Scheduling for Multi-channel Wireless Networks: Small Delay with Polynomial Complexity Shreeshankar Bodas

Tara Javidi

Department of EECS Massachusetts Institute of Technology [email protected]

Department of ECE University of California, San Diego [email protected]

Abstract—Scheduling for multi-channel (e.g., OFDM-based) wireless downlink systems is considered with the objective of providing low delay performance to users with real-time and stochastic traffic. The main contribution is the design of a lowcomplexity scheduling algorithm for the system with desired performance (in a large deviations sense). In particular, as the number of users and channels grows, the algorithm ensures an exponential decay of the probability of encountering significant delay at a near optimal decay rate when the arrivals are symmetric and the channel follows an ON-OFF model with multipacket reception. The algorithm also provides consistently good performance in the larger set up by guaranteeing throughput optimality, and a non-zero decay rate if it is possible under any other algorithm. Index Terms—Scheduling algorithms, wireless downlink networks, small delay, low complexity

I. I NTRODUCTION The advent of OFDM (Orthogonal Frequency Division Multiplexing)-based 4G wireless downlink systems (e.g., WiMax [1] and LTE [2]) has rekindled interest in the design of scheduling algorithms for wireless networks. In an OFDMbased wireless system, the available bandwidth is partitioned into hundreds to thousands of orthogonal sub-channels, and each sub-channel can be allocated to one of the many users in the coverage area. The data-rates that the sub-channels support are functions of time, user locations, and other random phenomena such as fading and shadowing. A scheduling rule that assigns the sub-channels to users must take into account the (time-varying) channel realizations and the user backlogs to provide a good quality of service. In this paper, we investigate the scheduling problem from the perspective of a multi-queuing multi-server system with random connectivities and arrivals. Queuing theoretic analysis of wireless single-hop networks is a well-investigated topic. While the earlier work on the topic provided an analysis of queue dynamics under specific and known policies [3], [4], [5], the seminal work by Tassiulas and Ephremides [6], [7] considered the problem of optimal single server allocation with random connectivity. They showed that the simple policy of serving the longest connected queue is throughput optimal (renders queues stable whenever it is possible to do so). Furthermore, they showed that under a symmetric arrival and connectivity model, this policy minimizes the expected delay. Subsequently, researchers have

considered various extensions of the model and the analysis of related policies [8], [9], [10], [11], [12], [13], [14], [3], [15]. Perhaps the most relevant papers to our work involve the multi-server generalization of the original model by Tassiulas and Ephremides: in [16], the authors consider a multi-server generalization in which the connectivity model is of a vector form indicating which queues are available for service at any given time (connectivity of a given queue is identical across all servers). In this setup, which was motivated by the multichannel allocation in satellite communication, [16] devised an optimal server allocation algorithm with respect to the average per-user delay under various technical assumptions. The authors of [17] considered a matrix connectivity profile whose entries indicate the maximum number of packets that can be served by a given server allocated to a given queue (generalizing [16]). In this setup, [17] presents a scheduling algorithm that optimizes delay in a strong stochastic sense (including in expectation) for the binary connectivity model in the special cases of 1) a two user (queue) system and 2) a server sharing model (fluid-model). In [18] and [19], the authors study the problem posed in [17] in a large deviations setting where the system has a large number of users and proportionally large number of channels, and present an algorithm with an optimal decay rate function (the exponent of the decay in the probability of encountering significant delay as the number of users grow). In this paper, we consider the multi-queue multi-server model of [17] and [18]. The expected delay optimality in [17] and the rate-function results in [18] and [19] assume that each of the sub-channels can serve either 0 or 1 packets per time slot. For a system where the sub-channels can serve multiple packets per timeslot, [20] presents an optimization problem (the computation of DeepestDrain, Chapter 4) whose solution results in an “almost” optimal rate-function performance for a wide class of systems. However, the low complexity (polynomial time) construction of the solution to the optimization problem remained unresolved in [20]. In this paper, taking cue from the construction of the optimal policy in [21], we present a scheduling algorithm that solves the aforementioned optimization problem in polynomial time. Carefully appropriating results in [17] and [20], hence, the main contribution of the present work (see Theorem 4) is to provide a low complexity algorithm (polynomial in the size of the input) which ensures:

(a) for a certain set of system parameters, an “almost” optimal decay in the probability of encountering significant delay as the number of users and channels grow, and (b) a strictly positive rate of decay of the probability of encountering a significant delay, if it is possible to do so under any algorithm. Before we close, we bring to the readers’ attention a growing body of work on rate vector allocation in the context of wireless multi-user scheduling [22], [23], [24]. In this approach, the wireless multi-user scheduling (in the presence or the absence of channel state information) is mapped into an interacting queuing problem in which the service rate vector is dynamically selected within the capacity or feasible rate region of the channel while minimizing the expected delay. When dealing with a polytope rate region, the transmission rate vector selection becomes equivalent to a multi-server allocation problem with a general connectivity matrix (our setup), albeit in a continuous time framework. The continuous time models are then analyzed to arrive at the characterization of the optimal allocation [22], or its structural properties [23], [24]. Interestingly, perhaps not surprising though, the results across discrete time and continuous time models seem to coincide and are of similar flavor despite the distinctly different mathematical definition and setup. II. S YSTEM M ODEL A. Model and Notations We consider a multi-queue multi-server system with stochastic connectivities as shown in Figure 1. There are X11 (t)

A1 (t)

S1

j=1

B. Problem Formulation

Q1 A2 (t)

