Designing multihop wireless backhaul networks with delay guarantees

Abstract

As wireless access technologies improve in data rates, the problem focus is shifting towards providing adequate backhaul from the wireless access points to the Internet. Existing wired backhaul technologies such as copper wires running at DSL, T1, or T3 speeds can be expensive to install or lease, and are becoming a performance bottleneck as wireless access speeds increase. Longhaul, non-line-of-sight wireless technologies such as WiMAX (802.16) hold the promise of enabling a high speed wireless backhaul as a cost-effective alternative. However, the biggest challenge in building a wireless backhaul is achieving guaranteed performance (throughput and delay) that is typically provided by a wired backhaul. This paper explores the problem of efficiently designing a multihop wireless backhaul to connect multiple wireless access points to a wired gateway. In particular, we provide a generalized link activation framework for scheduling packets over this wireless backhaul, such that any existing wireline scheduling policy can be implemented locally at each node of the wireless backhaul. We also present techniques for determining good interference-free routes within our scheduling framework, given the link rates and cross-link interference information. When a multihop wireline scheduler with worst case delay bounds (such as WFQ or Coordinated EDF) is implemented over the wireless backhaul, we show that our scheduling and routing framework guarantees approximately twice the delay of the corresponding wireline topology. Finally, we present simulation results to demonstrate the low delays achieved using our framework.

Appendices

Appendix A: hardness proof for routing

A feasible routing was defined in Sect. 4.

Theorem 4.1

Given the link capacities and the connection bit rates, determining if there is a feasible routing is NP-complete.

Proof

Clearly checking if a routing is feasible can be done in polynomial time. We show that the problem is NP-hard by a reduction from the known NP-complete problem called 3-Partition [13]. An instance of 3-Partition consists of a set A of 3m elements, a positive integer B and a positive integer size s(a) for each a ∈ A where B/4 < s(a) < B/2 and such that ∑_a∈A s(a) = mB. The goal is to find a partition of A into m disjoint sets $A_1, A_2,\ldots,A_m$ such that for 1 ≤ i ≤ m, $\sum\nolimits_{a\in A_{i}}s(a) =B.$ Notice that if such a partition exists, then each A _i contains exactly 3 elements of A. This problem is NP-complete in the strong sense.

Consider an instance I of 3-Partition. We now describe an instance J of the feasible routing problem based on I that can be constructed in time polynomial in I and such that I has a solution if and only if J does.

The construction is basically shown in Fig. 14. There is a root node R, a node labeled a _i for each a _i  ∈ A, nodes labeled $A_1, A_2,\ldots,A_m$ representing a partition of A and dummy nodes labeled $D_1,D_2,\ldots, D_m.$ Each node a _i has a connection in the up link direction with rate s(a _i ). These are the only connections.

Fig. 14

Illustrating the NP-completeness construction

Links of the form (a _i , A _j ) or (A _j , D _j ) have capacity 2B while links of the form (D _j , R) have capacity 2mB. All other links (not shown) have capacity 0.

Suppose more connections with total rate greater than B are routed through A _j . Then the congestion at A _j due to the link (A _j , D _j ) is more than B/2B = 1/2 and hence this would not be a feasible routing. Thus for any feasible routing each A _j must have connections routed through it with total rate exactly B. It’s easy to check that any routing that results in every node A _j have connections with total rate B routed through it is a feasible routing. Hence this feasible routing instance has a solution if and only if the 3-Partition instance has a solution.$\hfill\square$

Appendix B: propagation delay

We use the framework of GR server in [23] to prove Lemma 5.5 for WFQ. Let us first define guaranteed rate (GR) server.

Definition 2 (Definition 2.1.1 [23])

Consider a server that serves a connection. Packets are numbered in the order of arrival. Let a _n  ≥ 0 and d _n  ≥ 0 be the arrival and departure times of the n packet. A server is called a guaranteed rate server for this flow with rate r and latency ℓ if it guarantees that d _n  ≤ f _n  + ℓ where f _n is defined below and L is the packet size of the connection.

$$ \left\{\begin{array}{llll} f_0&=&0&\\ f_n&=&\max\{a_n,f_{n-1}\}+L/r & \forall n > 0 \end{array}\right. $$

Fact B.1

Consider a server with capacity c and a set of connections C that go through the server. Let$R_i=c\cdot \frac{\rho_i}{\sum\nolimits_{j\in C} \rho_j}$for connection i ∈ C,

• 1. Under the GPS scheduler the server is a GR server with rateR _i and latency 0 for connectioni.

• 2. Under the WFQ scheduler the server is a GR server with rateR _i and latencyL_max/c for connectioni, whereL_max is the maximum packet size among all connections inC.

• 3. Under the WFQ scheduler the server with propagation delayg is a GR server with rateR _i and latencyg + L_max/c for connectioni.

We note that Fact B.1.1 follows from the definition of the GPS protocol. Fact B.1.2 follows from Proposition 2.1.1 in [23]. Fact B.1.3 says that propagation delay defers the finishing time by g.

Theorem B.1 (Theorems 1.4.2, 1.7.2 and 2.1.3 [23])

If a GR server has rate r and latency ℓ for connection i then the delay bound for this connection is

$$ \frac{\sigma_i}{r} + \ell, $$

where σ _i is the burst size of connection i.

