problems of concern in Sect. 4.

Proposition 3

The average paging cost C(k) and the average paging delay L(k) are convex functions in k.

Proof

Appendix.

3.4 Monotonicity of the average paging cost and delay functions for the unconstrained optimal clustering

As assumed, the sequence of subnets investigated in our analysis is ordered in non-increasing location probabilities of the subnets. A partition then is completely specified by a partition vector v.

Let C*(k) = minvC(k,v), v*(k) = argminvC(k,v) where the number of clusters m is fixed. We claim that this optimal paging cost function with respect to v is monotonous in k in the following proposition.

Proposition 4

For a fixed number of clusters, C*(k) is a decreasing function of k.

Proof

Let v*(k) be defined by a = (Ai), and v*(k + 1) be defined by \( \tilde{a} = (\tilde{A}_{i} ) \) where 1 ≤ i ≤ m. Let b = (Bi) and \( \tilde{b} = (\tilde{B}_{i} ) \) be associated group paging cost vectors.

By Proposition 1, we have:
$$ \begin{aligned} C^{*} (k + 1) = & C(k + 1,v^{*} (k + 1))\\ = & {\frac{1}{k + 1}}\sum\limits_{i = 1}^{m} {\left( {\tilde{A}_{i}^{k + 1} - \tilde{A}_{i - 1}^{k + 1} } \right)\tilde{B}_{i} } \le {\frac{1}{k + 1}}\sum\limits_{i = 1}^{m} {\left( {A_{i}^{k + 1} - A_{i - 1}^{k + 1} } \right)B_{i} } \\ \le & {\frac{1}{k}}\sum\limits_{i = 1}^{m} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)B_{i} } = C(k,v^{*} (k)) = C^{*} (k) \\ \end{aligned} $$

Before going to the joint optimization problem, we present one more property of the sequential group paging with aggregation. The paging cost and paging delay with the optimal clustering are also monotonous with respect to the number of clusters m, which is presented in the following proposition.

Proposition 5

Letv*(m) = argminvC(k,v(m)), where k is fixed. The optimal average paging cost is decreasing in m: C(k,v*(m))  C(k,v*(m + 1)) while the optimal average paging delay is increasing in m: L(k,v*(m))  L(k,v*(m + 1)).

Proof

Let v(m + 1) be the partition obtained from v*(m) by moving the last subnet sN from the last cluster of v*(m) to the new (m + 1)th cluster. Suppose v*(m) is defined by vectors (Ai) or (Bj).
$$ \begin{aligned} C(k,{\mathbf{v}}(m + 1)) = & \sum\limits_{i = 1}^{m - 1} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)} B_{i} + \left( {\left( {A_{m} - P(s_{N} )} \right)^{k} - A_{m - 1}^{k} } \right)\left( {B_{m} - 1} \right) + \left( {1 - \left( {A_{m} - P(s_{N} )} \right)^{k} } \right)B_{m} \\ = & \sum\limits_{i = 1}^{m - 1} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)} B_{i} - \left( {A_{m} - P(s_{N} )} \right)^{k} - A_{m - 1}^{k} \left( {B_{m} - 1} \right) + B_{m} \\ = & \sum\limits_{i = 1}^{m - 1} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)} B_{i} + B_{m} \left( {1 - A_{m - 1}^{k} } \right) + A_{m - 1}^{k} - \left( {A_{m} - P(s_{N} )} \right)^{k} \\ \end{aligned} $$
$$ \begin{aligned} C\left( {k,{\mathbf{v}}(m)} \right) = & \sum\limits_{i = 1}^{m - 1} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)} B_{i} + \left( {A_{m}^{k} - A_{m - 1}^{k} } \right)B_{m} \\ = \sum\limits_{i = 1}^{m - 1} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)} B_{i} + \left( {1 - A_{m - 1}^{k} } \right)B_{m} \\ \end{aligned} $$
So
$$ C \left( {k,v(m)} \right) - C \left( k,{v^\prime} (m + 1) \right) =\left( {A_{m} - P(s_{N} )} \right)^{k} - A_{m - 1}^{k} \ge 0 $$
because Am − P(SN) ≥ Am−1 ≥ 0. Finally, we have the following as required:
$$ C\left( {k,v(m)} \right) \ge C\left( {k,v^{\prime } (m + 1)} \right) \ge C\left( {k,v(m + 1)} \right) $$

3.5 Recursive structure of the optimal clustering

In the next part, we investigate the structure of the optimal clustering with and without constraint on the average paging delay. These structures are essential in solving the unconstrained and constrained optimization problem of clustering for a given k using recursive algorithms. Due to the fact that the average paging cost and paging delay for a (m + 1)-partition can be separated into the paging cost and delay of m-partition and an extra term, the optimization problem can be solved recursively for the constrained case. The recursive structures presented in this section might be considered generalizations of those discussed in [18] where no aggregation of paging requests is applied.

The following proposition states the structure of the optimal clustering for the unconstrained optimization case.

Proposition 6

For a given aggregation factor k, ifv*(m) = (v1*,…,vm*) is the optimal m-group clustering of subnets without a constraint on the average paging delay, then\( v^{*} (m^{\prime}) = \left( {v^{*}_{ 1} , \ldots ,v^{*}_{m^{\prime} } } \right)\)is the optimal m′-group clustering of the first\( \sum\nolimits_{i = 1}^{m'} {v_{i}^{*} } \)subnets, where 1  m′  m.

Proof

By induction, if it holds for m′ = m − 1, then it will hold for any 1 ≤ m′ ≤ m. Thus, it suffices to prove for m′ = m − 1.

Suppose that there exists a partition v(m − 1) of the first \( \sum\nolimits_{i = 1}^{m '} {v_{i}^{*} } \) subnets such that C(k,v(m − 1)) < C(k,v*(m − 1)). Let \( v(m) = \left( {v(m - 1),v^{*}_{m} (m)} \right), \) that is a m-group partition vector with the first m − 1 elements are the elements of v(m − 1) and the last group is the last element of v*(m). A contradiction will result with the assumption that v*(m) is the optimal partition.
$$ \begin{aligned} C\left( {k,v^{*} (m)} \right) = & {\frac{1}{k}}\left( {\sum\limits_{i = 1}^{m - 1} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)B_{i} } + \left( {A_{m}^{k} - A_{m - 1}^{k} } \right)B_{m} } \right) \\ = & C\left( {k,v^{*} (m - 1)} \right) + {\frac{1}{k}}\left( {A_{m}^{k} - A_{m - 1}^{k} } \right)B_{m} \\ > & C\left( {k,v(m - 1)} \right) + {\frac{1}{k}}\left( {A_{m}^{k} - A_{m - 1}^{k} } \right)B_{m} = C\left( {k,v(m)} \right) \\ \end{aligned} $$
Next, we consider the structure of the optimal clustering for the constrained optimization problem. Denote n as the number of the first subnets out of our total N subnets ordered in descending sequence of subnet location probabilities. Let \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{C} (k, m, n, \theta ) \) be the optimal paging cost of paging n first subnets with the number of clusters m, the constraint on the average paging delay θ, i.e.
$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{C} (k,m,n,\theta ) = \mathop {\min }\limits_{{{\mathbf{v}} \in \Uplambda (m,n)}}\;C(k,m,n,v,\theta ){\text{ subject to }}L(k,m,n,v) \le \theta $$
where Λ(m, n) is a set function that denotes the set of all partitions of the first n subnets with m clusters. The form of optimal clustering is stated in the following proposition:

Proposition 7

The following recursive equation holds:
$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{C} (k,m + 1,n,\theta ) = \mathop {\min }\limits_{j = m,..,n} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{C} (k,m,j,\theta - \left( {m + 1} \right)\left( {A_{m + 1} - A_{m} } \right)) + {\frac{1}{k}}\left( {A_{m + 1}^{k} - A_{m}^{k} } \right)n} \right) $$
(8)

