# The analysis of batch sojourn-times in polling systems

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## Abstract

We consider a cyclic polling system with general service times, general switch-over times, and simultaneous batch arrivals. This means that at an arrival epoch, a batch of customers may arrive simultaneously at the different queues of the system. For the exhaustive service discipline, we study the batch sojourn-time, which is defined as the time from an arrival epoch until service completion of the last customer in the batch. We obtain exact expressions for the Laplace–Stieltjes transform of the steady-state batch sojourn-time distribution, which can be used to determine the moments of the batch sojourn-time and, in particular, its mean. However, we also provide an alternative, more efficient way to determine the mean batch sojourn-time, using mean value analysis. We briefly show how our framework can be applied to other service disciplines: locally gated and globally gated. Finally, we compare the batch sojourn-times for different service disciplines in several numerical examples. Our results show that the best performing service discipline, in terms of minimizing the batch sojourn-time, depends on system characteristics.

## Keywords

Queueing models Polling models Mean value analysis## Mathematics Subject Classification

60K25 68M20## 1 Introduction

Polling models are multi-queue systems in which a single server cyclically visits queues in order to serve waiting customers, typically incurring a switch-over time when moving to the next queue. Polling systems have been extensively used for decades to model a wide variety of applications in areas such as computer and communication systems, production systems, and traffic and transportation systems [1, 19]. In the majority of the literature on polling systems, it is assumed that in each queue, new customers arrive via independent Poisson processes. However, in many applications, these arrival processes are not necessarily independent; customers arrive in batches, and batches of customers may arrive at different queues simultaneously [21]. It is important to consider the correlation structure in the arrival processes for these applications, because neglecting it may lead to strongly erroneous performance predictions and, consequently, to improper decisions about system performance. In this paper, we study the *batch sojourn-time* in polling systems with simultaneous arrivals, that is, the time until all the customers in a single batch are served after an arrival epoch.

Batch sojourn-times are of great interest in many applications of polling systems with simultaneous arrivals. Below we describe two examples in manufacturing and communication. The first example is the *stochastic economic lot scheduling problem*, which is used to study the production of multiple products on a single machine with limited capacity, under uncertain demands, production times, and setup times [9, 24]. In the case of a cyclic policy, there is a fixed production sequence such that the order in which products are manufactured is always known to the manufacturer. Whenever a customer has placed an order for one or multiple products, the machine starts production. After the requested number of products has been produced, including possible demand for the same product from orders that just came in, the machine starts to process the next product in the sequence. In this way, the machine polls the buffers of the different product categories to check whether production is required. In this example, the server represents the machine, a customer represents a unit of demand for a given product, and a batch arrival corresponds to the order itself. The batch sojourn-time is defined as the total time required for manufacturing an entire order.

The second example from the area of computer communication systems is an *I/O subsystem* of a web server. Web servers are required to perform millions of transaction requests per day at an acceptable quality of service (QoS) level in terms of client response time and server throughput [22]. When a request for a web page from the server is made, several file-retrieval requests are made simultaneously (for example, text, images, and multimedia). In many implementations, these incoming file-retrieval requests are placed in separate I/O buffers. The I/O controller continuously polls, using a scheduling mechanism, the different buffers to check for pending file-retrieval requests to be executed. The web page will be fully loaded when all its file-retrieval requests are executed. In this application, the server represents the I/O controller, a customer represents an individual file-retrieval request, a batch of customers who arrive simultaneously corresponds to each web page request, and the batch sojourn-time is the time required to fully load a web page.

The objective of this paper is to analyze the batch sojourn-time in a cyclic polling system with simultaneous batch arrivals. The contribution of this paper is that we obtain exact expressions for the Laplace–Stieltjes transform of the steady-state batch sojourn-time distribution for exhaustive service, which can be used to determine the moments of the batch sojourn-time and, in particular, its mean. However, we provide an alternative, more efficient way to determine the mean batch sojourn-time by extending the mean value analysis (MVA) approach of Winands et al. [23]. We briefly show how our framework can be applied to other service disciplines that satisfy the branching property [16], i.e., locally gated and globally gated. We compare the batch sojourn-times for the different service disciplines in several numerical examples and show that the best performing service discipline, minimizing the batch sojourn-time, depends on system characteristics. From the results, we conclude that there is no unique best service discipline that minimizes the expected batch sojourn-time. As such, our results provide a starting point for a framework to minimize batch sojourn-times for a given polling system.

