# Load Balancing with Migration Penalties

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2 the job and send K jobs to the queue of another server We will examine two routing strategies: ABOUNCEC,d: This policy decides to share based on an additive threshold When a job arrives at a server i at time t, the server samples d other queues with replacement Let i be the index of the shortest of these queues If q i t 1 > q i t 1 C, then the job is bounced to queue i Otherwise, the job joins the queue at server i We require that C K MBOUNCEα,d: This policy employs a multiplicative threshold to decide when to share This threshold has been slightly modified so as not to bounce jobs to another queue when that queue would subsequently have more jobs than the original queue In this scheme, d queues are also sampled as above However, the job is only bounced to queue i if q i t 1 > maxαq i t 1, q i t 1 K Here, we require α > 1 The switching region under the ABOUNCE policy is far greater than that under MBOUNCE refer to the caricature in Figure 2 in the following sense The fraction of states in which no migration occurs under the ABOUNCE policy goes to whereas it remains strictly positive under the MBOUNCE policy Intuitively, this suggests that the ABOUNCE policy creates too much work and can therefore be unstable while the MBOUNCE policy can be stable One major purpose of this paper is to rigorously prove these statements Note that in both strategies we allow for d = ; in which case the shortest sampled queue is assumed to be the shortest queue in the system Fig 2 No-migration region shaded for n = 2 under ABOUNCE and MBOUNCE respectively q ABOUNCE1,d 8 1 q 1 q III STABILITY MBOUNCE16,d 8 1 q 1 In this section, we will examine the stability policies of the various load balancing strategies In particular, for each strategy, we are concerned with establishing the set of arrival rates λ such that the resulting system in positive recurrent First, consider the case where no sharing is performed Here, the system trivially splits into n independent M/M/1 queues with arrival rate λ and service rate 1 Hence, the system is positive recurrent if λ < 1 Ideally, the system with sharing should not sacrifice throughput and, thus, we would hope for the same stability region Additive Thresholds For the ABOUNCEC,d strategy, by numerical simulation, we can see that the stability region is strictly smaller than the system with no sharing For example, For example, in Figure 3, we see two sample paths for the ABOUNCE2,1 system with K = 2 and n = 1 The system is clearly stable when λ = 77, but unstable when λ = 79 Fig 3 Two sample paths for ABOUNCE2, 1 with n = 1 that suggest instability for λ 79 max i q i λ = 79 λ = t/3 The ABOUNCEC,d policy, however, is dominated by a system with no sharing where each job requires service time which is the sum of K exp1 random variables Such a system would be stable if λ < 1/K By making this coupling precise, we can establish the following theorem Theorem 1: If λ < 1/K, then the ABOUNCEC,d system is positive recurrent Further, in the case of 2 queues and d =, we can theoretically establish shrinkage of the stability region Theorem 2: If n = 2, the ABOUNCEC, system is null recurrent or transient if λ > λ, where λ < 1 is defined by λ = KC 1 KC 1 2 4K 1C 2C 3 2K 1C 2 Proof: For the λ > λ case, we appeal to the variation of Foster s Criterion in Proposition 54 in 7] In particular, given a function V : Z 2 R, define the drift operator V q = E V q1 q = q] V q To establish that the system is null recurrent or transient, it suffices to find a test function V and a finite set E Z 2 such that: I For all q / E, V q II There exists some q / E so that for all q E, V q > V q III V is bounded below and for some constant A < and all q E, E V q1 V q q = q] < A