Proof

$$ \begin{aligned} C(k,m + 1,n,\theta ) = & {\frac{1}{k}}\left( {\sum\limits_{i = 1}^{m} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)B_{i} } + \left( {A_{m + 1}^{k} - A_{m}^{k} } \right)B_{m + 1} } \right) \\ = & {\frac{1}{k}}\sum\limits_{i = 1}^{m} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)B_{i} } + {\frac{1}{k}}\left( {A_{m + 1}^{k} - A_{m}^{k} } \right)n \\ \end{aligned} $$
$$ \begin{gathered} L(k,m + 1,n,\theta ) = \sum\limits_{i = 1}^{m} {\left( {A_{i} - A_{i - 1} } \right)i} + \left( {A_{m + 1} - A_{m} } \right)\left( {m + 1} \right) + {\frac{k - 1}{2\lambda }} \le \theta \hfill \\ \sum\limits_{i = 1}^{m} {\left( {A_{i} - A_{i - 1} } \right)i} + {\frac{k - 1}{2\lambda }} \le \theta - \left( {A_{m + 1} - A_{m} } \right)\left( {m + 1} \right) \hfill \\ \end{gathered} $$
In order for C(k, m + 1, n, θ) to be minimized, it is necessary that the first term in brackets be minimized, i.e. the partition of all subnets in the first m clusters must be optimal subject to the varied constraint θ − (Am+1 − Am)(m + 1). □

4 Optimization problems and solutions

In the following parts, we consider several optimization problems and propose different approaches using the optimality properties and structures discussed in Sect. 3:
  • Given a clustering and location probabilities of subnets, find the optimal aggregation factor k under a constraint on the average paging delay.

  • Optimization of a joint objective function of both paging cost and paging delay.

  • Are aggregation factor optimization and clustering optimization independent?

  • Finding the optimal clustering given a specific value of the aggregation factor k.

  • Constrained joint optimization of aggregation factor and clustering.

4.1 Optimal aggregation factor under a constraint on the average paging delay with a fixed partition

Optimization problem in this section is to minimize C(k) subject to L(k) ≤ θ. We can solve this problem for the continuous variable x and then take \( k = \left\lfloor x \right\rfloor \). In this case, the problem is to minimize C(x) subject to L(x) ≤ θ.

As shown above, this is a convex optimization problem with Slater condition [21] easily satisfied for θ > minL(k) = 1. Note that L(k) is minimized when there is no aggregation, i.e. k = 1 and the blanket polling is performed, i.e. the number of groups m = 1.

So, we can solve it using standard procedure of duality without duality gap. However, this optimization problem can be solved more effectively for C(k) is monotonically decreasing and L(k) is monotonically increasing.

As a result, the optimization problem k* = argminC(k) subject to L(k) ≤ θ becomes the following:
$$ k^{*} = \max \left\{ {k|L(k) \le \theta } \right\} $$
(9)

Here L(k) is an affine function of k, so the solution is readily obtained.

4.2 Joint objective function of paging cost and paging delay with a fixed partition

For a total cost function that takes into account both the average paging cost and delay, if the total cost function preserves the convexity, then the optimization problem under a constraint on the average paging delay can be formulated as a convex optimization problem. For example, that is to minimize F(k) = αC(k) + βL(k) subject to L(k) ≤ θ, where α, β ≥ 0.

In the literature, a utility function is commonly assumed a concave and increasing function. In our case this is equivalent to a convex, decreasing total cost function or disutility function. So, in fact, the assumption of convexity of the total cost function is very reasonable and widely used in practice.

In this case, F(k) is convex. For unconstrained problem, we can solve by gradient descent method since F(x) is differentiable. Finding the suitable step for the gradient descent algorithm might be tough. Also, if the utility function is differentiable, solving the equation \( {\frac{{{\text{d}}F(k)}}{{{\text{d}}k}}} = 0 \) might not be straightforward.

For the constrained problem, the optimal solution can be found by taking duality approach. Here the Slater condition for zero duality gap easily holds.

Owing to the convexity of F(k) and k is an integer, we can use the following simple iterative algorithm to find the optimal k as follows:

We define a cost difference function between aggregation factor k and (k − 1) as follows:
$$ \delta (k) = C(k) - C(k - 1) $$
Then the optimal kopt is defined as the following:
$$ k_{\text{opt}} = \left\{ {\begin{array}{*{20}c} {1,} & {{\text{if }}\delta (2) > 0} \\ {\min \left\{ {k^{*} ,\max \left\{ {k:\delta (k) \le 0} \right\}} \right\},} & {\text{otherwize}} \\ \end{array} } \right. $$
(10)
where k* is defined in (9).

4.3 Optimal clustering under limited aggregation factor k, unconstrained expected paging delay

The constrained aggregation factor is due to the limited capacity of the paging agent, for instance, the limited buffer to store incoming paging requests.

When there is no constraint on the number of groups m, the monotonicity in m of the optimal policies results in a trivial scheme where m = N. So we are more interested in the optimization problem where both k and m are constrained. Here is our optimization problem:
$$ \mathop {\min }\limits_{k,m,v} C(k,m,N,v){\text{ subject to }}v \in \Uplambda (m,N),\quad 1 \le k \le k_{\max } ,1 \le m \le m_{\max } $$
By Proposition 5, due to the monotonicity of the optimal average paging cost function in m, this optimal problem simply becomes:
$$ \mathop {\min }\limits_{{{\mathbf{v}} \in \Uplambda (m_{\max } ,n)}} C(k_{\max } ,m_{\max } ,N,v) $$
With the structure of the optimal clustering as described in Proposition 6, the solution to this optimization problem can be solved via a quadratic-time algorithm of dynamic programming. Let \( \tilde{C}(k,m,n) \triangleq \mathop {\min }\limits_{v \in \Uplambda (m,n)} \,C(k,m,n,v) \). Due to the form of optimality in Proposition 6, we immediately have the following recursive equation ∀m, 1 ≤ m ≤ n − 1:
$$ \tilde{C}(k,m + 1,n) = \mathop {\min }\limits_{j = m,..,n} \left[ {\tilde{C}(k,m,j) + {\frac{1}{k}}\left( {A_{m + 1}^{k} - A_{m}^{k} } \right)B_{m + 1} } \right] $$
(11)
where \( A_{m + 1} = \sum\nolimits_{t = 1}^{n} {P_{S} (s_{t} )} \), \( A_{m} = \sum\nolimits_{t = 1}^{j} {P_{S} (s_{t} )} \), and \( B_{m + 1} = \sum\nolimits_{t = 1}^{m + 1} {C_{G} (g_{t} )} = n \).
When there is only one cluster of all n subnets, we have the initial condition for our recursive equation:
$$ \tilde{C}(k,1,n) = {\frac{1}{k}}n\left[ {\sum\limits_{t = 1}^{n} {P_{S} (s_{t} )} } \right]^{k} = {\frac{n}{k}}A_{m + 1}^{k} $$
(12)
Again this algorithm can be seen as a generalization of the algorithm in [18].

The optimization problem here is in fact a joint optimization problem without a constraint on the expected paging delay. As we point out in the next section, optimization of the aggregation factor and optimization of clustering are not separable. Fortunately, the monotonicity of the optimal policy dictates that the constraints on k and m must be binding, and therefore we need to solve only for the optimal clustering given k and m.

4.4 Joint optimal policy under constrained expected paging delay

The first question might be of our concern is if the optimization of the aggregation factor and optimization of clustering are coupled or independent? Contrary to possible intuition, they are in fact not separable. We can see this by simply taking an example for the case of uniform distribution of subnets. An example to illustrate this is described in Sect. 5.