The organization of this paper is as follows. In Sect. 2, the literature review is given. In Sect. 3, a detailed description of the model and the corresponding notation used in this paper is given. Section 4 analyzes the batch sojourn-time for exhaustive service, the analysis for locally gated service and globally gated service is shown in the appendix. We extensively analyze the results of our model in Sect. 5 via computational experiments for a range of parameters. Finally, in Sect. 6, we conclude and suggest some further research topics.

## 2 Literature review

In the literature, polling systems with simultaneous arrivals have not been studied intensively. Shiozawa et al. [17] studies a two-queue polling system where customers arrive at each station according to an independent Poisson process and, in addition, customers can arrive in pairs at the system and each join a different queue. The authors derive the Laplace–Stieltjes transform of the waiting time distribution of an individual customer and the response time distribution of a pair of customers who arrive simultaneously. Levy and Sidi [14] studies polling models with simultaneous batch arrivals. For models with gated or exhaustive service, they derive a set of linear equations for the expected waiting time at each of the queues. They also provide a pseudo-conservation law for the system, i.e., an exact expression for a specific weighted sum of the expected waiting times at the different queues. Chiarawongse and Srinivasan [5] also derives pseudo-conservation laws, but in their model all customers in a batch join the same queue. Finally, Van der Mei [20] considers an asymmetric cyclic polling model with mixtures of gated and exhaustive service and general service time and switch-over time distributions and studies the heavy traffic behavior. The results were further generalized in [21].

## 3 Model description

Consider a polling system consisting of \(N\ge 2\) infinite buffer queues \(Q_{1},\dots ,Q_{N}\) served by a single server that visits the queues in a fixed cyclic order. For ease of presentation, all references to queue indices greater than *N* or less than 1 are implicitly assumed to be modulo *N*, for example, \(Q_{N+1}\) is understood as \(Q_{1}\). Assume that a new batch of customers arrives according to a Poisson process with rate \(\lambda \). Each batch of customers is of size \(\varvec{K}=\left( K_{1},\dots ,K_{N}\right) \), where \(K_{i}\) represents the number of customers entering the system at \(Q_{i}\), \(i=1,\dots ,N\). The random vector \(\varvec{K}\) is assumed to be independent of past and future arrival epochs and at least one element of vector \(\varvec{K}\) is larger than 0 and the other elements are larger than or equal to 0, i.e., each batch contains at least one customer. The set of all possible realizations of \(\varvec{K}\) is denoted by \(\mathcal {K}\), and let \(\varvec{k}=\left( k_{1},\dots ,k_{N}\right) \) be a realization of \(\varvec{K}\). The joint probability distribution of \(\varvec{K}\), \(\pi \left( \varvec{k}\right) =\mathbb {P}\left( K_{1}=k_{1},\dots ,K_{N}=k_{N}\right) \) is arbitrary, and its corresponding probability generating function (PGF) is given by \(\widetilde{K}\left( \varvec{z}\right) =E\left( z_{1}^{K_{1}}z_{2}^{K_{2}}\dots z_{N}^{K_{N}}\right) \). The PGF of the marginal batch size distribution at \(Q_{i}\) is denoted by \(\widetilde{K}_{i}\left( z\right) =\widetilde{K}\left( 1,\dots ,1,z,1,\dots ,1\right) \), \(\left| z\right| \le 1\), where the *z* occurs at the *i*th entry. The arrival rate of customers to \(Q_{i}\) is \(\lambda _{i}=\lambda E\left( K_{i}\right) \), and let \(E\left( K_{ij}\right) =E\left( K_{i}K_{j}\right) \) for \(i\ne j\) and \(E\left( K_{ii}\right) =E\left( K_{i}^{2}\right) -E\left( K_{i}\right) \). The total arrival rate of customers arriving in the system is given by \(\varLambda =\sum _{i=1}^{N}\lambda _{i}\).

