Improve Rate-Base Pull an Push Strategies in Large Distribute Networks Wouter Minnebo an Benny Van Hout Department of Mathematics an Computer Science University of Antwerp - imins Mielheimlaan, B-00 Antwerp, Belgium Email: {wouter.minnebo,benny.vanhout}@ua.ac.be Abstract Large istribute systems benefit from the ability to exchange jobs between noes to share the overall workloa. To exchange jobs, noes rely on probe messages that are either generate by lightly-loae or highly-loae noes, which correspons to a so-calle pull or push strategy. A key quantity of any pull or push strategy, that has often been neglecte in prior stuies, is the resulting overall probe rate. If one strategy outperforms another strategy in terms of the mean elay, but at the same time requires a higher overall probe rate, it is unclear whether it is truly more powerful. In this paper we introuce a new class of rate-base pull an push strategies that can match any preefine maximum allowe probe rate, which allows one to compare the pull an push strategy in a fair manner. We erive a close form expression for the mean elay of this new class of strategies in a homogeneous network with Poisson arrivals an exponential job urations uner the infinite system moel. We further show that the infinite system moel is the proper limit process over any finite time scale as the number of noes in the system tens to infinity an that the convergence extens to the stationary regime. Simulation experiments confirm that the infinite system moel becomes more accurate as the number of noes tens to infinity, while the observe error is alreay aroun % for systems with as few as 00 noes. I. INTRODUCTION One of the key features of a contemporary istribute network is its ability to (reistribute the workloa among a large number of processing noes. Jobs in such a network can either enter the network via one (or multiple centralize job ispatchers [], [] or may enter via the processing noes themselves [3] [6]. In the latter case, the workloa is reistribute by the exchange of jobs between the processing noes. Two important families of strategies have been ientifie for reistributing jobs: the pull an push strategy. Uner the pull strategy lightly-loae noes try to attract work from highlyloae noes, a strategy that is also known as work stealing. Uner the push strategy the highly-loae noes take the initiative to transfer jobs to lightly-loae noes. Many performance stuies of pull an push strategies have been presente over the years. The performance of two traitional pull an push strategies in homogeneous networks with Poisson arrivals an exponential job urations was analyze in [3] an extene to heterogeneous networks in [7]. Both stuies showe that the pull strategy is superior uner high loa conitions, while the push strategy results in a lower mean elay uner low to meium loas. More recent analytic stuies of the performance of pull an push strategies inclue [5], [6]. All of these stuies provie valuable insights with respect to the performance of pull an push strategies. However, they also pai harly any attention to the probe rate, that is, the number of probe messages that the strategies uner consieration generate per time unit. These probe messages are use to enable the exchange of jobs an thus to balance the loa. Typically, when a noe wants to pull or push a job it probes another noe at ranom to see whether a job can be transferre. Some of the probes result in a job transfer, while others o not. Clearly, sening more probe messages tens to result in more job transfers an therefore in lower mean elays. As ifferent strategies ten to have ifferent probe rates that epen to a large extent on the arrival rate λ, it is typically not possible to aapt the parameters of the strategies uner consieration such that they generate the same overall probe rate (for arbitrary λ, making any comparison biase. Further, some so-calle optimal parameter settings also result in the highest probe rate, which makes the optimality questionable (e.g., the hybri pull/push strategy in [4]. To mitigate this, we introuce a class of rate-base pull an push strategies in [8] that make use of a single parameter r. Uner the pull strategy ile noes generate probe messages at rate r, while uner the push strategy probe messages at rate r are generate by the noes with at least queue jobs (incluing the one in service. As shown in [8], these ratebase strategies can match any preefine maximum allowe probe rate R by setting r in the appropriate manner an this for any arrival rate λ, allowing a fair comparison between the pull an push strategy. The main results in [8] showe that the rate-base push strategy results in a lower mean elay if an only if (R + + 4(R + (R + λ <, uner the so-calle infinite system moel an that a hybri pull/push strategy is always inferior to the pure pull or push strategy. To evaluate an compare the ifferent strategies consiere in this paper we introuce an infinite system moel, the evolution of which is escribe by a set of orinary ifferential