From the user viewpoint, in many scenarios, an important metric of QoS is the maximal expected paging delay. Thus, solving the optimization problem under this constraint is of our main concern in this section. However, this optimization requires much more computation complexity since the monotonicity of the optimal policy in k, m is lost. Moreover, even for a fixed m, optimization of v is not independent of k. Even worse, the convexity of C(k,v*(k, m), m), where v*(k, m) is the optimal clustering given k under a constrained expected paging delay, might not be maintained.

The constrained joint optimization of aggregation factor k and partition vector v can be formulated as:
$$ \mathop {\min }\limits_{k,v,m} C(k,v(k,m),m){\text{ subject to }}L(k,v(k,m),m) \le \theta $$
Here we assume for practical purpose that \( \theta \in D \), where D is a finite set of discrete values of time.
To find the constrained optimal solution, it is necessary to locate a feasible range of aggregation factor k. Apparently, given a number of clusters m, the average paging delay is maximized with the partition vmax(m) = (1,…,1, N – m + 1). From the constraint on the average paging delay and the monotonicity of L(k), the maximal value of k is given as follows:
$$ k_{\max } (m) = \max \left\{ {k|L(k,v_{\max } (m)) \le \theta } \right\} $$
(13)

Note that given a number of clusters m, L(k, vmax(m)) is monotonically increasing with respect to k, thus finding kmax(m) is straightforward. An algorithm then only needs to work with the range of aggregation factor: 1 ≤ k ≤ kmax.

By Proposition 7, for a given (k, m), solution for v*(k, m) can be derived with polynomial complexity by the following recursive equations with initial condition:
$$ \begin{array}{lll}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{C} (k,1,j,d) =&\left\{ {\begin{array}{*{20}c} {{\frac{j}{k}}\left( {\sum\nolimits_{i = 1}^{j} {P_{S} (s_{i} )} } \right)} & {{\text{if }}L(k,1,j) = \sum\nolimits_{i = 1}^{j} {P_{S} (s_{i} )} + {\frac{k - 1}{2\lambda }} \le d} \\ \infty & {\text{otherwise}} \\ \end{array} } \right. \hfill \\\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{C} (k,m + 1,n,d) =& \mathop {\min }\limits_{j = m,..,n} \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{C} (k,m,j,d - \left( {m + 1} \right)\left( {A_{m + 1} - A_{m} } \right) + {\frac{n}{k}}\left( {A_{m + 1}^{k} - A_{m}^{k} } \right)} \right) \hfill \\\end{array}$$
(14)
where \( A_{m + 1} = \sum\nolimits_{t = 1}^{n} {P_{S} (s_{t} )} \), \( A_{m} = \sum\nolimits_{t = 1}^{j} {P_{S} (s_{t} )} \), \( d \in D \).

Hence, the procedure we can do is to check all vectors (k,m) in the range: 1 ≤ m ≤ N, 1 ≤ k ≤ kmax(m) and solve for v*(k, m) with each pair and calculate the corresponding paging cost. Finally, find the policy with the minimal paging cost among all these. Note that our algorithm is still in polynomial time.

The method by using an approximate continuous probability density function as presented in [17] does not work for a general value of k because the Lagrange function is not always convex. This fact will be discussed in more detailed in Sect. 6. Moreover, the optimal structure [18] is difficult to be obtained for a general k.

5 Numerical examples

In this section, we implement our algorithms that use the forms of optimality described in Sects. 3 and 4, in particular, the monotonic and recursive forms of optimal clustering. We show some examples which use the combinations of two types of location probability distribution (uniform and binomial) and two types of optimization problems (unconstrained and constrained). In these examples, we basically consider a PA that consists of N = 10 subnets. The incoming calls destined for idle mobile nodes in the PA arrive according to a uniform or a Poisson point process with the average rate λ.

5.1 Examples of unconstrained optimization

First, we show the numerical examples of unconstrained optimization for two types of location probability distributions, uniform and binomial. The paging request rate is λ = 2.0.

Optimal clusterings for different numbers of clusters with the uniform location probability distribution are shown in Table 2. For k = 1, the clusterings are consistent with the results in [19] where it is stated that for the case of uniform distribution of location probabilities and no paging request aggregation, the optimal clustering is balanced.
Table 2

Optimal clusterings with uniform location probability distribution

Value of k

Patterns of optimal clustering

1

(10), (5,5), (4,3,3), (3,3,2,2), (2,2,2,2,2), (2,2,2,2,1,1), (2,2,2,1,1,1,1), (2,2,1,1,1,1,1,1), (2,1,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1,1)

2

(10), (7,3), (5,3,2), (4,2,2,2), (4,2,2,1,1), (4,2,1,1,1,1), (3,2,1,1,1,1,1), (3,1,1,1,1,1,1,1), (2,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1,1)

3

(10), (7,3), (6,2,2), (6,2,1,1), (5,2,1,1,1), (4,2,1,1,1,1), (4,1,1,1,1,1,1), (3,1,1,1,1,1,1,1), (2,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1,1)

4

(10), (8,2), (7,2,1), (6,2,1,1), (6,1,1,1,1), (5,1,1,1,1,1), (4,1,1,1,1,1,1), (3,1,1,1,1,1,1,1), (2,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1,1)

Figures 2 and 3 depict average paging costs and paging delays against the number of groups for optimal clusterings, respectively.
https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig2_HTML.gif
Fig. 2

Average paging cost (uniform)

https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig3_HTML.gif
Fig. 3

Average paging delay (uniform)

Optimal clusterings for different number of clusters with the binomial location probability distribution are shown in Table 3.
Table 3

Optimal clustering with binomial location probability distribution

Value of k

Patterns of optimal clustering

1

(10), (4,6), (2,3,5), (1,2,2,5), (1,2,2,2,3), (1,1,1,2,2,3), (1,1,1,1,1,2,3), (1,1,1,1,1,1,1,3), (1,1,1,1,1,1,1,2,1), (1,1,1,1,1,1,1,1,1,1)

2

(10), (5,5), (3,2,5), (3,2,2,3), (2,1,2,2,3), (2,1,1,1,2,3), (2,1,1,1,1,1,3), (1,1,1,1,1,1,1,3), (1,1,1,1,1,1,1,2,1), (1,1,1,1,1,1,1,1,1,1)

3

(10), (5,5), (5,2,3), (3,2,2,3), (3,1,1,2,3), (3,1,1,1,1,3), (2,1,1,1,1,1,3), (2,1,1,1,1,1,2,1), (2,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1,1)

4

(10), (5,5), (5,2,3), (3,2,2,3), (3,1,1,2,3), (3,1,1,1,1,3), (3,1,1,1,1,2,1), (2,1,1,1,1,1,2,1), (2,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,1,1,1,1)

Figures 4 and 5 depict average paging costs and paging delay against the number of groups for optimal clusterings, respectively.
https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig4_HTML.gif
Fig. 4

Average paging cost (binomial)

https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig5_HTML.gif
Fig. 5

Average paging delay (binomial)

These examples obviously show advantages of paging with aggregation. While average paging cost can reduce several times, the corresponding average paging delay may vary much less for a high value of λ = 2.0. In particular, the average paging delay is not necessarily increasing in k since the paging delay element Ls is in fact reducing in k. Figures 2 and 4 confirm Proposition 4 stating that the optimal paging cost function is decreasing in k. Proposition 5 is confirmed via Figs. 2, 3, 4, and 5 since the optimal paging cost function is strictly decreasing in the number of clusters m, while the corresponding paging delay is strictly increasing in m. Examination of the optimal clusterings for different values of k shows that group location probabilities tend to concentrate more on first groups as k increases.