The cycle time \(C_{i}\) of \(Q_{i}\) is defined as the time between two successive visits of the server at this queue. A cycle consists of *N* visit periods each followed by a switch-over time; \(V_{i},S_{i},V_{i+1},\dots ,V_{i+N-1},S_{i+N-1}\) (see Fig. 1). A visit period, \(V_{i}\), starts whenever there are customers waiting at \(Q_{i}\) with a service beginning and ends with a service completion. Its duration equals the sum of service times of the customers served during the current visit to \(Q_{i}\). By definition, a visit beginning always corresponds to a switch-over completion, whereas a visit completion corresponds to a switch-over beginning. In the case where there are no customers waiting at \(Q_{i}\), these two epochs coincide. It is well-known that the mean cycle length is independent of the queue involved (and the service discipline considered in this paper) and is given by (see, for example, [18]) \(E\left( C\right) =E\left( S\right) /\left( 1-\rho \right) \).

*exhaustive policy*, when a visit beginning starts at \(Q_{i}\), the server continues to work until the queue becomes empty. Any customer who arrives during the server’s visit to \(Q_{i}\) is also served within the current visit. However, under the

*locally gated policy*, the server only serves the customers who were present at \(Q_{i}\) at its visit beginning; all customers who arrive during the course of the visit are served in the next visit to \(Q_{i}\). The final policy is the

*globally gated policy*; according to this policy, the server will only serve the customers who were present at all queues at the visit beginning of a reference queue, which is normally assumed to be \(Q_{1}\). Customers arriving after this visit beginning will only be served after the server has finished its current cycle. This policy strongly resembles the locally gated policy, except that all queues are gated at the same time instead of one per visit beginning.

The batch sojourn-time of a specific customer batch \(\varvec{k}\), denoted by \(T_{\varvec{k}}\) and its LST by \(\widetilde{T}_{\varvec{k}}\left( .\right) \), is defined as the time between its arrival epoch until the service completion of the last customer in the arrived batch; see Fig. 2. In this example, assume that when the server is in a visit period of \(Q_{j}\), a batch of three customers arrives in \(Q_{1}\) and \(Q_{i}\). Then the batch sojourn-time of this batch equals the residual time in \(V_{j}\), switch-over times \(S_{j},\dots ,S_{i-1}\), visit periods \(V_{j+1},\dots ,V_{i-1}\), and the time until service completion of the last customer of the batch in \(V_{i}\). By definition, the batch sojourn-time corresponds to the sojourn-time of the last customer who is served within the batch. It is important to realize that the queue where the batch finishes service *depends* on the location of the server on the arrival of the batch, and there is no fixed order in which the customers need to be served. The order in which the customers are served in this example is the same for the three service policies, but varies between disciplines depending on the location of the server. Finally, the batch sojourn-time of an arbitrary customer batch is denoted by *T* and its corresponding LST by \(\widetilde{T}\left( .\right) \).

*cyclic sum*and, analogously, a

*cyclic product*as [3]and alternatively,Finally, let \(\mathcal {K}_{i,j}\) be a subset of \(\mathcal {K}\) where the last customer of an arbitrary arriving customer batch is served in \(Q_{j}\) and all its other customers are served in \(Q_{i},\dots ,Q_{j}\). By definition, a batch will complete its service in one of the queues, such that \(\bigcup _{j=1}^{N}\mathcal {K}_{i,j}=\mathcal {K}\), \(i=1,\dots ,N\). The corresponding probability of subset \(\mathcal {K}_{i,j}\) is given by

## 4 Exhaustive service

In this section, we start by deriving the LST of the batch sojourn-time distribution of a specific batch of customers in the case of exhaustive service. The batch sojourn-time distribution is found by conditioning on the numbers of customers present in each queue at an arrival epoch and then studying the evolution of the system until all customers within the batch have been served. For this analysis, we first study the joint queue-length distribution at several embedded epochs in Sect. 4.1. We use these results to determine the LST of the batch sojourn-time distribution for both a specific and an arbitrary batch of arriving customers in Sect. 4.2, and present a MVA to calculate the mean batch sojourn-time in Sect. 4.3.