X22 (t)

S2

Q2 X2n (t)

An (t) Xnn (t)

For the system described above, we wish to determine a Markovian server  allocation policy  that maximizes −1 I(b) := lim inf n log P max Qi (0) > b , for a fixed, fin→∞

Sn

Qn

Fig. 1.

a centralized resource manager. The resource manager has perfect knowledge of the current queue backlogs and the connectivities which are assumed constant during a timeslot but varying independently over timeslots (e.g., block fading model). We do not allow sharing of any servers and assume no error in the transmission. The following notations are used throughout the paper: • Q(t) = (Q1 (t), . . . , Qn (t)): Backlogs (in units of packets) of each queue at the end of timeslot t. • A(t) = (A1 (t), . . . , An (t)): Stochastic number of fixedlength packets arrived to the queues at the beginning of timeslot t. We assume that random variables Ai (t) is i.i.d. across queues and of finite support for which P{Ai (t) = m} = pm , for 0 ≤ m ≤ M. • X(t) = [Xij (t)]: the n-by-n random connectivity matrix at timeslot t where Xij (t) ∈ {0, 1, . . . , K} (with K < ∞) denotes the maximum number of packets server Sj can serve from queue Qi at time t. We assume that Xij (t) is i.i.d. and P{Xij (t) = k} = qk for 0 ≤ k ≤ K. • Y(t) = [Yij (t)]: the n-by-n allocation matrix at the beginning of timeslot t where Yij (t) ∈ {0, 1}, and P i Yij (t) ≤ 1. Here, Yij (t) = 1 denotes that server Sj is assigned to serve queue Qi during time t. Let a+ := max(a, 0). The dynamics of the queue length vectors under an allocation Y(t) is described by the following equation: for all i and all t, +  n X Xij (t)Yij (t) . (1) Qi (t) = Qi (t − 1) + Ai (t) −

System Model

n queues (users) and n servers (OFDM subcarriers).1 The system is time-slotted. The set of queues is denoted by Q = {Q1 , Q2 , . . . , Qn } and the set of servers, S = {S1 , S2 , . . . , Sn }. Fixed-size packets arrive stochastically for each user and are transmitted over a set of channels (servers) with varying rates and connectivities. Each user has an infinite buffer to store the data packets that cannot be immediately transmitted. The users have the same priority and are symmetric, i.e., they have statistically identical arrival and channel connectivity processes. At the beginning of each timeslot, the assignment of servers to users is instantaneous and made by 1 The case where the number of queues and subcarriers scale in some proportion with each other can be handled as well.

1≤i≤n

nite integer b ≥ 0. Here Qi (0) denotes the length of Qi at time 0, i.e., under the stationary probability measure (imagine starting the system at time −∞.) The quantity I(b), referred to as the rate function in the large deviations theory, indicates the exponential rate of decline in the probability of a queue being larger than b as the number of severs and queues go to ∞. For large  n, it captures the probability of the “overflow” event max Qi (0) > b up to sub-exponential 1≤i≤n

(in n) multiplicative factors.

C. K-MTLB: A Server Allocation Algorithm For the server allocation problem in Section II, we propose the following algorithm called K-MTLB, which stands for Maximum Throughput and Load Balancing: 1) At the beginning of timeslot t, update the queue-lengths to account for arrivals, i.e., for all 1 ≤ i ≤ n compute Li := Qi (t − 1) + Ai (t) as the initial queue-length in timeslot t, after arrivals. Let H(Q ∪ S, E) denote the undirected bipartite graph where an edge is present

between the nodes Qi and Sj if the corresponding channel supports a rate Xij (t) = K. 2) For each 1 ≤ j ≤ n, allocate the server Sj to the longest queue that it can potentially serve, breaking ties between multiple longest queues arbitrarily. 3) For all i and j, if the queue Qi is allocated the server Sj , then mark the edge eij as a forward edge, else a backward edge. Let G denote the resultant directed graph. 4) Let ℓi denote the effective length of the queue Qi , defined as

possible allocations starting with the same initial queuelength vector and given 0-K channel connectivities (Theorem 2). 4) The algorithm renders the queues stable for all admissible arrivals under MaxWeight (Theorem 3). 5) The algorithm yields a strictly positive value of the rate function for the system (Theorem 4). 0-K channel connectivity model describes the special case where the channels are “on-off” and in the ”on” state channel can serve K packets, i.e. Xij (t) ∈ {0, K} for all i, j, t. That is, qk = 0 for k ∈ / {0, K}. This special case is of interest due to the following reasons: ℓi = Li −K·number of servers currently allocated to Qi . 1) For the system described above, the rate-function is M Note that ℓi can be negative. Define r = 1. P i ≤ 1 (see [25], pi nonzero for all b ≥ 0 only if th 5) Find the queue with the r smallest effective length, say K i=1 Qx . Formally, Qx is the rth queue in a sorted vector Lemma 2). In other words, the maximum channel-rate of effective queue-lengths, where ties between queues K determines the overall behavior of the rate-function, of equal effective length are broken arbitrarily. Let Qy that is, whether it is possibly nonzero under any algobe the queue with the longest effective length that is rithm. (Compare this with the stability condition: for n reachable from Qx in the directed graph G. large, the system is stable under any algorithm only if M P a) [Reallocation step] If ℓy −ℓx ≥ K +1, then reverse pi (i/K) ≤ 1.) all the edges on the path from Qx to Qy , thus i=1 2) If the channel-rates are in the set {0, K}, there exremoving one server from those serving Qx and ist scheduling algorithms that achieve a nonzero rateadding a server to those serving Qy . (If there are function as long as the above condition is met with strict multiple paths from Qx to Qy , choose any one path inequality (see [25], Theorem 5 or [20], Section 4.4). arbitrarily.) Go to step 4. b) If ℓy − ℓx ≤ K, then set r := r + 1. If r < n, then Thus, the case of 0-K channels essentially captures the ratefunction behavior of the system, at least to a first order. go to step 5. 6) “Read off” and use the server allocation: a forward edge Therefore, at times, we restrict our attention to the case of Qi → Sj denotes that Sj is allocated to Qi . ⋄ 0-K channels. We now establish the properties of K-MTLB one-by-one Here is a brief description of the algorithm in words: in below, starting with certain basic properties of the algorithm every timeslot, compute the final allocation in multiple rounds (Lemmas 1- 4) and culminating in the computational comof balancing the queue-lengths. First allocate each of the plexity result (Theorem 1). Next we establish the throughput servers to a longest connected queue. This might result in (Theorem 3) and rate-function (Theorem 4) performance of draining the longest queues much more than warranted by the algorithm. good load-balancing. To correct for this, if a server S is j