Given a location probability distribution of subnets, we conjecture that for optimal clusterings, the group location probability distribution with k1 is stochastically larger than the group location probability distribution with k2 if k1 < k2. A formal definition and good treatment of stochastic order in usual sense can be found in [22]. Intuitively, this property, if it holds, implies that group location probabilities concentrate more on first groups for a larger value of k.

5.2 Examples of constrained optimization

Next, we show the numerical examples of constrained optimization for two types of location probability distributions, uniform and binomial, like as in the examples of unconstrained optimization. In these examples, we assume that due to hardware constraint, the paging agent can aggregate at most 5 paging requests, i.e. aggregation factor k ≤ 5. The incoming paging rate λ = 2.0 and the step of discrete levels of time is 0.25.

The optimal clusterings with uniform location probability distribution are presented in Table 4. Figures 6 and 7 show the optimal average paging cost and optimal aggregation factor vs. upper bounds of the average paging delay, respectively.
Table 4

Optimal clustering with uniform location probability distribution

Delay upperbound (step 0.25)

Optimal clustering

0–0.75

None

1.00–2.00

(10)

2.25

(9,1)

2.5

(8,2)

2.75

(7,2,1)

3–3.25

(7,1,1,1)

3.5–3.75

(6,1,1,1,1)

4–4.50

(5,1,1,1,1,1)

4.75–5.5

(4,1,1,1,1,1,1)

5.75–6.25

(3,1,1,1,1,1,1,1)

6.5–7.25

(2,1,1,1,1,1,1,1,1)

7.5 and above

(1,1,1,1,1,1,1,1,1,1)

https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig6_HTML.gif
Fig. 6

Average paging costs (uniform)

https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig7_HTML.gif
Fig. 7

Optimal aggregation factors (uniform)

The optimal clusterings with binomial location probability distribution are presented in Table 5. The minimal average paging costs and the optimal aggregation factors for different upper bounds of average paging delay are plotted in Figs. 8 and 9, respectively. When the upper bound is 1, average paging cost is minimized with k = 1 and the optimal clustering is 1-group clustering for both examples. Obviously, the optimal average paging costs are decreasing in upper bounds of the average paging delay.
Table 5

Optimal clustering with binomial location probability distribution

Delay upper bound (step 0.25)

Optimal clustering

Optimal k

Optimal paging cost

0–0.75

None

  

1.00

(10)

1

10.0000

1.25

(10)

2

4.9902

1.50

(10)

3

3.3236

1.75

(10)

4

2.4902

2.00

(10)

5

1.9903

2.25

(5,5)

4

1.7038

2.50

(6,4)

5

1.4199

2.75

(5,2,3)

5

1.2278

3.00

(5,1,1,3)

5

1.1725

3.25

(4,1,1,1,3)

5

1.1420

3.50

(3,1,1,1,1,3)

5

1.1176

3.75

(3,1,1,1,1,2,1)

5

1.0990

4.00–4.50

(3,1,1,1,1,1,1,1)

5

1.0899

4.75–5.50

(2,1,1,1,1,1,1,1,1)

5

1.0861

5.75 and above

(1,1,1,1,1,1,1,1,1,1)

5

1.0859

https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig8_HTML.gif
Fig. 8

Average paging costs (binomial)

https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig9_HTML.gif
Fig. 9

Optimal aggregation factors (binomial)

We can see that with aggregation we can control both the aggregation factor k and the number of clusters m to minimize average paging cost while satisfying the constrained average paging delay. The aggregate paging mechanism offers a very low and stable paging cost in both Figs. 6 and 8.

Figure 9 shows that for constrained optimization, the optimal aggregation factor is not necessarily increased in the upper bound of the average paging delay, given a discrete step of time. This is due to the dependency of the average paging cost and delay on k, the clustering of the PA, and the tradeoff between the average paging cost and delay. For a given discrete step of time, when the upper bound of delay is increased step by step, sometimes the average paging cost can be reduced more by changing the clustering than by increasing k. We have also observed that a stochastic order of group location probability distribution as k increases may not exist.

Now, we demonstrate the impact of aggregation with another example where the paging agent has a larger capacity that allows it to aggregate up to 10 paging requests, i.e. aggregation factor k ≤ 10, and the average incoming paging rate is lower λ = 0.5. The location probability distribution is binomial.

In this case, when k increases, the aggregation delay greatly increases, consequently, the average paging delay is increased significantly. We plot the optimal average paging cost and aggregation factors in Figs. 10 and 11, respectively.
https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig10_HTML.gif
Fig. 10

Average paging costs (binomial, k ≤ 10, λ = 0.5)

https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig11_HTML.gif
Fig. 11

Optimal aggregation factors (binomial, k ≤ 10, λ = 0.5)

From Figs. 8 and 10, we can see that the benefit of aggregation is less for a lower paging request rate. Yet, aggregation paging is still very effective because it can bring about great paging cost reduction for any upper bounds of average paging delay that is larger than or equal to 1.5 where 1 is the minimal paging delay. The aggregation factor k also grows slowly as the upper bound of average paging delay increases.

Note that, without aggregation, the paging cost for any individual MN is bounded below by 1. However, with aggregation, the average paging cost per MN is calculated by dividing the paging cost incurred when paging an aggregate paging request by the aggregation factor k. This point is demonstrated in (2). As a result, the average paging cost (per MN) could be smaller than 1, and this clearly shows a benefit of aggregation.

The processing capability of the paging agent is an important factor to aggregate paging system, which can be measured by the step of discrete levels of time it uses for calculation and optimization, and the capacity of storing and aggregating multiple paging requests. To investigate the impact of the processing capacity of the paging agent, we further calculate the results for the discrete steps 0.1 and 0.01, and the maximal values of the aggregation factor 5 and 10. As described in our algorithms, the computational complexity is therefore increasing in the maximal value of k and decreasing in the value of time step. Our results for these scenarios are presented in Figs. 12, 13, 14, 15, 16, 17, 18, 19.
https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig12_HTML.gif
Fig. 12

Average paging costs (binomial, k ≤ 5, λ = 2.0, step 0.1)

https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig13_HTML.gif
Fig. 13

Optimal aggregation factors (binomial, k ≤ 5, λ = 2.0, step 0.1)

https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig14_HTML.gif
Fig. 14

Average paging costs (binomial, k ≤ 5, λ = 2.0, step 0.01)

https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig15_HTML.gif
Fig. 15

Optimal aggregation factors (binomial, k ≤ 5, λ = 2.0, step 0.01)

https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig16_HTML.gif
Fig. 16

Average paging costs (binomial, k ≤ 10, λ = 0.5, step 0.1)

https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig17_HTML.gif
Fig. 17

Optimal aggregation factors (binomial, k ≤ 10, λ = 0.5, step 0.1)

https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig18_HTML.gif
Fig. 18

Average paging costs (binomial, k ≤ 10, λ = 0.5, step 0.01)

https://static-content.springer.com/image/art%3A10.1007%2Fs11276-009-0212-z/MediaObjects/11276_2009_212_Fig19_HTML.gif
Fig. 19

Optimal aggregation factors (binomial, k ≤ 10, λ = 0.5, step 0.01)

Examination of these graphs and our corresponding program running times reveals an important point: although with a smaller step of time the results are smoother, the average paging cost experiences only very minor changes while the computational complexity increases significantly. In fact, a paging system requires real time processing, thus high computational complexity is not justifiable. Our aggregate paging system works pretty well with a coarse step of 0.25 or higher (and for a much larger number of subnets in the PA). In these cases, the optimal clusterings can be computed instantly.