equations (ODEs as in [5], [8]. To assess the mean elay an overall probe rate of a strategy, we efine a set of ODEs, give an explicit expression for its unique fixe point an express the mean elay an probe rate using this fixe point. To guarantee all trajectories converge to the fixe point, we prove that the fixe point is a global attractor. We also show that the set of ODEs captures the evolution of the limit process of a family of ensity epenent Markov chains as introuce by Kurtz in [9], [0]. Simulation experiments confirm that the infinite system moel becomes exact as the number of noes in the system tens to infinity, while the error is about % for systems with as few as 00 noes. This paper makes the following contributions: We introuce a more general class of rate-base pull an push strategies that rely on two parameters T an r an that coincie with the strategies introuce in [8] when T =. Close form results for the mean elay of this new class of pull an push strategies are presente (uner the infinite system moel. We show that setting T > reuces the mean elay of the rate-base push strategy (for larger λ an smaller R values. This is in contrast to earlier finings for the traitional strategy [4], [7], for which smaller T values result in higher probe rates, making the comparison biase. For the rate-base pull strategy we show that setting T = is optimal. 3 We introuce the so-calle max-push strategy an erive a close form expression for its mean elay (uner the infinite system moel. We show that the max-push strategies further reuce the mean elay of the best ratebase pull an push strategies with T for certain combinations of (λ, R. 4 Finally, we prove that the infinite system moels introuce in this paper are the proper limit processes of the finite stochastic systems with N noes as N tens to infinity over any finite time scale. In aition, we prove that the convergence extens to the stationary regime (i.e., the ODEs have a global attractor. The paper is structure as follows. Section II introuces the rate-base pull an push strategies. For the rate-base strategies with T we present the infinite system moel in Section III. In Section IV we valiate this moel using simulation results an present some numerical examples that compare the performance of the rate-base pull an push strategy. Section V introuces the max-push strategy an its infinite system moel, while numerical results for the the maxpush strategy are presente in Section VI. II. PULL AND PUSH STRATEGIES We consier a continuous-time system consisting of N queues, where each queue consists of a single server with an infinite buffer. As in [3], [5] [7], jobs arrive locally accoring to a Poisson process with rate λ <, an have an exponentially istribute uration with mean. Servers process jobs in a first-come first-serve orer. Servers can sen probe messages to each other to query for queue length information, an to transfer jobs. We assume that the time require to transfer probe messages an jobs is sufficiently small in comparison with the processing time of a job, i.e., transfer times are consiere zero. We consier the following loa balancing strategies that all make use of two parameters: an integer T an a real number r > 0. We note that the rate-base strategies with T = were initially introuce in [8]. Rate-base Push: As soon as the queue length excees T, a server starts to generate probe messages accoring to a Poisson process with rate r. Whenever the queue length rops below T, this process is interrupte until the queue length excees T again. The noe that is probe is selecte at ranom an is only allowe to accept a job if it is ile. Rate-base Pull: Whenever a server is ile it generates probe messages accoring to a Poisson process with rate r. This process is interrupte whenever the server is busy. The noe that is probe is selecte at ranom an is only allowe to transfer one of its jobs if its queue length excees T. For each of the above strategies transferre jobs are immeiately serve by the accepting noe, hence any job transfer results in a reuction of the mean elay. To make the comparison fair the mean overall probe rate R shoul be ientical. The rate R is efine as the mean number of probes that is sent by a server per time unit irrespective of its queue length, where R is clearly less than r. Further