allocated to Qx but can also serve Qy , and if the effective length of Qx is less than that of Qy by an amount K + 1 or more, then revise the allocation so that Sj serves Qy and not Qx . The algorithm allows for more complex reallocations of servers, taking into account a sequence of servers that can be shuffled to balance the queue-lengths. An example of KMTLB is shown in Figure 2 for the case K = 2.

Lemma 1. The proposed algorithm terminates in a finite number of steps. Proof: Please see Appendix A.

III. A NALYSIS OF K-MTLB

Lemma 2. If a queue Qx of effective length ℓx =: L loses a server to a queue Qy of effective length ℓy ≥ L + K + 1, then in the subsequent operations of the algorithm, no queue of effective length strictly less than L can lose a server. Proof: Please see Appendix B.

Some of the salient structural features of K-MTLB allocation algorithm are as follows: 1) This is a causal, online algorithm: it does not need to know (or learn) the arrival or channel process statistics. Further, it is agnostic to the parameter b, the “overflow level.” 2) The algorithm can be implemented in O(n3 ) computations per timeslot (Theorem 1). 3) The algorithm returns lexicographically the smallest possible queue-length vector (see Definition 1) among all

Corollary 1. If a queue Qx loses a server in the reallocation step (step 5a) of the algorithm, then it cannot subsequently receive a server from any other queue. Proof: If possible, let a queue Qx lose a server when its effective length is ℓx =: L, so that its new effective length is L + K, and then (without losing any more servers) the queue Qx receives a server. For Qx to receive a server, a queue of effective length L − 1 or less must lose a server, which is impossible by the monotonicity of the effective lengths of the queues that lose server(s), as a result of Lemma 2.

8

4

S1 Q1

7

5

S2 Q2

7

5

S3

Temporary allocation: Q1 → S1 , Q1 → S2 , Q2 → S3

Fig. 2.

S2 Q2

Q3

Path to reverse: Q1 → S1 → Q2 → S3 → Q3

S1 Q1

S2

7

S3

(The graph H )

6

Q2

Q3 Original queue-lengths and connectivities

S1 Q1

5

S3 Q3

Final allocation: Q1 → S2 , Q2 → S1 , Q3 → S3 (No path to reverse)

The balancing operation

Informally, once a queue loses a server in step 5a, it is never “long enough” again to receive a server. We now show that if a queue receives a server as a result of the reallocation step, then it does not subsequently lose a server. Lemma 3. If a queue Qy receives a server as a result of the reallocation step, then it does not subsequently lose a server. Proof: Please see Appendix C. Corollary 1 and Lemma 3 help us limit the complexity of the proposed algorithm. These results say that there are no frivolous reassignments of servers under the proposed algorithm: once a server is allocated to a queue under the reallocation, its server-count does not decrease (however, the actual servers serving the queue may change as a result of the subsequent reallocations), and similarly a queue that loses a server cannot subsequently receive a server. The statement of the next lemma roughly says, “if a queue has been considered for losing a server, but could not lose a server because all the queues reachable from it were not long enough, then that queue need not again be considered for losing a server.” Lemma 4. Consider two queues Qx and Qy with the same effective length L under some allocation, and they both have the smallest effective lengths in the system. Suppose that all the queues reachable from Qx have an effective length at most L + K. Among the queues reachable from Qy , suppose Qz is the longest (in terms of the effective length), and has a length at least L + K + 1. If the edges on a path P from Qy to Qz are reversed, the queue Qx continues to have no directed path to any queue of length at least L + K + 1. Proof: Please see Appendix D. Theorem 1 (Complexity of K-MTLB). K-MTLB can be implemented in O(n3 ) computations per timeslot. Proof: Please see Appendix E. For the subsequent analysis, we need the following definition of lexicographic (or dictionary) ordering: Definition 1 (Lexicographic ordering). Given two vectors u, v ∈ ℜn , we say that u is lexicographically smaller than v and write u ≺D v if: 1) The vectors u′ , v ′ are sorted (in the decreasing order) copies of the vectors u, v respectively. 2) There exists i ∈ {1, 2, . . . , n} such that u′ (i) < v ′ (i) and u′ (j) = v ′ (j) for all 1 ≤ j < i. ⋄