It is worth noting that the graphs of the optimal k are stepped curves in Figs. 11, 13, 15, 17 and 19, while they are almost monotonous in Figs. 7 and 9. In our optimization problem described in Sect. 4.4, there is a basic tradeoff: the average paging cost C(k) will be reduced by increasing k, but at the same time the average paging delay will be increased. In (5) of the average paging delay, the second term is independent of k, but dependent of the clustering of the PA. For a fixed clustering, when k increases, the first term increases at the rate that depends on the value of λ For a large (e.g. λ = 2), L(k) increases slowly in k (e.g. with the marginal rate of 0.25 for λ = 2, which is equal to the discrete time step of Figs. 7 and 9). For a small λ (e.g. λ = 0.5), L(k) increases sharply in k (e.g. with the marginal rate of 1.0 for λ = 0.5).

Now, the relationship between the marginal rate of L(k) and the discrete time step used by the paging agent will make graphs step-wise or rapid rise. If the marginal step is smaller or equal to the discrete time step as in Figs. 7 and 9, then when the upper bound of average paging delay is increased by one step, the average cost can be brought down by increasing k (at least by 1) if keeping the clustering unaltered. If the marginal step is larger than the steps of discrete time as in Figs. 11, 13, 15, 17 and 19, then when the upper bound of average paging delay is increased by one step of time, increasing k to reduce C(k) will violate the constraint, hence resulting in stepped curves.

6 Discussion

6.1 Optimization of clustering and aggregation factor

We verify the correlation between the optimal clustering and the aggregation factor by a simple example. Let’s investigate an example with N = 10, m = 2, k = 2. The optimal clustering is v*(k) = (n1*, n2*).

For k = 1, we have
$$ \begin{gathered} C(1,v) = {\frac{1}{N}}\sum\limits_{i = 1}^{2} {n_{i} } \sum\limits_{j = 1}^{i} {n_{j} } = {\frac{1}{N}}\left( {n_{1} n_{1} + n_{2} (n_{1} + n_{2} )} \right) = {\frac{1}{N}}\left( {(n_{1} + n_{2} )^{2} - n_{1} n_{2} } \right) = {\frac{1}{N}}(N^{2} - n_{1} n_{2} ) \hfill \\ C(i,v^{*} ) \le C(i,v) \hfill \\ \Leftrightarrow n_{1}^{ * } n_{2}^{ * } \ge n_{1} n_{2} \hfill \\ \end{gathered} $$
It is easy to verify that \( v^{*} (1) = \arg \min C(1,v) = \{ v|(n_{1} ,n_{2} ) = \arg \max n_{1} n_{2} \} = (5,5). \)

This confirms the fact that the optimal clustering is balanced for the uniform distribution of location probability.

For k = 2,
$$ \begin{aligned} C(2,v^{*} ) \le & C(2,v) \\ \Leftrightarrow & n_{1}^{ * 2} n_{1}^{ * } + \left( {(n_{1}^{ * } + n_{2}^{ * } )^{2} - n_{1}^{ * 2} } \right) \left( {n_{1}^{ * } + n_{2}^{ * } } \right) = n_{1}^{ * 3} + N^{3} - Nn_{1}^{ * 2} \le n_{1}^{3} + Nn_{1}^{2} \\ \Leftrightarrow & n_{1}^{ * 2} (n_{1}^{ * } - N) \le n_{1}^{2} (n_{1} - N) \\ \Leftrightarrow & n_{1}^{ * 2} n_{2}^{ * } \ge n_{1}^{2} n_{2} \\ \end{aligned} $$
From this, the optimal partition vector v can be readily derived: v* = (7,3). Apparently, the optimal clustering has been changed when k varies from 1 to 2. Note that these optimal partitions are consistent with our numerical results in Sect. 5.

6.2 Overhead paging cost

Without generality, let’s normalize the cost of paging 1 subnet with k = 1 to be 1. Assume that the cost of paging a subnet is proportional to the length of the paging message. The cost of paging a subnet with aggregation factor k could be calculated as: 1 + α(k − 1) where
$$ \alpha = {\frac{\text{size of an IP address}}{{{\text{size of a paging message with }}k = 1}}}. $$

According to the structures of paging request message and paging message in [20], α is pretty small fraction. Note that there is no change in the formulas of L(.) when we relax the assumption of negligible overhead paging cost, and hence, all the properties of L(.) we have stated so far are still valid.

Let Bi(k) be the cost of paging i groups from 1 to i with aggregation factor k and \( B_{i} \triangleq B_{i}^{(1)} \), we have Bi(k) = [1 + α(k − 1)]Bi. Let CH(.) be the function of average paging cost, taking into account the overhead paging cost. Equation (6) now becomes:
$$ \begin{aligned} C^{H} (k) = & {\frac{1}{k}}\sum\limits_{i = 1}^{m} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)B_{i}^{(k)} } = {\frac{1}{k}}\sum\limits_{i = 1}^{m} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)B_{i}^{(1)} \left[ {1 + \alpha (k - 1)} \right]} \\ = & \left[ {1 + \alpha (k - 1)} \right]C(k) \\ = & {\frac{1 - \alpha }{k}}\sum\limits_{i = 1}^{m} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)B_{i} } + \alpha \sum\limits_{i = 1}^{m} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)B_{i} } \\ \end{aligned} $$
(15)

Unfortunately, the monotonicity of CH(k) in k, as stated in Propositions 1 and 4, and the convexity of CH(k) with respect to k, as stated in Proposition 3, do not hold in general. However, the recursive forms of optimality for unconstrained optimization problem and constrained optimization problem, as stated in Propositions 6 and 7, and the monotonicity of the optimal average paging cost function in the number of groups, as stated in Proposition 5, continue to hold. The proofs are pretty simple with a little modification from the case of negligible overhead paging cost. Due to space limitation, we omit these proofs in this paper.

The algorithms for the optimization problems in Sects. 4.1 and 4.2 now requires a check over all possible values of k: 1,…, k*, where k* can be specified similarly as in (13). Of course the computational complexities are still polynomial. The optimization problem in Sect. 4.3 also requires a check over all possible values of k: 1,…, kmax. For the problem in Sect. 4.4, since there is no optimal structure concerning aggregation factor k in this problem, the algorithm there is still applicable, i.e. we still can solve the optimization problems in polynomial time.

Given possible huge benefit of aggregation while the overhead cost is just a small fraction of average paging cost, we argue that aggregation is recommendable for a high incoming paging rate and a large PA.

6.3 Inter-PA mobility and node missing problem

It should be an interesting and practical research problem to consider the case where node missing and inter-PA mobility are included. In this scenario, the node missing probability depends on paging delay and therefore it is a function of the aggregation factor k. To investigate this problem, some mobility model, PA shape and size need to be assumed. Also we need to investigate neighbor PAs of the current PA. The objective is to evaluate boundary moving in and moving out rates. However, our model of location probability can no longer be used. We need to develop a new model to investigate clustering and paging sequence for dynamic sets of subnets.