3 Consider the test function V q = q 1 q 2 1 λ 1 λ C q 1 q 2 l, l= and set E = {} This function clearly satisfies conditions II and III Define the following sets, for l C, A = { q Z 2 E : q 1 q 2 > C }, B l = { q Z 2 E : q 1 q 2 = l, q 1, q 2 > } By separately considering the drift V q in each of these sets, it can be verified that for λ > λ, condition I also holds In a system with n = 2 and K = 2 under the ABOUNCE2, policy, for example, Theorems 1 and Theorem 2 indicate stability when λ < 5 and instability when λ > 89 Multiplicative Thresholds The MBOUNCEα,d strategy, on the other hand, intuitively bounces less and less as the system gets more and more loaded see Figure 2 To see this, consider the n = 2 case If one queue is more loaded than the other, bounces will occur and the queues will be equalized As the levels of the queues increase, it takes more and more time for them to become unequal by a constant fraction again, and hence bouncing decreases As the following theorem establishes that, because of this behavior, MBOUNCEα,d maintains the same stability region as not sharing The proof is deferred until the appendix Theorem 3: MBOUNCEα,d is stable for all λ < 1 IV DELAY - FORMAL MEAN FIELD ANALYSIS We now turn our attention to queuing delay We approach this in two ways We will present simulation results in the next section Here we will present a formal mean-field analysis Such an analysis permits us to answer questions about rare events in large systems which is difficult to do via simulation In particular, we study tail behavior for systems with n large In this section we revert to considering continuous time Markov-chains as opposed to the embedded chains of the previous section Multiplicative Thresholds Consider the Markov chain corresponding for a system of n queues under the MBOUNCEd,α policy Let m i τ denote the number of queues with at least i jobs at time τ and observe that the sequence m i i suffices to describe the state of the chain at time τ For ease of exposition we make the assumption that α is integral It then follows that for i > K and τ small, Em i τ τ m i τ m i τ] λ τ m i 1 τ m i τ m i/α 1 τ/n d λ τ m αi Km τ m i Km τ/n d m i Km1 τ/n d] τ m i τ m i1 τ, where the first term in the summation corresponds to arrivals at a queue with i 1 jobs, the second term corresponds to jobs that are bounced to queues with between i K and i 1 jobs, whereas the last term corresponds to departures from a queue with i jobs Setting τ = 1/n, we have: Em i τ τ/n m i τ/n m i τ] τ λ τ m i 1 τ/n m i τ/n m i/α 1 τ/n d λ τ m αi Km τ/n m i Km τ/n d m i Km1 τ/n d] τ m i τ/n m i1 τ/n Letting lim n Em i τ/n] = s i τ, and noting that by the Law of Large Numbers, lim n m i τ/n s i τ, we formally have that for large n, the MBOUNCEd, α system is described by the following system of ODEs: ṡ i τ = λ s i 1 τ s i τ s d i/α 1 τ λ s αi Km τ s d i Kmτ s d i Km1τ s i τ s i1 τ, i > K, τ >, ṡ i τ = λ s i 1 τ s i τ λ i 1 s maxmk,αm τ s d mτ s d m1τ s i τ s i1 τ, < i K, τ >, s τ =, τ >, s = 1, s i =, i > Fixed points of this system of ODEs have the following property which suggests that for large n, the MBOUNCEd, α system has far lighter tails than a system with no sharing Lemma 1: Any fixed point of the system of equations 1, satisfying i= s i < and s i i must also satisfy: s i = O for arbitrary ɛ > Additive Thresholds λ1 ɛ ilog α d1 1 d Similar reasoning as the multiplicative case suggests the following system of differential equations for the tails of an ABOUNCEd system with n large: ṡ i τ = λ s i 1 τ s i τ s d i K1τ λs im τ s d i Kmτ s d i Km1τ s i τ s i1 τ, i > K, τ >, 1

4 ṡ i τ = λ s i 1 τ s i τ i 1 λs mk τ s d mτ s d m1τ s i τ s i1 τ, < i K, τ >, s τ =, τ >, s = 1, s i =, i > Fixed points of this system of ODEs have the following property which suggests that for large n, the ABOUNCEd system has lighter tails than both the system with no sharing as well as the MBOUNCEd, α system for large n Of course, we also know from the previous section that such an improvement in tail behavior comes at a cost to throughput Lemma 2: Any fixed point of the system of equations 2, satisfying i= s i < and s i i must also satisfy: s i = O for arbitrary ɛ > λ1 ɛ d1 i/2k1 1 d V DELAY - NUMERICAL RESULTS In Figures 4 and 5, we can see empirical queue size tails for the ABOUNCEC,d and MBOUNCEα,d policies as compared with no sharing In both cases, a significant Fig 4 Queue-size tails for ABOUNCE1, C vs tails under no sharing n = 1, λ = 75, K = improvement is seen versus the policy of not sharing Further, the multiplicative threshold policies offer better performance than additive policies for this system Fig 6 Empirical expected delay for MBOUNCEα,1 policy with varying α versus no sharing n = 1, K = 2 Expected Delay λ = 7 MBOUNCE λ = 75 MBOUNCE λ = 8 MBOUNCE λ = 85 MBOUNCE λ = 7 No Sharing λ = 75 No Sharing λ = 8 No Sharing λ = 85 No Sharing α VI CONCLUSION We have presented two algorithms for load balancing when there is a cost to transferring jobs The use of the MBOUNCE algorithm provably entails no loss in throughput Simulation results, as well as a formal mean-field analysis strongly suggest that this algorithm affords improvements in waiting time as well as lighter queue-tails than the no-sharing case Several issues remain: 1 Can the mean field analysis of section IV be rigorized? The large deviations techniques used in 2] do not appear to extend 2 Is there an optimal decision boundary to decide between sharing and not-sharing a job? Prq 1 i C = 2 C = 3 C = C = 5 No sharing i ACKNOWLEDGMENTS The first author was supported by a supplement to NSF Grant ECS provided by the MKIDS Program The second author was supported by a Benchmark Stanford Graduate Fellowship The authors wish to thank Devavrat Shah and Chandra Nair for helpful discussions REFERENCES Fig 5 Queue-size tails for MBOUNCE1, α vs tails under no sharing n = 1, λ = 75, K = 2 Prq 1 i α = 14 α = 1 6 α = 26 No sharing i 1] M Mitzenmacher, The power of two choices in randomized load balancing, PhD dissertation, University of California, Berkeley, ] M Mitzenmacher, B Prabhakar, and D Shah, Load balancing with memory, in Proceedings of FOCS, 2 3] N Vvedenskaya, R Dobrushin, and F Karpelevich, Queueing system with selection of the shortest of two queues: An asymptotic approach, Problems of Information Transmission, vol 32, no 1, pp 15 29, ] C Nair, B Prabhakar, and D Shah, The randomness in randomized load balancing, in Proceedings of the 39th Annual Allerton Conference on Communication, Control and Computing, October 1, pp ] Y Azar, A Z Broder, A R Karlin, and E Upfal, Balanced allocations, SIAM J Comput, vol 29, no 1, pp 18, 6] M S Squiillante and E D Lazowska, Using processor-cache affinity information in shared-memory multiprocessor scheduling, IEEE Trans Parallel Distrib Syst, vol 4, no 2, pp , ] S Asmussen, Applied Probability and Queues, 2nd ed Springer-Verlag, 3