We use the symbols ≻D , D , D and =D with the obvious meanings. We now establish that when the proposed algorithm terminates, it returns a smallest queue-length vector among all the queue-length vectors that can possibly result from all possible allocations of the servers to the queues. This proof is on the similar lines as that of Proposition 4 in [17]. Lemma 5. Given the initial queue-lengths and the server connectivities (the 0-K connectivity model), the K-MTLB algorithm returns a server allocation such that under that allocation, the resultant effective queue-length vector is lexicographically less than or equal to that under any other allocation. Proof: Please see Appendix F. We now deal with the actual queue-lengths (as opposed to the effective queue-lengths), given by Equation (1), and establish that the proposed algorithm results in lexicographically the smallest queue-length vector, starting from any initial queuelengths and server connectivities. Theorem 2. For a system with 0-K connectivities, K-MTLB returns an allocation that results in a queue-length vector that is lexicographically the smallest possible, among all possible allocations. Proof: Follows from Lemma 5 and noting that if u D v, then max(u, 0) D max(v, 0). Now we analyze the throughput and rate-function performance of the proposed algorithm. Note that the result of Theorem 3 does not require the 0-K connectivity model, or even the i.i.d. channel assumption, and holds for every value of n (as opposed to that of Theorem 4, which holds asymptotically as n → ∞). Theorem 3 (Throughput-optimality of K-MTLB). Given any arrival rate vector, the system is stable (i.e., positive recurrent queue-length process with finite mean) under the proposed algorithm if it is stabilizable under the MaxWeight algorithm [7]. That is, the proposed algorithm is throughput-optimal. Proof: First consider the case when the channel rates are 0-K. Let the term “weight of a schedule” denote the sum of the queue-length weighted channel-rates corresponding to n n P P (Qi (t − 1) + the allocation under the algorithm, i.e. i=1 j=1

Ai (t))Xij (t)Yij (t). It is known that the weight of a schedule

is the negative of the drift in the quadratic Lyapunov function modulo a bounded quantity. If under the MaxWeight algorithm the server Sj serves a queue Qi of length L, then under our algorithm, it must serve a queue of length at least L − nK, since otherwise a reassignment of Sj to Qi results in more balanced queues (in terms of their effective lengths) because there are only n servers and K is the maximum number of packets that can be served by a given server. Thus under the same initial conditions (at the beginning of a timeslot), the weight of the schedule under our algorithm differs from that under the throughput-optimal MaxWeight algorithm by at most n2 K 2 . When the queues are sufficiently large, the additive constant is negligible and the Lyapunov drift under our algorithm becomes negative when MaxWeight stabilizes the queues, implying stability for the 0-K channel case. The same argument can be extended with minor modifications to the algorithm when the channels are not just ON-OFF. We omit the details for the sake of brevity and refer the reader to Lemma 4.7 in [20]. Theorem 4 (Rate-function performance of K-MTLB). Consider the special case M ≤  K, and any fixed  integer b ≥ 0.

Let I(b) := lim inf n→∞

−1 n

log P

max Qi (0) > b . Then under

1≤i≤n

K-MTLB,

1 b+1 log ≤ I(b) ≤ M 1 − q0



b M



 + 1 log

1 . 1 − q0

Further, for  any values of the system parameters such that PM i ≤ 1, the proposed algorithm results in a strictly i=1 pi K positive value of the rate function. Proof: The proof follows from Theorem 2 here, and Theorems 4.9 and 4.10 in [20]. Note that for the case where b+1 is divisible by M, for M ≤ K, the two bounds on the rate-function match, implying the rate-function optimality of the proposed algorithm for certain system parameters. IV. C ONCLUSIONS We considered the problem of scheduling for multi-channel (e.g., OFDM-based) wireless downlink systems, with the aim of designing a scheduling algorithm that results in a good (low) delay performance for the users. This problem is of interest where the users request delay-sensitive data (such as voice or video). We analyzed the problem in a large deviations setting with a large number of users and a proportionally large bandwidth, which is the typically anticipated scenario for the next generation wireless systems. A key observation in this context is that the small-delay performance is essentially characterized by the maximum rate that a channel can support, and the intermediate channel rates are as good as zero. With this observation, we developed a scheduling algorithm that makes the system stable whenever it is possible to do so, and results in a good delay performance for the users for the system model under consideration. The novelty is an explicit, polynomial-time characterization of the