6.4 Convexity of paging cost function in approximation method

Now we verify the convexity in the approximation method [17] for a general k. Given a location probability distribution sorted in descending order, we can approximate it with a decreasing probability density function h(x) which is comparable to the given discrete distribution, and h(x) is differentiable. The detail of this approximation is discussed in [17]. Now the average paging cost and paging delay can be calculated as:
$$ \begin{gathered} C(x) = \sum\limits_{i = 1}^{m} {x_{i} \left( {\left( {\int\limits_{0}^{{x_{i} }} {h(t){\text{d}}t} } \right)^{k} - \left( {\int\limits_{0}^{{x_{i - 1} }} {h(t){\text{d}}t} } \right)^{k} } \right)} \hfill \\ L(x) = d + \sum\limits_{i = 1}^{m} {i\int_{{x_{i - 1} }}^{{x_{i} }} {h(t){\text{d}}t} } \hfill \\ \end{gathered} $$
where d = (k − 1)/2λ. Let \( {\frac{{{\text{d}}H(x)}}{{{\text{d}}x}}} = h(x) \), we can present paging cost and delay functions as:
$$ \begin{gathered} C(x) = \sum\limits_{i = 1}^{m} {x_{i} \left( {H^{k} (x_{i} ) - H^{k} (x_{i - 1} )} \right)} \hfill \\ L(x) = \sum\limits_{i = 1}^{m} {i\left( {H(x_{i} ) - H(x_{i - 1} )} \right)} + d \hfill \\ \end{gathered} $$
Now we form the Lagrange function and seek to minimize it over x: Q = C(x) + α(L(x) − θ), α ≥ 0.
Differentiate Q with respect to xn and set to zero, we have:
$$ \begin{aligned} {\frac{\partial Q}{{\partial x_{i} }}} = & H^{k} (x_{i} ) - H^{k} (x_{i - 1} ) + x_{i} kH^{k - 1} (x_{i} )h(x_{i} ) - x_{i + 1} kH^{k - 1} (x_{i} )h(x) - \alpha h(x_{i} ) \\ = & H^{k} (x_{i} ) - H^{k} (x_{i - 1} ) + h(x_{i} )\left( {kH^{k - 1} (x_{i} )x_{i} - kH^{k - 1} (x_{i} )x_{i + 1} - \alpha } \right) = 0 \\ \end{aligned} $$
However, the most important point is to check if Q is convex.
$$ {\frac{\partial Q}{{\partial x_{i} \partial x_{j} }}} =\left\{ {\begin{array}{lll}&{kH^{k - 1} (x_{i} )h(x_{i} ) + h'(x_{i} )\left( {kH^{k - 1} (x_{i} )x_{i} - kH^{k - 1} (x_{i} )x_{i + 1} - \alpha } \right)} & {} \\&{\,\,\,\,\, + h(x_{i} )\left( {k(k - 1)H^{k - 2} (x_{i} )h(x_{i} )(x_{i} - x_{i + 1} ) + kH^{k - 1} (x_{i} )} \right),} & {j = i} \\&{ - kH^{k - 1} (x_{i} )h(x_{i} ),} & {j = i + 1} \\&{ - kH^{k - 1} (x_{i - 1} )h(x_{i - 1} ),} & {j = i - 1} \\&{0,} & {\text{otherwise}} \\ \end{array} } \right. $$
Let z(t) = tx + (1 − t)y where t ∈ [0,1] and x, y be two m-dimensional vectors. Q(x) is convex iff Q(z(t)) = Q(t) is convex in t. We convert the m-dimensional convex problem into one dimensional problem. Let ∆ = x − y, then z = t(x − y) + y = t∆ + y.
$$ \begin{aligned} {\frac{{\partial^{2} Q(t)}}{{\partial t^{2} }}} = & \sum\limits_{i,j = 1}^{m} {{\frac{{\partial^{2} Q(z)}}{{\partial x_{i} \partial x_{j} }}}} \Updelta_{i} \Updelta_{j} = \sum\limits_{i = 1}^{m} {{\frac{{\partial^{2} Q(z)}}{{\partial x_{i}^{2} }}}} \Updelta_{i}^{2} + 2\sum\limits_{i = 1}^{m - 1} {{\frac{{\partial^{2} Q(z)}}{{\partial x_{i} \partial x_{i + 1} }}}} \Updelta_{i} \Updelta_{i + 1} \\ = & \sum\limits_{i = 1}^{m} {\left\{ \begin{gathered} kH^{k - 1} (x_{i} )h(x_{i} ) + h'(x_{i} )\left( {kH^{k - 1} (x_{i} )x_{i} - kH^{k - 1} (x_{i} )x_{i + 1} - \alpha } \right) \hfill \\ \quad + h(x_{i} )\left( {k(k - 1)H^{k - 2} (x_{i} )h(x_{i} )(x_{i} - x_{i + 1} ) + kH^{k - 1} (x_{i} )} \right) \hfill \\ \end{gathered} \right\}} \Updelta_{i}^{2} - 2\sum\limits_{i = 1}^{m - 1} k H^{k - 1} (x_{i} )h(x_{i} )\Updelta_{i} \Updelta_{i + 1} \\ = & \sum\limits_{i = 1}^{m} {h'(x_{i} )\left( {kH^{k - 1} (x_{i} )x_{i} - kH^{k - 1} (x_{i} )x_{i + 1} - \alpha } \right)} \Updelta_{i}^{2} + \sum\limits_{i = 1}^{m} {2kH^{k - 1} (x_{i} )h(x_{i} )} \Updelta_{i}^{2} \\ & + \sum\limits_{i = 1}^{m} {k\left( {k - 1} \right)} H^{k - 2} (x_{i} )h^{2} (x_{i} )(x_{i} - x_{i + 1} )\Updelta_{i}^{2} - 2\sum\limits_{i = 1}^{m - 1} k H^{k - 1} (x_{i} )h(x_{i} )\Updelta_{i} \Updelta_{i + 1} \\ = & \sum\limits_{i = 1}^{m} {h'(x_{i} )\left( {kH^{k - 1} (x_{i} )x_{i} - kH^{k - 1} (x_{i} )x_{i + 1} - \alpha } \right)} \Updelta_{i}^{2} + kH^{k - 1} (x_{1} )h(x_{1} )\Updelta_{1}^{2} + kH^{k - 1} (x_{m} )h(x_{m} )\Updelta_{m}^{2} \\ & + \sum\limits_{i = 1}^{m - 1} k H^{k - 1} (x_{i} )h(x_{i} )(\Updelta_{i} - \Updelta_{i + 1} )^{2} + \sum\limits_{i = 1}^{m} {k(k - 1)} H^{k - 2} (x_{i} )h^{2} (x_{i} )(x_{i} - x_{i + 1} )\Updelta_{i}^{2} \\ \end{aligned} $$
For k = 1, we have
$$ {\frac{{\partial^{2} Q(t)}}{{\partial t^{2} }}} = \sum\limits_{i = 1}^{m} {h'(x_{i} )\left( {x_{i} - x_{i + 1} - \alpha } \right)} \Updelta_{i}^{2} + h(x_{1} )\Updelta_{1}^{2} + h(x_{m} )\Updelta_{m}^{2} + \sum\limits_{i = 1}^{m - 1} {h(x_{i} )\left( {\Updelta_{i} - \Updelta_{i + 1} } \right)^{2} } $$
h(x) is a decreasing function so \( {\frac{{{\text{d}}h(x)}}{{{\text{d}}x}}} \le 0 \). Since xi ≤ xi+1 and α ≥ 0, the first term is non-negative and hence \( {\frac{{\partial^{2} Q(t)}}{{\partial t^{2} }}} \ge 0 \) and Q(t) is convex in t. This is consistent with the result in [17].
However, we can show that this is not true in general.
$$ \begin{aligned} {\frac{{\partial^{2} Q(t)}}{{\partial t^{2} }}} = & \sum\limits_{i = 1}^{m} {h'(x_{i} )\left( {kH^{k - 1} (x_{i} )(x_{i} - x_{i + 1} ) - \alpha } \right)} \Updelta_{i}^{2} + kH^{k - 1} (x_{1} )h(x_{1} )\Updelta_{1}^{2} \\ & + kH^{k - 1} (x_{m} )h(x_{m} )\Updelta_{m}^{2} + \sum\limits_{i = 1}^{m - 1} k H^{k - 1} (x_{i} )h(x_{i} )\left( {\Updelta_{i} - \Updelta_{i + 1} } \right)^{2} \\ & + \sum\limits_{i = 1}^{m} {k\left( {k - 1} \right)} H^{k - 2} (x_{i} )h^{2} (x_{i} )\left( {x_{i} - x_{i + 1} } \right)\Updelta_{i}^{2} \\ \end{aligned} $$

If we choose x and y arbitrarily close, then |∆i| and |∆i − ∆i+1| become arbitrarily small. The last term is a quadratic of k, while the other terms are linear of k. When k getting large enough, the last term will dominate the expression, and since this term is negative, the sum becomes negative with a sufficiently large k.