Theorem B.2 (Theorem 2.1.5 [23])

ConsiderK _i servers where each server 1 ≤ k ≤ K _i is GR with ratesr _k and latencies ℓ _k for connectioni. The concatenation of theseK _i servers is GR with rater = min _k r _k and latency$\ell=\sum\nolimits_{1\le k\le K_{i}} \ell_k +\sum\nolimits_{1\le k\le K_{i}-1} L_i/r_k$for connectioni.

The combination of Theorem B.1 and Theorem B.2 allows us to derive the following delay bound for K _i concatenated GR servers.

$$ \frac{\sigma_i}{\min_{1\le k\le K_{i}} r_k}+\sum_{1\le k\le K_{i}} \ell_k + \sum_{1\le k\le K_{i}-1} \frac{L_i}{r_k} $$

(9)

Under WFQ, for connection i each server k has rate $r_k=c_k\cdot \frac{\rho_i}{\sum_{j} \rho_j}$ and latency $\ell_k=L_{\max}/c_k$ by Fact B.1.2 where c _k is the capacity of serve k. Due to the admissibility of the traffic, we have r _k  ≥ ρ _i for all i and k which implies r ≥ ρ _i . Therefore, Eq. 9 implies the end-to-end delay under WFQ with zero propagation is

$$ \frac{\sigma_i}{\rho_i}+\sum_{1\le k\le K_{i}} \frac{L_{\max}}{c_k} + (K_i-1)\cdot \frac{L_i}{\rho_i}. $$

Note that the above bound is given in [28]. By Fact B.1.3, we have $\sum\nolimits_{1\le k\le K_{i}}\ell_k = g(i)+\sum\nolimits_{1\le k\le K_{i}}L_{\max}/c_k$ where g(i) is the total propagation delay along the path for connection i. Therefore, the end-to-end delay under WFQ with propagation is

$$ \left(\frac{\sigma_i}{\rho_i}+\sum_{1\le k\le K_{i}} \frac{L_{\max}}{c_k} + (K_i-1)\cdot \frac{L_i}{\rho_i}\right)+g(i). $$

Therefore, Lemma 5.5 holds for WFQ.

The Coordinated Earliest Deadline First (CEDF) scheduling protocol was described and analyzed in [2] for zero propagation delay. Consider a packet p from connection i. In [2] the authors define the following deadline D _k , 1 ≤ k ≤ K _i , for each link that p goes through. When multiple packets compete for a link, the scheduler gives priority to the packet that has the earliest deadline. The schedule is “coordinated” since the deadlines are incremented along the way.

$$ \left\{\begin{array}{ll} D_1 = \tau+ G_{{m_{1} }} \\ D_k = D_{k-1}+ G_{{m_{k} }} \end{array}\right. $$

(10)

Roughly speaking, τ is a random number which serves the purpose of spreading out packets that are injected in a burst. The value of τ is upper bounded by $t_{inj}+O((\sigma_i+L_i)/\rho_i)$ where t _inj is the injection time of packet p. The value of $G_{{m_{k} }}$ is proportional to $L_{\max}/c_{{m_{k} }}$ where m _k is the kth link along the path for connection i. (See [2] for the detailed definition of τ and $G_{{m_{k} }}.$) To prove a delay bound for CEDF, [2] shows the following two lemmas.

Lemma B.3 (Lemma 2 [2])

With high probability, for everyt that is a potential deadline the time interval [$t - G_{{m_{k} }} ,t$] has at most$G_{{m_{k} }}$bits of packets that have deadlines within the interval at linkm _k .

Lemma B.4 (Lemma 3 [2])

If the condition in Lemma B.2 holds, then every packet meets all its deadlines.

These two lemmas immediately imply that the end-to-end delay is $\tau-t_{inj}+\sum\nolimits_k G_{{m_{k} }}.$ Combined with the definitions of τ and $G_{{m_{k} }},$ [2] states

Theorem B.5 (Theorem 1 [2])

With high probability, the end-to-end delay under CEDF is upper bounded by

$$ \frac{\sigma_i+4L_i/\varepsilon}{\rho_i} + \sum_{1\le k\le K_{i}}\frac{L_{\max}}{ c^{{m_{k} }}}\log(\cdot). $$

If link e has propagation delay g _e , we redefine the deadlines as follows.

$$\begin{aligned} \left\{ \begin{array}{ll} D_1 = \tau+G_{{m_{1} }}+g_{{m_{1} }}\\ D_k = D_{k-1}+G_{{m_{k} }}+g_{{m_{k} }} \end{array} \right.\end{aligned}$$

(11)

That is, each deadline is postponed by the corresponding propagation delay. Both Lemmas B.3 and B.4 hold for identical proofs under the new definitions (11) of the deadlines. Therefore, with link propagation delays the end-to-end delay becomes $\tau-t_{inj}+\sum\nolimits_k G_{{m_{k} }}+\sum\nolimits_k g_{{m_{k} }}.$ In other words, if connection i has total propagation delay of g(i) its delay bound under CEDF becomes,

$$ \left(\frac{\sigma_i+4L_i/\varepsilon}{\rho_i} + \sum_{1\le k\le K_{i}}\frac{L_{\max}}{ c^{{m_{k} }}}\log(\cdot)\right) + g(i).$$

(12)

Lemma 5.5 holds for CEDF.