algorithm that requires only the current queue-length and channel-rate information (as opposed to an abstract characterization in [20]). V. ACKNOWLEDGMENTS This work was in part supported by a University of California Discovery Grant, Nokia Siemens, QUALCOMM, ViaSat, the Center for Wireless Communications at the University of California, San Diego, and the DARPA ITMANET program. R EFERENCES [1] W. Forum, “Mobile WiMAX Part I: A technical overview and performance evaluation,” March 2006, white Paper. [2] G. T. 25.913, “Requirements for Evolved UTRA (E-UTRA) and Evolved UTRAN (E-UTRAN),” March 2006. [3] N. Bambos and G. Michailidis, “On the stationary dynamics of parallel queues with random server connectivities,” in 34th IEEE Conference on Decision and Control, vol. 4, Dec. 1995, pp. 3638 –3643 vol.4. [4] M. Carr and B. Hajek, “Scheduling with asynchronous service opportunities with applications to multiple satellite systems,” IEEE Transactions on Automatic Control, vol. 38, no. 12, pp. 1820 –1833, Dec. 1993. [5] Y. Chandramouli, M. Neuts, and V. Ramaswami, “A queuing model for meteor burst packet communication systems,” IEEE Transactions on Communications, vol. 37, no. 10, pp. 1024 –1030, Oct. 1989. [6] L. Tassiulas and A. Ephremides, “Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks,” IEEE Trans. Automat. Contr., vol. 4, pp. 1936–1948, December 1992. [7] ——, “Dynamic server allocation to parallel queues with randomly varying connectivity,” IEEE Trans. Inform. Theory, vol. 39, pp. 466– 478, March 1993. [8] A. Eryilmaz, R. Srikant, and J. Perkins, “Stable scheduling policies for fading wireless channels,” IEEE/ACM Trans. Network., vol. 13, pp. 411– 424, April 2005. [9] M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, R. Vijayakumar, and P. Whiting, “CDMA data QoS scheduling on the forward link with variable channel conditions,” Bell Labs Tech. Memo, April 2000. [10] S. Shakkottai and A. Stolyar, “Scheduling for multiple flows sharing a time-varying channel: The exponential rule,” Ann. Math. Statist., vol. 207, pp. 185–202, 2002. [11] M. J. Neely, E. Modiano, and C. E. Rohrs, “Power and server allocation in a multi-beam satellite with time varying channels,” in Proc. IEEE Infocom, vol. 3, New York, NY, June 2002, pp. 1451–1460. [12] S. Shakkottai, R. Srikant, and A. Stolyar, “Pathwise optimality of the exponential scheduling rule for wireless channels,” Ann. Appl. Prob., vol. 36, no. 4, pp. 1021–1045, December 2004. [13] L. Ying, R. Srikant, A. Eryilmaz, and G. Dullerud, “A large deviations analysis of scheduling in wireless networks,” IEEE Trans. Inform. Theory, vol. 52, no. 11, pp. 5088–5098, November 2006. [14] K. Kar, X. Luo, and S. Sarkar, “Throughput-optimal scheduling in multichannel access point networks under infrequent channel measurements,” IEEE Transactions on Wireless Communications, vol. 7, no. 7, pp. 2619 –2629, Jul. 2008. [15] C. Lott and D. Teneketzis, “On the Optimality of an Index Rule in MultiChannel Allocation for Single-Hop Mobile Networks with Multiple Service Classes,” Probability in the Engineering and Informational Sciences, vol. 14, no. 3, pp. 259 – 297, Jul. 2000. [16] A. Ganti, E. Modiano, and J. N. Tsitsiklis, “Optimal Transmission Scheduling in Symmetric Communication Models With Intermittent Connectivity,” IEEE Trans. Inform. Theory, vol. 53, no. 3, pp. 998– 1008, Mar. 2007. [17] S. Kittipiyakul and T. Javidi, “Delay-Optimal Server Allocation in Multiqueue Multiserver Systems with Time-Varying Connectivities,” IEEE Trans. Inform. Theory, vol. 55, no. 5, pp. 2319 – 2333, May 2009. [18] S. Bodas, S. Shakkottai, L. Ying, and R. Srikant, “Scheduling in MultiChannel Wireless Networks: Rate Function Optimality in the SmallBuffer Regime,” in Proc. SIGMETRICS/Performance Conf., Jun. 2009. [19] ——, “Low-complexity Scheduling Algorithms for Multi-channel Downlink Wireless Networks,” in Proc. IEEE Infocom, Mar. 2010.

[20] S. R. Bodas, “High-performance Scheduling Algorithms for Wireless Networks,” Ph.D. dissertation, The University of Texas at Austin, Dec. 2010. [21] S. Kittipiyakul and T. Javidi, “A fresh look at optimal subcarrier allocation in ofdma systems,” in 43rd IEEE Conference on Decision and Control, vol. 3, Dec. 2004, pp. 3289 –3294 Vol.3. [22] E. Yeh, “Delay optimal multiaccess communication for general packet length distributions,” in IEEE International Symposium on Information Theory, Jun. 2004, p. 247. [23] N. Ehsan and T. Javidi, “Delay optimal transmission policy in a wireless multiaccess channel,” IEEE Transactions on Information Theory, vol. 54, no. 8, pp. 3745 –3751, Aug. 2008. [24] J. Yang and S. Ulukus, “Delay minimization with a general pentagon rate region,” in IEEE International Symposium on Information Theory, Jun. 2010, pp. 1808 –1812. [25] S. Bodas, S. Shakkottai, L. Ying, and R. Srikant, “Scheduling for Small Delay in Multi-rate Multi-channel Wireless Networks,” in Proc. IEEE Infocom, Apr. 2011. [26] J. Kleinberg and E. Tardos, Algorithm Design. Pearson Education, 2006.

A PPENDIX C P ROOF OF L EMMA 3

A PPENDIX A P ROOF OF L EMMA 1 In the step 5a of the algorithm, let the queues Qx and Qy be such that ℓy ≥ ℓx + K + 1 and there is a directed path in the graph G from Qx to Qy . Once the step 5a is executed, (new) the new effective lengths of Qx and Qy are: ℓx = ℓx + K (new) and ℓy = ℓy − K. The effective lengths of all the other queues are unchanged. Thus, (ℓ(new) )2 + (ℓ(new) )2 x y

Qz ∈ Q∗ , then the edge-configuration (i.e., the directions of the edges to the different queues) of Sj does not change as a result of the reallocation. Thus, the effective lengths of the queues reachable from Q∗ remain unchanged as a result of the reallocation. Consider the next time some queue Qa loses a server. Immediately before losing the server, if ℓa < L, then Qa ∈ Q∗ , which is impossible since the set Q∗ remains unchanged as a result of Qx losing a server, and the effective queuelengths of all the queues reachable from any queue in Q∗ are unchanged as well. Thus, ℓa ≥ L and the effective length of the queues losing server(s) in the reallocation step (step 5a) of the algorithm is a non-decreasing function of time (or the number of reallocations under the algorithm), as claimed.