on we will show that r can be set in such a manner that it can match any preefine R 0. III. PERFORMANCE OF RATE-BASED STRATEGIES WITH T In this section we introuce the infinite system moel to assess the performance of the rate-base strategies with T. This moel, the evolution of which is capture by a set of ODEs, is valiate by simulation in Section IV, while in Section VII it is argue to be the proper limit process of the stochastic finite system moel with N noes as N tens to infinity. The evolution of both the rate-base pull an push strategy moel is given by a set of ODEs enote as tx(t = F (x(t, where x(t = (x (t, x (t,... an x i (t represents the fraction of the number of noes with at least i jobs at time t. As explaine below, this set of ODEs can be written as x (t t = (λ + rx T + (t( x (t (x (t x (t, x i (t = λ(x i (t x i (t (x i (t x i+ (t, ( t for i T an x i (t t = λ(x i (t x i (t (x i (t x i+ (t, ( r( x (t(x i (t x i+ (t, (3
3 for i > T. The terms λ(x i (t x i (t an x i (t x i+ (t, for i, correspon to arrival an service completions, respectively. Uner the pull strategy probes are sent at rate r( x (t an a probe is successful with probability x T + (t, while uner the push strategy the probe rate equals rx T + (t an a probe is successful with probability ( x (t. Hence, for both strategies queues of length are create by job transfers at rate rx T + (t( x (t. Similarly, job transfers reuce the number of queues with exactly i jobs, for i > T, at rate r( x (t(x i (t x i+ (t uner both strategies. Let E = {(x, x,... x i [0, ], x i x i+, i, j x j < }. The next two theorems show that this set of ODEs is Lipschitz on E an it has a unique fixe point in E. Theorem. The function F is Lipschitz on E. Proof: F is Lipschitz provie that for all x, y E there exists an L > 0 such that F (x F (y L x y. By efinition of F (x one fins F (x F (y (λ + + r x y + r i>t x (x i x i+ y (y i y i+. The above sum can be boune by (x y (x i x i+ + y (x i x i+ y i + y i+, i>t which is boune by x y on E. Hence, F is Lipschitz by letting L = λ + + 8r. As E is a Banach space the Lipschitz conition of F suffices to guarantee that the set of ODEs tx(t = F (x(t, with x(0 E, has a unique solution φ t (x(0 [, Section.]. Theorem. The set of ODEs given by ( to (3 has a unique fixe point π = ( π, π,... with i π i <. Let η i = π i π i+ an η 0 = λ, then the fixe point can be expresse as λ( + ( λr λ η = + ( λ T, r η i = η λ i, i T, ( i T λ η i = η T, i > T. + ( λr Proof: Assume π is a fixe point with i π i <, meaning F i ( π = 0 for i, where F (x = (F (x, F (x,.... When i π i <, we can simplify i F i(π = 0 to λ π = 0. Hence, π must equal λ. The expressions for η i then reaily follow from the conitions F i ( π = 0, for i. This fixe point is also the unique solution of the Kolmogorov ifferential equation for a state epenent M/M/ queue with λ 0 = λ + r π T +, λ i = λ, for i, µ i =, for i =,..., T, an µ i = + ( λr, for i T +. The The solution φ t(x belongs to the class of continuously ifferentiable functions as in the finite imensional case. arrival process of such an M/M/ queue is Poisson with rate λ i an the service is exponential with rate µ i whenever the queue length equals i. The set of ODEs in ( to (3 escribes the transient evolution of the infinite system, while we are in fact intereste in its behavior as t goes to infinity. Thus, we are intereste in the limit of all the trajectories of this set of ODEs. In Appenix A we prove the following theorem: Theorem 3. All the trajectories of the set of ODEs given by ( to (3, starting from x E converge towars the unique fixe point π. Due to the above theorem, we can now express the main performance measures of the push an pull strategies with T via Theorem : Corollary. The mean elay D of a job uner the push or pull strategy equals ( D both = rλt λ λ ( λ(+r + T + r( λ T. Proof: Using Theorem, one can apply Little s law to express the mean response time D as i π i/λ. Corollary. Given a preefine maximum allowe probe rate R, the rate r must be set as r pull = R λ, (4 r push = R λ T + ( λ T R, (5 with r push = for R > λ T + /( λ T. Hence, if the preefine value of R excees λ T + /( λ T, the rate r push can be set arbitrarily high. Proof: From the relationships R = ( π r pull an R = r push π T +, we fin an R = R = ( λr pull, λ T + ( λ T + /r push. Theorem 4. The mean response time D of a job uner the push stategy equals D push = R λ RT λ + R λ T ( λ(λ + R, if R λ T + /( λ T, while for R > λ T + /( λ T the rate r push =, an the mean elay D push is given by: D push r= = λ T λt λ T. Remark, D push r= = for T =. Uner the pull strategy the mean response time equals D pull = λ RλT λ ( λ+r + T λ + R( λ T.