### 4.1 The joint queue-length distribution

*visit*beginnings and completions at \(Q_{i}\), where \(\varvec{z}=\left( z_{1},\dots ,z_{N}\right) \) is an

*N*-dimensional vector with \(\left| z_{i}\right| \le 1\). Similarly, let \(\widetilde{LB}^{\left( S_{i}\right) }\left( \varvec{z}\right) \) and \(\widetilde{LC}^{\left( S_{i}\right) }\left( \varvec{z}\right) \) be the joint queue-length PGFs at

*switch-over*beginnings and completions at \(Q_{i}\), respectively. Because of the branching property [16], these PGFs can be related to each other as follows:

*laws of motion*. The interpretation of (2) is that the queue-length in \(Q_{j}\), \(j\ne i\), at the end of visit period \(V_{i}\) is given by the number of customers already at \(Q_{j}\) at the visit beginning plus all the customers who arrive in the system during visit period \(V_{i}\). For \(Q_{i}\), all customers who are already in \(Q_{i}\) or arrive during \(V_{i}\) will be served before the end of the visit completion, and therefore, \(Q_{i}\) will contain no customers at the end of the visit period. Equation (3) simply states that the PGF of a visit completion corresponds to the PGF of the next switch-over beginning (see also Fig. 1). Finally, the queue-length vector at a switch-over completion corresponds to the sum of customers already present at the switch-over beginning plus all the customers who arrive during this switch-over period (4), and by definition the queue-length vector at a switch-over completion is the same for the next visit beginning (5). Note that Eqs. (2)–(5) can be differentiated with respect to \(z_{1},\dots ,z_{N}\) to compute moments of the queue-length distributions on embedded points [14] or numerically inverted for the queue-length probability distributions (for example, [6] for the case for non-simultaneous arrivals).

*service*beginnings and completions at \(Q_{i}\). Eisenberg [8] proved that besides the laws of motion, there exists a simple relation between the joint queue-length distributions at

*visit-*and

*service*beginnings and completions. He observed that each visit beginning either starts with a service beginning, or with a visit completion in the case where there are no customers at the queue. Similarly, each visit completion coincides with either a visit beginning or a service completion. Eisenberg [8] only considered polling systems either with exhaustive or gated service at all queues and individual arriving customers, but [4] has proven that the relation is not restricted to a particular service discipline and also holds for general branching-type service disciplines. In this section, we generalize this result for the case of simultaneous batch arrivals. Similarly to [8], the four PGFs are related as follows:

### 4.2 Batch sojourn-time distribution

In order to determine the LST of the steady-state batch sojourn-time distribution, we follow the method of Boon et al. [2] by conditioning on the location of the server and determining the time it takes until the last customer in a specific batch is served. These results are then used to determine the batch sojourn-time distribution of an arbitrary batch. Boon et al. [2] developed this method to study the steady-state waiting time distribution for polling systems with rerouting. For these kinds of models, the distributional form of Little’s Law [10] cannot be applied, since the combined processes of internal and external arrivals do not necessarily form a Poisson process. However, by studying the evolution of the system after a customer arrival, this problem can be avoided and the waiting time distribution can be obtained. Important in their analysis is the concept of *descendants* from the theory of branching processes, which are defined as all the customers who arrive during the service of a tagged customer, plus the customers who arrive during the service of those customers, etc. (i.e., the total progeny of the tagged customer).

*given*that the batch arrived during \(V_{j}\), and where \(\widetilde{T}_{\varvec{k}}^{\left( S_{j}\right) }\left( .\right) \) is

*given*that the customer batch arrived during \(S_{j}\). The remainder of this section will focus on how to determine \(\widetilde{T}_{\varvec{k}}^{\left( V_{j}\right) }\left( .\right) \), \(\widetilde{T}_{\varvec{k}}^{\left( S_{j}\right) }\left( .\right) \), and the LST of an arbitrary batch \(\widetilde{T}\left( .\right) \).