=

(ℓx + K)2 + (ℓy − K)2

=

ℓ2x + ℓ2y − 2K (ℓy − ℓx − K) | {z } ≥1

≤ ℓ2x + ℓ2y − 2K, Pn implying that the quantity i=1 ℓ2i strictly decreases (by at least 2) as a result of every execution of step 5a. It follows that the algorithm terminates in a finite number of steps. A PPENDIX B P ROOF OF L EMMA 2 Suppose the queue Qx loses a server to the Qy as a result of the reallocation step (step 5a) of the algorithm. Immediately before the edges are reversed according to the step, let Q∗ be the set of queues with effective length strictly less than ℓx =: L. Then any queue Qz ∈ Q∗ does not have a directed path to a queue of effective length ℓz + K + 1 or more, since that would contradict the choice of Qx as the shortest queue with a balancing path in the step 5a of the algorithm. As a result of Qx losing a server to Qy , the new effective (new) length of Qy is ℓy = ℓy − K ≥ L + 1. Thus, the set Q∗ of queues of effective length strictly less than L remains unchanged as a result of the reallocation. Let the directed path (before reversal) from Qx to Qy be Qx → Sv1 → Qu2 → Sv2 → · · · → Svk → Qy . None of the queues in the set Q∗ has a directed path to any of the servers Sv1 , Sv2 , . . . , Svk , for if a queue Qz ∈ Q∗ has a directed path to Svi for 1 ≤ i ≤ k, then Qz has a directed path to Qy and ℓy ≥ L + K + 1 > ℓz + K + 1, contradicting the choice of Qx as the shortest queue with a balancing path in the step 5a of the algorithm. Thus, if a server Sj is reachable from a queue

Let a queue Qy be involved in the following event: E ∗ = {Qy receives a server from Qx by reversing the path Qx = Qu1 → Sv1 → Qu2 → · · · → Svk → Quk+1 = Qy }. Let Q∗ be the set of queues reachable from Qx immediately before the event E ∗ , and Q† be the set of queues reachable from Qy immediately after the event E ∗ , under the respective directed graphs. Claim 1 establishes that Q∗ = Q† . Since Qy is a queue with the maximum effective length that is reachable from Qx immediately before the event E ∗ , it (Qy ) is at most K shorter than the longest queue (in terms of the effective length) in the set Q∗ = Q† , immediately after the event E ∗ . That is, for all Qj ∈ Q† , we have ℓj ≤ ℓy + K immediately after the event E ∗ . We prove by induction that this property continues to hold during the subsequent operations of the algorithm, where Q† is the set of queues reachable from Qy under the directed graph at that time (i.e., the set Q† can change with time or the number of reallocation operations). Let S † denote the set of servers allocated to any queue in Q† immediately after the event E ∗ . Case 1: Consider a reallocation event (after the event E ∗ ) when a certain queue Qa loses a server and a queue Qb receives a server. If the path (to be reversed as a result of the reallocation) from Qa to Qb does not include any queue from the set Q† , then it also does not include any server from the set S † , since any server has at most one incoming edge (in the case of the servers in S † , the edge originates at a queue in Q† ). Thus, Q† = the set of queues reachable from Qy (and their lengths) remains unchanged, we have ℓj ≤ ℓy + K for all Qj ∈ Q† , and the induction continues. Case 2: Consider the first time after the event E ∗ that a (to-be-reversed) path P from a queue Qa to Qb includes a queue from the set Q† , say Qui . Let ℓa , ℓb , ℓy respectively denote the effective lengths of Qa , Qb , Qy immediately before the reallocation. Since Qui is reachable from Qy , and Qb from Qui , we have Qb ∈ Q† , implying ℓb ≤ ℓy + K before the reallocation (by the induction hypothesis). Further, ℓa + K + 1 ≤ ℓb from the reallocation condition (step 5a of the algorithm).