5 and APPENDIX: PROOF OF THEOREM 3 We use the Lyapunov function V q = V 1 qv 2 q where, V 1 q = qi 2, V 2 q = 1 α 1 j=i1 q i q j 2 To prove positive recurrence ie stability for the MBOUNCE Markov chain see Proposition 53 in 7], it will suffice to show for some ɛ > that V q ɛ for all q / K where K is some compact set For ease of exposition, we will assume for the balance of this proof that α = 2 The same technique can be applied for other α We also assume, without loss of generality, that the queues are numbered so that at time, q i q j for i j; in the event of ties we assign the queue with a higher number prior to re-numbering, the higher number In particular, after renumbering q 1 is the longest queue, while q n is the shortest We define the event E n = {q 1 < 2q n } {q 1 q n < K} That is, E n is the event that no transfer or arriving jobs is possible at the next epoch It is then straightforward to check that for q E n, V q = 2 N λ 1 q i C 1, 3 where C 1 is a constant depending on n, λ Now, for m {1,, n 1}, define E m = {q m 2q n, q m q n K} {q m1 < 2q n } {q m1 q n < K} Essentially, E m is the event wherein a transfer of job is possible and the shortest queue from which a transfer is possible is q m Further, define, p j,m = P argmin q i = j E m, i S where S is the set of indexes of the d queues sampled in the event of an arrival Thus p j,m is the probability that the shortest of the d queues sampled is queue j breaking ties in favor of the higher numbered queue An elementary argument then gives p j,m p j1,m Also, let p B = n p j,m It is readily checked that for q E m : 1 λ 1 V 1 q n q i i=m1 m ] 2p j,m Kq j 21 p B q i 4 1/n ] 2q i C 1,m We now consider V 2 q In evaluating V 2 q for q E m, we make the following observations: I It is straightforward to check that departures from q i, 1 i n or arrivals to q i, m 1 i n contribute at most a constant to the expectation II Consider an arrival to q i, 1 i m We get the following positive contribution to the expectation for when it is unsuccessful in transferring its job: 1 p B j=i1 2q i q j However, this contribution is negated by arrivals to queues numbered j > i, 21 p B λ m ] q i q j q i q j n j=i1 Now, consider the situation where i is successful at transferring its job to some queue This transfer reduces the difference in lengths between i and the queue it transfers its job to, which results in the following negative contribution 2p j,m Kq j The difference in queue lengths between the queue to which i transferred its job and the queues below it increases, which contributes the following positive term to the expectation: p j,m n l=j1 2Kq j q l 5 This term is negated since transfers from i reduce the difference in queue lengths between queues receiving a transfer and larger queues in {m 1,, n} contributing: p j,m j 1 l=m1 2Kq j q l 6 We conclude 5 6 using p j,m p j1,m Putting these observations together, we have that an arrival to q i, 1 i m contributes to the expectation, a term upper bounded by: 2p j,m Kq j C Putting our observations together yields that for q E m m V 2 q 2p j,m Kq j C 2,m 7 From 4 and 7, we have that for q E m, 1 λ 1 m V q n q i p ] B/n 2q i i=m1 C 1,m C 2,m Of course, 8 holds for all m, 1 m n 1, and so together with 3 we have V q ɛ, for n q i sufficiently large { n q i N ɛ }, say Since the complement of this set is compact, we are done 8

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