7 Conclusions

Sequential group paging with aggregation mechanism has been demonstrated to save the paging cost critically and be very implementable. However, the performance of these systems is very sensitive to the aggregate factor and subnet clustering. This fact calls for a demand to find the optimal aggregate factor and optimal clustering from different perspectives requirements. It turns out that the optimization problem of these systems, which involves the trade-off between paging cost and paging delay and the interdependence of optimal aggregate factor and optimal subnet clustering, is quite complicated.

In this paper, we have formulated and solved different optimization problems of sequential group paging with aggregation. The average paging cost function C(k,v) is proven to be a decreasing function of the aggregation factor k, while the average paging delay L(k,v) is an increasing function of k. These two functions are shown to be convex in k. These features help us find the optimal k under a maximal paging delay constraint easily. For a total cost function that preserves convexity, we can readily formulate and solve a convex optimization problem. An important property of sequential paging with aggregation about the optimal paging order of subnets is verified. Paging in the descent order of location probabilities of subnets is shown to minimize both the average paging cost and paging delay. With this, we only need to work with partition vectors.

It has been shown in this paper that the optimization of aggregation factor k and clustering are not separable. Therefore, we need to address a joint optimization problem. Using the properties of optimal schemes, namely monotonicity in the aggregation factor and the number of groups and the recursive structure, the joint optimization problem without a constraint on the expected paging delay can be solved in polynomial time. The joint optimization problem under constrained expected paging delay is much harder to solve due to the loss of monotonicity and convexity. However, by specifying the feasible ranges and employing the recursive form stated in Proposition 7, the problem is still solvable in polynomial time.

As shown in our numerical examples, by aggregation, the paging cost can be reduced significantly (several times) while the average paging delay does not increase much for a high or moderate paging rate. This huge benefit makes aggregation paging justifiable and the assumption of negligible overhead paging cost less influential to our results.