4 Mean Delay (D push 0 9 8 7 6 5 4.046 λ = 0.95 λ = 0.9.0.03.0045.003.006.00064.053.06.03.0064.0033.007.00065 Mean Delay (D push 4 0 8 6.07.0.0068.0038.007.00033.0003 λ = 0.95 λ = 0.9.03.07.0083.004.009.00.00057 3.055 λ = 0.85.08.04.007.0035.007.00085.05.04 λ = 0.8.0.0059.003.005.00073 5 50 00 00 400 800 600 Infinity Number of Noes (N 4.035 λ = 0.85.07.0085.0045.00.00.0005.0.009.004.00.00 λ = 0.8.0005.0006 5 50 00 00 400 800 600 Infinity Number of Noes (N Fig.. Simulate mean elay for a finite system of varying size, using a ratebase push strategy with T =, matching an overall request rate of R =. The relative error, shown above a simulate point, becomes smaller when simulating larger systems. In aition, the infinite system moel approximates systems of moerate size well. Fig.. Simulate mean elay for a finite system of varying size, using a rate-base push strategy with T = 4, matching an overall request rate of R = 0.5. The relative error, shown above a simulate point, becomes smaller when simulating larger systems..5 λ = 0.8 Proof: The expressions for D push an D pull are foun using Corollary, by plugging in the appropriate value for r, given by Corollary, in the expression for D both. Theorem 5. The optimal choice for the rate-base pull strategy is T =. Proof: It can be verifie that increasing T by one will increase D pull if an only if λ/( λ + R + T λ + R + T + λ + R( λ T λλ/( λ + R( λ T +, which is equivalent to stating T ( λ + R( λ λr( λ T. This conition can be rewritten as T ( λ + R R T λ i, i= which hols as λ i <, for i =,..., T. Hence, increasing T by one always increases the mean elay of the rate-base pull strategy. Although it may at first seem sensible to steal jobs from long queues only if the maximum allowe probe rate R is low, the pull strategy is in some sense blin as it also nees to sen probes to locate these long queues. This is contrary to the push strategy, where noes will only probe if their queue is long, as a result setting T = is not always optimal for the rate-base push strategy (see numerical results in Section IV. IV. NUMERICAL RESULTS FOR RATE-BASED STRATEGIES WITH T A. Valiation In this section we present valiation results for the ratebase push strategy with T as the moel for both ratebase strategies with T = was alreay valiate in [8] an Probe Rate (R..05 λ = 0.85 λ = 0.9 λ = 0.95 5 50 00 00 400 800 600 Infinity Number of Noes (N Fig. 3. Request rate for the finite system using a rate-base push strategy with T =. the mean elay of the pull strategy is minimize for T =. The infinite system moel an simulation setup only iffer in the system size. The rate r push in the simulation experiments is inepenent of N an was etermine by λ an R using the expression for R in (5. Each simulate point in the figures represents the average value of 5 simulation runs. Each run has a length of 0 6 (where the service time is exponentially istribute with mean an a warm-up perio of length 0 6 /3. Figure epicts the mean elay as a function of N for T =, R = an λ = 0.8, 0.85, 0.9 an 0.95, while Figure epicts the same for T = 4, R = 0.5. In both cases the relative error shown above the simulation results ecreases as N tens to infinity. The error for a system with as few as 00 noes is only slightly above % when T =. We shoul note that the actual overall probe rate observe in the finite system excees R for smaller N values as shown in Figure 3 an 4. In other wors, the relation between R an r push given by (5 is not very accurate for small N values as the