*N*-dimensional vector defined as follows:

*N*-dimensional vector defined as

We first focus on the batch sojourn-time of a customer batch that arrives during a visit period. Assume than an arriving customer batch \(\varvec{k}\) enters the system while the server is currently within visit period \(V_{j}\) and the last customer in the batch will be served in \(Q_{i}\). Formally, this means \(k_{i}>0\) and all the other customer arriving in the same batch should be served before the next visit to \(Q_{i}\); \(k_{l}\ge 0\), \(l=j,\dots ,i-1\), and \(k_{l}=0\) elsewhere. Whenever all the customers arrive in the same queue that is currently visited, then \(k_{i}=k_{j}>0\), and \(k_{l}=0\) elsewhere.

*and*the residual service time; \(\widetilde{L}^{\left( V_{i}\right) }\left( \varvec{z},\omega \right) \). First, since the number of customers who arrive in the elapsed and residual part of the service time are independent of each other and from the queue-lengths at a service beginning, we can write the LST of the joint distribution of \(\widetilde{B}_{j}^{P}\left( .\right) \) and \(\widetilde{B}_{j}^{R}\left( .\right) \) as [7]

### Proposition 1

### Proof

Now, consider a customer batch that arrives during a switch-over period. Assume an arriving customer batch \(\varvec{k}\) enters the system while the server is currently within switch-over period \(S_{j-1}\) and the last customer in the batch will be served in \(Q_{i}\). The reason that we consider \(S_{j-1}\) is that batch \(\varvec{k}\) will finish service in the same queue had it arrived in \(V_{j}\) because of the exhaustive service discipline.

*and*the residual switch-over time \(\widetilde{S}_{j-1}^{R}\left( .\right) \). From (11), we have the joint queue-length distribution at a switch-over beginning, \(\widetilde{LB}^{\left( S_{j-1}\right) }\left( .\right) \), and the number of customers who arrived in the elapsed part of the switch-over time, \(\widetilde{S}_{j-1}^{P}\left( .\right) \). Similarly to \(\widetilde{B}_{j}^{PR}\left( .\right) \), we define \(\widetilde{S}_{j-1}^{PR}\left( \omega _{R},\omega _{P}\right) \) as the LST of the joint distribution of the elapsed and residual switch-over time \(S_{j-1}\) as

### Proposition 2

### Proof

Similarly to Proposition 1, we condition on the number of customers present in the system before the arrival of batch \(\varvec{k}\) and the number of customer who enter the system per queue that arrived in batch \(\varvec{k}\). Then, studying the contribution of each customer to the batch sojourn-time, we obtain (23). \(\square \)

*and*a term that corresponds to the additional contribution batch \(\varvec{k}\) makes to the batch sojourn-time: where \(1_{\left( \varvec{k}\in \mathcal {K}_{j,i}\right) }\) is an indicator function that is equal to one if all customers in batch \(\varvec{k}\) are served in \(Q_{j},\dots ,Q_{i}\) and the last customer will be served in \(Q_{i}\), and zero otherwise. The terms \(\widetilde{W}_{i}^{\left( V_{j}\right) }\left( \omega \right) \) and \(\widetilde{W}_{i}^{\left( S_{j-1}\right) }\left( \omega \right) \) can be considered as the time between the batch arrival epoch and the service completion of the last customer in \(Q_{i}\) that was already in the system at the arrival of the customer batch, excluding batch \(\varvec{k}\) and any arrivals to \(Q_{i}\) after the arrival epoch, conditioned on the location of the server. In the case where there are only individually arriving customers, this would correspond to the LST of the waiting time distribution of a customer arriving in \(Q_{i}\) conditional on the server being in a visit or switch-over period. The LST of the batch sojourn-time distribution of a specific customer batch \(\varvec{k}\) can now be calculated using (13).