Let Q⋄ denote the set of queues reachable from Qa immediately before the reallocation, and reachable from Qb immediately after the reallocation (by Claim 1). Let Q‡ denote the set of queues reachable from Qy immediately after the reallocation. By Claim 2, Q‡ ⊆ Q⋄ ∪ Q† . We now show that for any Qj ∈ Q⋄ ∪Q† , the new effective lengths of the queues Qj and Qy satisfy ℓj ≤ ℓy + K. 1) Fix any queue Qj ∈ Q⋄ \{Qa , Qb }. Immediately before the reallocation, we have ℓj ≤ ℓb since Qb was a longest queue (in terms of the effective length) reachable from Qa . Further, ℓb ≤ ℓy + K (by the induction hypothesis). Since the effective length of Qj does not change as a result of the reallocation, we have ℓj ≤ ℓy + K after the reallocation. 2) If Qj = Qb , then before the reallocation, ℓb ≤ ℓy + K and after the reallocation, the effective length of Qb reduces by K, implying ℓb ≤ ℓy ≤ ℓy + K after the reallocation. 3) If Qj = Qa , then before the reallocation, we have ℓa + K + 1 ≤ ℓb ≤ ℓy + K implying ℓa + 1 ≤ ℓy . After the reallocation, the effective length of Qa increases by K, implying ℓa + 1 ≤ ℓy + K, and hence ℓa ≤ ℓy + K. 4) If Qj ∈ Q† \Q⋄ , then the effective length of Qj does not change as a result of the reallocation, and ℓj ≤ ℓy + K continues to hold by the induction hypothesis. Thus, we have shown that once a queue Qy receives a server, it is at most K shorter than any queue reachable from it (in terms of the effective length) under the subsequent operations of the algorithm, and thus cannot lose a server, completing the proof of Lemma 3, subject to Claims 1 and 2.  Claim 1. Consider a directed graph G(V, E) and a path P from a ∈ V to b ∈ V. Let Xi denote the set of nodes reachable from a node i ∈ V in the graph G, and Yi denote the set of nodes reachable from the node i when all the edges on the path P are reversed. Then Xa = Yb . Proof: We prove this statement by induction. Let f (n) denote the following statement: in any directed graph G(V, E), if a path from a node a to a node b contains exactly n edges, then Xa = Yb . Base case (n = 1): If the path from a to b has exactly one edge, then the edge (a, b) is the path. A node c reachable from a in the original graph is reachable via a path through b, or not. If c is reachable through b, then it remains reachable from b when the edge (a, b) is reversed, since the edge (a, b) cannot belong to a (WLG loop-free) path from b to c. If there is a path from a to c that does not contain the node b, then reversing the edge (a, b) makes c reachable from b. Hence Xa ⊆ Yb . Reversing the edge (b, a) gives back the original graph G, and the above argument implies Yb ⊆ Xa . Hence Xa = Yb . Induction step: Suppose f (n − 1) is true for some n, and we need to establish f (n). Let the path from a to b be P = a → a1 → a2 → · · · → an = b. We have Xa = Xa1 ∪ (Xa \ Xa1 ). Once the path P is reversed, by an argument similar to that in the base-case, Xa1 ⊆ Yb (by induction hypothesis) and Xa \ Xa1 ⊆ Yb . Thus Xa ⊆ Yb . By reversing the edges on

the path P again, and by the induction hypothesis, Yb ⊆ Xa , implying Xa = Yb and the proof is complete by induction. Claim 2. Consider a directed graph G(V, E). Fix any nodes a, b ∈ V and let A, B be the sets of nodes reachable from a and b respectively. Suppose c ∈ A ∩ B, and let P be a path from b to c. Let A′ denote the set of nodes reachable from a in the graph G′ (V, E ′ ), obtained from G by reversing the edges on the path P. Then A′ ⊆ A ∪ B. Proof: By Claim 1, B is precisely the set of nodes reachable from the node c in the graph G′ . For any x ∈ A′ , then there is a path in G′ from a to x that passes through the node c, or not. 1) If there is a path from a to x that passes through c, then x ∈ B. 2) If there is no path from a to x that passes through c, then x ∈ / B. Hence in the graph G, there is no directed path from b to x. Thus a path P ′ from a to x in the graph G (if any) cannot be destroyed by reversing the edges on the path P from b to c, since none of the edges on this path were part of P ′ . Thus x ∈ A. Thus A′ ⊆ A ∪ B and the proof is complete. A PPENDIX D P ROOF OF L EMMA 4 Let G denote the original graph and G′ be obtained from G by reversing the edges on the path P. Let Q∗x denote the set of queues reachable from Qx in the graph G. Before the reallocation of servers by reversing the path P from Qy to Qz , for all Qj ∈ Q∗x we have ℓj ≤ ℓx + K. Any queue Qi on the path P cannot belong to Q∗x , since (in the graph G) there is a directed path from Qi to Qz , whose effective length is at least L + K + 1. Thus, reversing the edges on the path P does not affect the edge-configurations of any of the queues in the set Q∗x , the set of reachable queues from Qx remains unchanged, and the proof is complete. A PPENDIX E P ROOF OF T HEOREM 1 Consider the following implementation of the algorithm: in place of steps 4 and 5 in the algorithm, 4’. Define the set of active queues, Q⋄ = Q. 5’. Compute the effective lengths of all the queues. Among the queues in Q⋄ , find a queue with the smallest effective length, say L. Let this queue be Q∗L . Find the set Q† of all queues to which Q∗L has a path. a) If the longest queue Q∗M (in terms of the effective length) in the set Q⋄ has an effective length at least L + K + 1, then reverse the directions of all the edges from Q∗L to Q∗M (thereby changing the server allocations). Define Q⋄ := Q⋄ \ {Q∗M }. If Q⋄ 6= ∅, go to step 5’. b) If the longest queue Q∗M (in terms of the effective length) in the set Q⋄ has an effective length at most L + K, then remove Q∗L from the set Q⋄ ,