5 Probe Rate (R 0.68 0.66 0.64 0.6 0.6 0.58 0.56 λ = 0.8 0.54 λ = 0.85 0.5 λ = 0.9 λ = 0.95 0.5 5 50 00 00 400 800 600 Infinity Number of Noes (N Fig. 4. Request rate for the finite system using a rate-base push strategy with T = 4. Mean Delay (D 5 4.5 4 3.5 3.5.5 R = Push T=6 Pull Push T= 0.5 0.5 0.6 0.7 0.8 0.9 Loa (λ Fig. 5. The mean elay for the rate-base pull (T = an push strategy (T =,..., 6 with R =. For the push strategy the part of the curve with r = is ashe. infinite moel is optimistic with respect to the queue length istribution. However, as soon as the system consists of several hunre noes, there is a fairly goo agreement. Similar results were observe for other parameter settings. B. Comparison of push an pull strategy The mean elay of the rate-base push (for T =,..., 6 an pull (for T = strategy is shown in Figure 5 as a function of λ for a mean overall probe rate R =. The curves for the push strategy consist of two parts an r push = for the ashe part of the curve. For these loas λ the rate r push can be set arbitrarily high without violating the maximum allowe probe rate R =. The results inicate that the optimal T value for the push strategy increases as λ increases (while R remains fixe. This is as expecte as small T values allow queues with a length below average to probe for ile servers, using part of the available probe rate. For the same reason smaller R values also give rise to larger optimal T values (for fixe λ. Figure 5 also inicates that setting T > implies that the rate-base push strategy can outperform the pull strategy Probe Rate (R.8.6.4. 0.8 0.6 0.4 0. Push T= Push T= Pull 0.5 0.6 0.7 0.8 0.9 Loa (λ Fig. 6. The optimal rate-base strategy with variable T as a function of the loa λ an the overall probe rate R. The pull strategy is optimal for high loas. The optimal T for the push strategy increases as R ecreases. for a larger range of loas λ. The rate-base strategy (with variable T that minimizes the mean elay for λ [0.5, 0.9] an R [0, ] is epicte in Figure 6. The pull strategy is superior for loas above 75 to 80%. For lower loas the push strategy prevails an lower maximum allowe probe rates R give rise to larger optimal T values. V. THE MAX-PUSH STRATEGY The mean elay uner the rate-base push strategy given in Theorem 4 can be further reuce as follows. Recall, whenever R > λ T + /( λ T, the rate r push can be chosen arbitrarily large (i.e., r push =. In other wors, even if requests are sent at infinite rate when the queue length excees T, the overall probe rate remains below R. Hence, in orer to use this remaining request rate, we introuce the max-push strategy when T > an λ T + /( λ T < R < λ T /( λ T. (6 Note, for any R > 0 an 0 < λ <, there exists a single T > 0 such that the above relationship hols. Uner the maxpush strategy we sen probes at an infinite rate whenever the queue length excees T an at rate r < if the queue length equals T. Note, uner this strategy jobs are instantaneously transferre to another queue if the queue length equals T upon arrival (at the expense of a number of probe messages. The evolution of the infinite system moel for this strategy is also reaily formulate as a set of ODEs tx(t = G(x(t, where x(t = (x (t,..., x T (t an x i (t represents the fraction of the number of noes with at least i jobs at time t: x (t t x i (t t = λ( x (t + x T (t (x (t x (t + rx T (t( x (t (7 = λ(x i (t x i (t (x i (t x i+ (t, (8
6 for i T an x T (t t = λ(x T (t x T (t x T (t( + r( x (t. The terms of the form λ(x i (t x i (t an (x i (t x i+ (t, for i T, are again ue to arrival an service completion events, respectively. Aitionally, queues of length are create at rate λx T (t ue to the instantaneous job transfers an rate rx T (t( x (t ue to successful probes sent by a queue of length T, while the latter event also reuces the number of queues of length T by one. Let E T = {(x,..., x T x x... x T 0}. The next two theorems show that this set of ODEs is Lipschitz on E T an it has a unique fixe point in E T. Theorem 6. The function G is Lipschitz on E T. Proof: G is Lipschitz provie that for all x, y E T there exists an L > 0 such that G(x G(y L x y. By efinition of G(x one fins G(x G(y (λ + + r x y. Hence, G is Lipschitz by letting L = 4λ + + 4r. As E T is a finite imensional space the Lipschitz conition of G suffices to guarantee that