Finally, we focus on the LST of the batch sojourn-time of an arbitrary batch \(\widetilde{T}\left( .\right) \).

### Theorem 1

### Proof

It can be easily seen that (26) follows by enumerating all possible realizations of customer batches and the law of total probability.

Differentiating (27) will give the mean batch sojourn-time; however, in the next section, an alternative, more efficient way to determine the mean batch sojourn-time is presented.

### 4.3 Mean batch sojourn-time

In this section, we derive the mean batch sojourn-time of a specific batch and an arbitrary batch using *MVA*. MVA for polling systems was developed by Winands et al. [23] to study mean waiting times in systems with exhaustive, gated service, or mixed service. The main advantage of MVA is that it has a pure probabilistic interpretation and is based on standard queueing results, i.e., the Poisson arrivals see time averages (PASTA) property [25] and Little’s Law [15]. Furthermore, MVA evaluates the polling system at arbitrary time periods and not on embedded points such as visit beginnings, like in the buffer occupancy method [18] and the descendant set approach [12].

*before*the server starts serving \(Q_{i}\). Let \(E\left( B_{j,j}\right) =E\left( B_{j}\right) \) and \(E\left( B_{j,j+1}\right) =E\left( B_{j}\right) /\left( 1-\rho _{j}\right) \) be the expected busy period initiated by a customer in \(Q_{j}\). Then, \(E\left( B_{j,j+2}\right) \) equals the busy period in \(Q_{j}\) plus all the customers who arrive during this busy period in \(Q_{j+1}\) and the busy periods that they trigger:

*before*the server starts serving \(Q_{i}\). Then \(E\left( S_{j,j+1}\right) =E\left( S_{j}\right) \) and, in general, for \(i\ne j+1\),Finally, \(E\left( B_{j,i}^{R}\right) \) is the mean residual service of a customer in \(Q_{j}\) and all its descendants

*before*the server starts serving \(Q_{i}\) and is given by replacing \(E\left( B_{j}\right) \) by \(E\left( B_{j}^{R}\right) =E\left( B_{j}^{2}\right) /2E\left( B_{j}\right) \) in \(E\left( B_{j,i}\right) \). In addition, \(E\left( S_{j,i}^{R}\right) \) is defined as \(E\left( S_{j,i}\right) \) and by replacing \(E\left( S_{j}\right) \) by \(E\left( S_{j}^{R}\right) =E\left( S_{j}^{2}\right) /2E\left( S_{j}\right) \).

## 5 Numerical results

In this section we investigate the batch sojourn-times for the three server disciplines. In Sect. 5.1 we study a symmetrical polling system with two queues and derive a closed-form solution for the expected batch sojourn-times and show under which parameters settings, which service discipline has the smallest expected batch sojourn-time. In Sect. 5.2 we study asymmetrical systems and show that the service discipline that achieves the shortest expected batch sojourn-time depends on the system parameters.

### 5.1 A symmetrical polling system with two exponential queues

Parameters for three polling models

\(Q_{i}\) | | | | ||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | |

\(E\left( B_{i}\right) \) | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.10 | 0.40 | 0.90 |

\(E\left( B_{i}^{\left( 2\right) }\right) \) | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 1.00 | 1.00 | 1.00 |

\(E\left( S_{i}\right) \) | 0.10 | 0.10 | 0.10 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

\(E\left( S_{i}^{\left( 2\right) }\right) \) | 0.02 | 0.02 | 0.02 | 2.00 | 2.00 | 2.00 | 1.00 | 1.00 | 1.00 |

\(\varvec{k}\in \mathcal {K}\) | \(\pi \left( 1,1,0\right) =1/4\) | \(\pi \left( 1,0,0\right) =1/3\) | \(\pi \left( 1,1,0\right) =4/5\) | ||||||

\(\pi \left( 3,0,1\right) =3/4\) | \(\pi \left( 0,1,0\right) =1/3\) | \(\pi \left( 1,0,3\right) =1/5\) | |||||||