Supplementary material

$$ \begin{aligned} C(k,{\user2{v}}) \ge & C(k,{\user2{v}}\prime ) \\ \Leftrightarrow & \sum\limits_{i = 1}^{m} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)B_{i} } \le \sum\limits_{i = 1}^{m} {\left( {A_{i}^{\prime k} - A_{i - 1}^{\prime k} } \right)B_{i} } \\ \Leftrightarrow & \sum\limits_{i = 1}^{j - 1} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)B_{i} } + \left( {A_{j}^{k} - A_{j - 1}^{k} } \right)B_{j} + \sum\limits_{i = j + 1}^{l - 1} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)B_{i} } + \left( {A_{l}^{k} - A_{l - 1}^{k} } \right)B_{l} + \sum\limits_{i = l + 1}^{m} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)B_{i} } \\ \le & \sum\limits_{i = 1}^{j - 1} {\left( {A_{i}^{\prime k} - A_{i - 1}^{\prime k} } \right)B_{i} } + \left( {A_{j}^{\prime k} - A_{j - 1}^{\prime k} } \right)B_{j} + \sum\limits_{i = m + 1}^{l - 1} {\left( {A_{i}^{\prime k} - A_{i - 1}^{\prime k} } \right)B_{i} } + \left( {A_{l}^{\prime k} - A_{l - 1}^{k} } \right)B_{l} \\ & + \sum\limits_{i = l + 1}^{m} {\left( {A_{i}^{'k} - A_{i - 1}^{'k} } \right)B_{i} } \\ \Leftrightarrow & \left( {A_{j}^{k} - A_{j - 1}^{k} } \right)B_{j} + \sum\limits_{i = j + 1}^{l - 1} {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right)B_{i} } + \left( {A_{l}^{k} - A_{l - 1}^{k} } \right)B_{l} \\ \le & \left( {A_{j}^{'k} - A_{j - 1}^{'k} } \right)B_{j} + \sum\limits_{i = j + 1}^{l - 1} {\left( {A_{i}^{'k} - A_{i - 1}^{'k} } \right)B_{i} } + \left( {A_{l}^{'k} - A_{l - 1}^{'k} } \right)B_{l} \\ \Leftrightarrow & \left( {A_{j}^{k} - A_{j}^{'k} } \right)B_{j} + \left( {A_{l - 1}^{'k} - A_{l - 1}^{k} } \right)B_{l} + \sum\limits_{i = j + 1}^{l - 1} {\left( {\left( {A_{i}^{k} - A_{i - 1}^{k} } \right) - \left( {A_{i}^{'k} - A_{i - 1}^{'k} } \right)} \right)B_{i} } \ge 0 \\ \end{aligned} $$$$ \begin{array}{ll} &\left( {A_{j} - A\prime_{j} } \right)B_{j} + \left( {A\prime_{l - 1} - A_{l - 1} } \right)B_{l} \ge 0 \hfill \\ &\quad\Leftrightarrow \left( {P_{G} (g_{j} ) - P_{G} (g_{l} )}\right)B_{j} + \left( {P_{G} (g_{l} ) - P_{G} (g_{j} )} \right)B_{l} \ge 0 \hfill \\&\quad\Leftrightarrow \left( {P_{G} (g_{l} ) - P_{G} (g_{j} )} \right)\left( {B_{l} - B_{j} } \right) \ge 0 \hfill \\\end{array} $$$$ a_{i} = A_{j - 1} + \sum\nolimits_{t = j + 1}^{i} {P_{G} (g_{t} )} . $$$$ \begin{gathered} \Leftrightarrow \left( {\left( {a_{j} + P_{G} (g_{j} )} \right)^{k} - \left( {a_{j} + P_{G} (g_{l} )} \right)^{k} } \right)B_{j} + \left( {\left( {a_{l - 1} + P_{G} (g_{l} )} \right)^{k} - \left( {a_{l - 1} + P_{G} (g_{j} )} \right)^{k} } \right)B_{l} \hfill \\ + \sum\limits_{t = j + 1}^{l - 1} {\left( {\left( {\left( {a_{t} + P_{G} (g_{j} )} \right)^{k} - \left( {a_{t - 1} + P_{G} (g_{j} )} \right)^{k} } \right) - \left( {\left( {a_{t} + P_{G} (g_{l} )} \right)^{k} - \left( {a_{t - 1} + P_{G} (g_{l} )} \right)^{k} } \right)} \right)B_{t} } \ge 0 \hfill \\ \end{gathered} $$$$ \begin{gathered} \left( {\left( {a_{j} + P_{G} (g_{j} )} \right)^{k} - \left( {a_{j} + P_{G} (g_{l} )} \right)^{k} } \right)B_{j} + \left( {\left( {a_{j} + P_{G} (g_{l} )} \right)^{k} - \left( {a_{j} + P_{G} (g_{j} )} \right)^{k} } \right)B_{l} \ge 0 \hfill \\ \Leftrightarrow \left( {\left( {a_{j} + P_{G} (g_{l} )} \right)^{k} - \left( {a_{j} + P_{G} (g_{j} )} \right)^{k} } \right)\left( {B_{l} - B_{j} } \right) \ge 0 \hfill \\ \end{gathered} $$$$ \begin{aligned} L(k,v) = & \sum\limits_{i = 1}^{m} {P_{G} (g_{i} )i} + {\frac{{L_{\text{aggr}} }}{2}} = \sum\limits_{i = 1}^{m} {\left( {A_{i} - A_{i - 1} } \right)i} + {\frac{{L_{{_{\text{aggr}} }} }}{2}} \\ \ge & L(k,v') = \sum\limits_{i = 1}^{m} {P_{G} (g'_{i} )i} + {\frac{{L'_{\text{aggr}} }}{2}} = \sum\limits_{i = 1}^{m} {\left( {A'_{i} - A'_{i - 1} } \right)i} + {\frac{{L'_{{_{\text{aggr}} }} }}{2}} \\ \Leftrightarrow & \left( {A_{j} - A_{j - 1} } \right)j + \left( {A_{l} - A_{l - 1} } \right)l \ge \left( {A'_{j} - A'_{j - 1} } \right)j + \left( {A'_{l} - A'_{l - 1} } \right)l \\ \Leftrightarrow & \left( {A_{j} - A'_{j} } \right)j + \left( {A'_{l - 1} - A_{l - 1} } \right)l \ge 0 \\ \Leftrightarrow & \left( {P_{G} (g_{l} ) - P_{G} (g_{j} )} \right)\left( {l - j} \right) \ge 0 \\ \end{aligned} $$$$ \begin{aligned} C(x) = & C(\alpha x_{1} + \left( {1 - \alpha } \right)x_{2} ) \\ = & {\frac{1}{{\alpha x_{1} + \left( {1 - \alpha } \right)x_{2} }}}\left( {\sum\limits_{i = 1}^{m} {\left( {A_{i}^{{\alpha x_{1} + (1 - \alpha )x_{2} }} - A_{i - 1}^{{\alpha x_{1} + (1 - \alpha )x_{2} }} } \right)B_{i} } } \right) \\ \le & {\frac{\alpha }{{x_{1} }}}\sum\limits_{i = 1}^{m} {\left( {A_{i}^{{x_{1} }} - A_{i - 1}^{{x_{1} }} } \right)B_{i} } + {\frac{1 - \alpha }{{x_{2} }}}\sum\limits_{i = 1}^{m} {\left( {A_{i}^{{x_{2} }} - A_{i - 1}^{{x_{2} }} } \right)B_{i} } \\ \Leftrightarrow & LHS \le {\frac{1}{{x_{1} x_{2} }}}\left( {\sum\limits_{i = 1}^{m} {\left( {A_{i}^{{x_{1} }} - A_{i - 1}^{{x_{1} }} } \right)\alpha x_{2} B_{i} + \sum\limits_{i = 1}^{m} {\left( {A_{i}^{{x_{2} }} - A_{i - 1}^{{x_{2} }} } \right)\left( {1 - \alpha } \right)x_{1} B_{i} } } } \right) \\ = & {\frac{1}{{x_{1} x_{2} }}}\sum\limits_{i = 1}^{m} {\left( {\left( {\alpha x_{2} A_{i}^{{x_{1} }} + \left( {1 - \alpha } \right)x_{1} A_{i}^{{x_{2} }} } \right) - \left( {\alpha x_{2} A_{i - 1}^{{x_{1} }} + (1 - \alpha )x_{1} A_{i - 1}^{{x_{2} }} } \right)} \right)} B_{i} \\ \Leftrightarrow & x_{1} x_{2} \sum\limits_{i = 1}^{m} {\left( {A_{i}^{{\alpha x_{1} + \left( {1 - \alpha } \right)x_{2} }} - A_{i - 1}^{{\alpha x_{1} + \left( {1 - \alpha } \right)x_{2} }} } \right)B_{i} } \\ \le & \left( {\alpha x_{1} + \left( {1 - \alpha } \right)x_{2} } \right)\sum\limits_{i = 1}^{m} {\left( {\left( {\alpha x_{2} A_{i}^{{x_{1} }} + (1 - \alpha )x_{1} A_{i}^{{x_{2} }} } \right) - \left( {\alpha x_{2} A_{i - 1}^{{x_{1} }} + (1 - \alpha )x_{1} A_{i - 1}^{{x_{2} }} } \right)} \right)} B_{i} \\ \Leftrightarrow & \sum\limits_{i = 1}^{m} {x_{1} x_{2} A_{i}^{{\alpha x_{1} + (1 - \alpha )x_{2} }} B_{i} } - \sum\limits_{i = 1}^{m} {x_{1} x_{2} A_{i - 1}^{{\alpha x_{1} + (1 - \alpha )x_{2} }} B_{i} } \\ \le & \sum\limits_{i = 1}^{m} {\left( {\alpha x_{1} + \left( {1 - \alpha } \right)x_{2} } \right)\left( {\alpha x_{2} A_{i}^{{x_{1} }} + (1 - \alpha )x_{1} A_{i}^{{x_{2} }} } \right)B_{i} } - \sum\limits_{i = 1}^{m} {(\alpha x_{1} + (1 - \alpha )x_{2} )\left( {\alpha x_{2} A_{i - 1}^{{x_{1} }} + (1 - \alpha )x_{1} A_{i - 1}^{{x_{2} }} } \right)B_{i} } \\ \Leftrightarrow & \sum\limits_{i = 1}^{m} {\left( {x_{1} x_{2} A_{i}^{{\alpha x_{1} + (1 - \alpha )x_{2} }} - \left( {\alpha x_{1} + (1 - \alpha )x_{2} } \right)\left( {\alpha x_{2} A_{i}^{{x_{1} }} + \left( {1 - \alpha } \right)x_{1} A_{i}^{{x_{2} }} } \right)} \right)} B_{i} \\ \le & \sum\limits_{i = 1}^{m} {\left( {x_{1} x_{2} A_{i - 1}^{{\alpha x_{1} + (1 - \alpha )x_{2} }} - \left( {\alpha x_{1} + \left( {1 - \alpha } \right)x_{2} } \right)\left( {\alpha x_{2} A_{i - 1}^{{x_{1} }} + (1 - \alpha )x_{1} A_{i - 1}^{{x_{2} }} } \right)} \right)B_{i} } \\ \end{aligned} $$$$ \sum\limits_{i = 1}^{m} {\left( {h(A_{i} ) - h(A_{i - 1} )} \right)} \le 0 $$$$ \begin{aligned} {\frac{{{\text{d}}h(A_{i} )}}{{{\text{d}}A_{i} }}} = & x_{1} x_{2} \left( {\alpha x_{1} + \left( {1 - \alpha } \right)x_{2} } \right)A_{i}^{{\alpha x_{1} + \left( {1 - \alpha } \right)x_{2} - 1}} - \alpha x_{1} x_{2} \left( {\alpha x_{1} + \left( {1 - \alpha } \right)x_{2} } \right)A_{i}^{{x_{1} - 1}} \\ & - \left( {\alpha x_{1} + \left( {1 - \alpha } \right)x_{2} } \right)\left( {1 - \alpha } \right)x_{1} x_{2} A_{i}^{{x_{2} - 1}} \\ = & {\frac{{x_{1} x_{2} \left( {\alpha x_{1} + \left( {1 - \alpha } \right)x_{2} } \right)}}{{A_{i} }}}\left( {A_{i}^{{\theta x_{1} + \left( {1 - \theta } \right)x_{2} }} - \left( {\alpha A_{i}^{{x_{1} }} + \left( {1 - \alpha } \right)A_{i}^{{x_{2} }} } \right)} \right) \\ \end{aligned} $$$$ A_{i}^{{\alpha x_{1} + (1 - \alpha )x_{2} }} \le \alpha A_{i}^{{x_{1} }} + \left( {1 - \alpha } \right)A_{i}^{{x_{2} }} $$

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