i.e., define Q⋄ := Q⋄ \ {Q∗L }. If Q⋄ 6= ∅, go to step 5’. This implementation of the algorithm is indistinguishable from the original one. More specifically, Lemmas 2 and 4 imply that once the algorithm fails to find a path from a given queue to a queue with the effective queue-length difference of at least K + 1, that queue need not be considered again for potentially losing a server. Further, if a queue Q∗M receives a server, it needs not be considered for losing a server (Lemma 3), so removing it from the set Q⋄ is justified. Now we bound the complexity of this implementation. 1) The queue-lengths can be updated in O(n) computations, and the undirected graph H can be constructed in O(n2 ) computations. Thus the step 1 of the algorithm can be implemented in O(n2 ) computations. 2) For any server Sj , the longest queue connected to it can be computed in O(n) computations. Since there are n servers, the step 2 of the algorithm can be implemented in O(n2 ) computations. 3) The maximum possible number of edges in the graph is n2 , so one execution the step 3 can be implemented in O(n2 ) computations. 4) The step 4’ (in the equivalent implementation) can be implemented in O(n) computations. Each execution of the step 5’ can be implemented in O(n2 ) computations, since all the nodes reachable from a given node in a graph can be found in O(n2 ) computations using the Depth-First-Search (DFS) algorithm ([26], Theorem 3.13). Further, the step 5’ needs to be executed at most n times, since each execution decreases |Q⋄ | by at least 1 and we start with |Q⋄ | = n. Thus, the overall complexity of the algorithm is O(n3 ) computations per timeslot. A PPENDIX F P ROOF OF L EMMA 5 Let W ⊆ E denote the allocation under the proposed (KMTLB) algorithm: if an edge eij ∈ E belongs to W, then the server Sj is allocated to queue Qi , etc. Let the resultant effective queue-length vector under the allocation W be L = [ℓ1 , ℓ2 , . . . , ℓn ]. Let W be a set of allocations such that for any allocation W ∗ ∈ W, the resultant effective queue-length vector L∗ = [ℓ∗1 , ℓ∗2 , . . . , ℓ∗n ] obeys L∗ ≺D L. WLG let every allocation W ∗ ∈ W be “locally unimprovable,” that is, in the graph obtained by marking the edges in W ∗ as forward edges and the remaining edges as backward edges, there is no directed path from any node Qi to another node Qj such that ℓ∗i ≤ ℓ∗j − (K + 1). (It follows from Lemma 1 and its proof in Appendix A that given any allocation, we can obtain a locally unimprovable allocation by eliminating one-by-one the pairs (i, j) with the conflicts, and get a resultant queue-length that is lexicographically smaller than the original (improvable) queue-length vector.) We show that if W = 6 ∅, then there exist queues Qx and Qy such that ℓx + K + 1 ≤ ℓy and that there exists a directed path from Qx to Qy in the final directed

graph when the algorithm terminates (i.e., where the edges belonging to W are forward-directed and those in E \ W are backward-directed), a contradiction to the stopping condition under the algorithm. Let W ∗ be an allocation in the set W for which the symmetric difference between the sets W and W ∗ , namely W ⊕W ∗ := (W \W ∗ )∪(W ∗ \W ) has the smallest cardinality. Let Gd be the directed graph defined by the edges in the set W ⊕ W ∗ , where the edges belonging to W \ W ∗ are marked as forward-edges, and those belonging to W ∗ \ W are marked as backward edges. We first show that for any directed path P in Gd from a node Qi to Qj , we have ℓ∗i ≤ ℓ∗j +K −1. To see this, note that if ℓ∗i > ℓ∗j +K, then the allocation W ∗ ⊕ P results in an effective queue-length vector that is lexicographically strictly smaller than L∗ , contradicting the choice of W ∗ as a locally unimprovable allocation. If ℓ∗i = ℓj + K, then by alternating the server assignment along the path P, we get an allocation W ∗∗ = W ∗ ⊕ P that results in a queue-length vector that is lexicographically equal to L∗ , but |W ⊕ W ∗∗ | < |W ⊕ W ∗ |, contradicting the choice of W ∗ . Similarly, it P follows that P Gd is acyclic.P n n n ∗ We have ℓ = i i=1 i=1 ℓi = i=1 xi − nK, where x denotes the length of Q before any allocations. Since i i Pn Pn ∗ ∗ i=1 ℓi = i=1 ℓi and L 6= L , there exists i such that ℓi ≤ ℓ∗i − K (since the difference between ℓi and ℓ∗i must be a multiple of K). Clearly there is an edge (in the graph Gd ) leaving the node Qi . Starting from Qi , we build a path P ′ in the graph Gd (starting with P ′ = ∅) as follows: 1) Add an arbitrary edge leaving Qi to the path P ′ . 2) If a server-node Sk is currently the last node on the path P ′ , add the single edge leaving Sk to P ′ . 3) If Qj is (currently) the last node of P ′ , if there is an edge leaving Qj and ℓj ≤ ℓi + K, add an arbitrary edge leaving Qj to P ′ , and go to step 2. If no such queue Qj can be found, then stop. Since Gd is acyclic, the path P ′ is well-defined and finite. Let Qm be the last vertex of P ′ . There are two possible cases: C1. ℓm ≥ ℓi +K +1. In this case, we have found the queues Qx = Qi , Qy = Qm such that ℓy ≥ ℓx + K + 1 and a directed path from Qx to Qy (namely, P ′ ), reaching the desired contradiction and completing the proof. C2. There is no edge leaving Qm . Thus, Qm is served at least K more packets under W ∗ than under W. Hence (a)

(b)

ℓm ≥ ℓ∗m + K. Thus, ℓm − K ≥ ℓ∗m ≥ ℓ∗i − (K − 1) ≥ (ℓi + K) − (K − 1), implying ℓm ≥ ℓi + K + 1. Here the inequality (a) follows because for any directed path in Gd that starts at a node Qi and ends at a node Qm , we have ℓ∗i ≤ ℓ∗m + K − 1. The inequality (b) follows from the choice of i. Thus, the desired contradiction is reached with Qx = Qi and Qy = Qm , completing the proof. Therefore, the proposed algorithm returns an effective queuelength vector that is lexicographically the smallest possible.