\(\pi \left( 0,0,1\right) =1/3\) |

### 5.2 Asymmetrical polling systems with multiple queues

In the previous section we have shown that depending on the system parameters, exhaustive service or locally gated service minimizes the expected batch sojourn-time. However, it can be shown that *any* of the three service disciplines studied in this paper can minimize the expected batch sojourn-time. In Table 1, the parameters of three systems with \(N=3\) are given. *Model a* has short switch-over times, *Model b* is a system with individual arriving customers and equal switch-over times and service times, and in *Model c* the last queue is the slowest and receives most of the work. Using the results of Sect. 4.3, and the online appendix the expected batch sojourn-times for the three different models can be calculated. The batch sojourn-times are shown in Fig. 5 for \(0\le \rho <1\). The results of *Model a* in Fig. 5a show that locally gated achieves the lowest expected batch sojourn-times, which is similar to Sect. 5.1 when the switch-over times were short. From the results of *Model b* shown in Fig. 5b, it can be seen that exhaustive service has the lowest expected batch sojourn-times. Here it is beneficial to serve a customer arriving to the same queue that is currently being served, since otherwise this customer has to wait a full cycle which increases the mean batch sojourn-time. Finally, *Model c* in Fig. 5c shows that globally gated service achieves the lowest expected batch sojourn-times, since for this policy the server will switch more often between the queues and finish service for all customers in a batch during one cycle, compared to the other disciplines.

## 6 Conclusion and further research

In this paper we analyzed the batch sojourn-time in a cyclic polling system with simultaneous batch arrivals and obtained exact expressions for the Laplace–Stieltjes transform of the steady-state batch sojourn-time distribution for the locally gated, globally gated, and exhaustive service disciplines. Also, we provided a more efficient way to determine the mean batch sojourn-time using MVA. We compared the batch sojourn-times for the different service disciplines in several numerical examples and showed that the best performing service discipline, minimizing the batch sojourn-time, depends on system characteristics.

A further research topic would be to determine, for each of the three policies, under what conditions on the system parameters its mean batch sojourn-time is smaller than that of the other two, and whether alternative service disciplines can achieve even lower batch sojourn-times. Another interesting further research topic would be to study how the customers of an arriving customer batch should be allocated over the various queues in order to minimize the batch sojourn-times.

## Supplementary material

## References

- 1.Boon, M.A.A., Van der Mei, R.D., Winands, E.M.M.: Applications of polling systems. Surv. Oper. Res. Manag. Sci.
**16**(2), 67–82 (2011)Google Scholar - 2.Boon, M.A.A., Van der Mei, R.D., Winands, E.M.M.: Waiting times in queueing networks with a single shared server. Queueing Syst.
**74**(4), 403–429 (2012). doi: 10.1007/s11134-012-9334-6 CrossRefGoogle Scholar - 3.Boxma, O.J., Groenendijk, W.P., Weststrate, J.A.: A pseudoconservation law for service systems with a polling table. IEEE Trans. Commun.
**38**(10), 1865–1870 (1990). doi: 10.1109/26.61458 CrossRefGoogle Scholar - 4.Boxma, O.J., Kella, O., Kosinski, K.M.: Queue lengths and workloads in polling systems. Oper. Res. Lett.
**39**(6), 401–405 (2011). doi: 10.1016/j.orl.2011.10.006 CrossRefGoogle Scholar - 5.Chiarawongse, J., Srinivasan, M.M.: On pseudo-conservation laws for the cyclic server system with compound Poisson arrivals. Oper. Res. Lett.
**10**(8), 453–459 (1991). doi: 10.1016/0167-6377(91)90022-H CrossRefGoogle Scholar - 6.Choudhury, G.L., Whitt, W.: Computing distributions and moments in polling models by numerical transform inversion. Perform. Eval.
**25**(4), 267–292 (1996). doi: 10.1016/0166-5316(95)00015-1 CrossRefGoogle Scholar - 7.Cohen, J.W.: The Single Server Queue. North-Holland, Amsterdam (1982)Google Scholar
- 8.Eisenberg, M.: Queues with periodic service and changeover time. Oper. Res.
**20**(2), 440–451 (1972). doi: 10.1287/opre.20.2.440 CrossRefGoogle Scholar - 9.Federgruen, A., Katalan, Z.: The impact of adding a make-to-order item to a make-to-stock production system. Manag. Sci.
**45**(7), 980–994 (1999)CrossRefGoogle Scholar - 10.Keilson, J., Servi, L.D.: A distributional form of Little’s law. Oper. Res. Lett.
**7**(5), 223–227 (1988). doi: 10.1016/0167-6377(88)90035-1 CrossRefGoogle Scholar - 11.Kleinrock, L., Levy, H.: The analysis of random polling systems. Oper. Res.
**36**(5), 716–732 (1988)CrossRefGoogle Scholar - 12.Konheim, A.G., Levy, H., Srinivasan, M.M.: Descendant set: an efficient approach for the analysis of polling systems. IEEE Trans. Commun.
**42**(234), 1245–1253 (1994). doi: 10.1109/TCOMM.1994.580233 CrossRefGoogle Scholar - 13.Levy, H., Sidi, M.: Polling systems: applications, modeling, and optimization. IEEE Trans. Commun.
**38**(10), 1750–1760 (1990)CrossRefGoogle Scholar - 14.Levy, H., Sidi, M.: Polling systems with simultaneous arrivals. IEEE Trans. Commun.
**39**(6), 823–827 (1991). doi: 10.1109/26.87170 CrossRefGoogle Scholar - 15.Little, J.D.C.: A proof of the queuing formula \(L=\lambda W\). Oper. Res.
**9**(3), 383–387 (1961)CrossRefGoogle Scholar - 16.Resing, J.A.C.: Polling systems and multitype branching processes. Queueing Syst.
**13**(4), 409–426 (1993). doi: 10.1007/BF01149263 CrossRefGoogle Scholar - 17.Shiozawa, Y., Takine, T., Takahashi, Y., Hasegawa, T.: Analysis of a polling system with correlated input. Comput. Netw. ISDN Syst.
**20**(1–5), 297–308 (1990). doi: 10.1016/0169-7552(90)90038-T CrossRefGoogle Scholar - 18.Takagi, H.: Analysis of Polling Systems. MIT Press, London (1986)Google Scholar
- 19.Takagi, H.: Analysis and application of polling models. In: Haring, G., Lindemann, C., Reiser, M. (eds.) Performance Evaluation: Origins and Directions. Lecture Notes in Computer Science, vol. 1769, pp. 424–442. Springer, Berlin (2000)Google Scholar
- 20.Van der Mei, R.D.: Polling systems with simultaneous batch arrivals. Stoch. Models
**17**(3), 271–292 (2001). doi: 10.1081/STM-100002274 CrossRefGoogle Scholar - 21.Van der Mei, R.D.: Waiting-time distributions in polling systems with simultaneous batch arrivals. Ann. Oper. Res.
**113**(1–4), 155–173 (2002). doi: 10.1023/A:1020918230560 Google Scholar - 22.Van der Mei, R.D., Hariharan, R., Reeser, P.K.: Web server performance modeling. Telecommun. Syst.
**16**(3–4), 361–378 (2001). doi: 10.1023/A:1016667027983 Google Scholar - 23.Winands, E.M.M., Adan, I.J.B.F., Van Houtum, G.J.: Mean value analysis for polling systems. Queueing Syst.
**54**(1), 35–44 (2006). doi: 10.1007/s11134-006-7898-8 CrossRefGoogle Scholar - 24.Winands, E.M.M., Adan, I.J.B.F., Van Houtum, G.J.: The stochastic economic lot scheduling problem: a survey. Eur. J. Oper. Res.
**210**(1), 1–9 (2011). doi: 10.1016/j.ejor.2010.06.011 CrossRefGoogle Scholar - 25.Wolff, R.W.: Poisson arrivals see time averages. Oper. Res.
**30**(2), 223–231 (1982). doi: 10.1287/opre.30.2.223 CrossRefGoogle Scholar

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