the set of ODEs t x(t = G(x(t, with x(0 E T, has a unique solution (ue to the Picar Linelöf theorem. Theorem 7. The set of ODEs given by (7-(9 has a unique fixe point π = ( π,..., π T in E T that can be expresse as π i = λ + ( λ i λ + r( λt i + ( λ λ + r( λt, for i T. Proof: Assume π is a fixe point with i π i T, meaning G i ( π = 0 for i, where G(x = (G (x, G (x,..., G T (x. When i π i T, we can simplify i G i( π = 0 to λ π = 0. Hence, π must equal λ. The expressions for π i then follow from the conition G i ( π = 0. In Appenix B we prove the following theorem: Theorem 8. All the trajectories of the set of ODEs given by (7-(9, starting from x E T converge towars the unique fixe point π. Due to the above theorem, we can now express the main performance measures of the max-push strategy via Theorem 7: Corollary 3. The mean elay D mp of a job uner the maxpush strategy equals D mp = λt + ( λ λ + r( T λt + (T λ T + r( λ( λ T λ T. A preefine overall probe rate R can be matche by setting r mp = (9 R λ T (R + λ R λ λ, (0 Mean Delay (D push.5.4.3.. λ = 0.75.033 λ = 0.7.0 λ = 0.65.09.0044.0004.000.e 05.6e 05.4e 06.0063.009.004.0007.00034.0007.0078.0037.008.0009.00047.0003 5 50 00 00 400 800 600 Infinity Number of Noes (N Fig. 7. Simulate mean elay for a finite system of varying size, using a max-push strategy with T = an R =. The relative error, shown above a simulate point, becomes smaller when simulating larger systems. In aition, the infinite system moel approximates systems of moerate size well. Probe Rate (R.8.7.6.5.4.3.. λ = 0.75 λ = 0.7 λ = 0.65 5 50 00 00 400 800 600 Infinity Number of Noes (N Fig. 8. Observe probe rate for the finite system using a max-push strategy with T =. where 0 r mp < for λ T + /( λ T R < λ T /( λ T. Proof: The mean response time D can be expresse as T i π i/λ by Little s law. For the max-push strategy the overall probe rate R equals ( λ R = π T λ + r mp, as the instantaneous transfer of an arrival to a queue with T jobs requires /( λ probe messages on average. VI. NUMERICAL RESULTS FOR THE MAX-PUSH STRATEGY A. Valiation In this section we valiate the infinite system moel for the max-push strategy using the same approach as in Section IV-A for the rate-base push strategy with T >. The rate r mp in the simulation was etermine using the relationship in (0. The mean elay as a function of the number of noes N an the relative error are shown in Figure 7 for T =,
7 Mean Delay (D push 5 4.5 4 3.5 3.5.5 R = 0.5 0.5 0.6 0.7 0.8 0.9 Loa (λ Fig. 9. The mean elay for the rate-base push strategy (full lines for T =,..., 6 with R =. For each strategy, the probe rate r = when λ is below the loa marke by a ot. The max-push strategy interconnects the ots, as shown by ash-otte curves. Probe Rate (R.8.6.4. 0.8 0.6 0.4 0. Push T= Push T Max Push T Pull 0.7 0.75 0.8 0.85 Loa (λ Fig. 0. The ifferent areas ientify the (λ, R combinations for which the rate-base pull strategy (with T = is outperforme by the rate-base push strategy with T =, by the rate-base push strategy with T an by the max-push strategy. R = an λ = 0.65, 0.7 an 0.75. Notice, the max-push strategy with T = an R = is only properly efine for λ [0.680, 0.7549] ue to (6, larger λ values woul result in the choice of a larger T value. The relative errors are small (below % for N = 00 noes an ecrease as N increases. We shoul note that the probe rates observe uring the simulation are well above for small N as illustrate in Figure 8. Hence, the relationship in (0 for the maxpush strategy is less accurate than (5 for the rate-base push strategy for small N. Nonetheless, the observe probe rate still seems to ecrease to as N tens to infinity. B. Comparison of pull an max-push strategy The mean elay of the max-push strategy is epicte in Figure 9. The ots represent the points where r = for the rate-base push strategy, i.e., the positive real roots of λ T + + (λ T R = 0. The max-push strategy nicely interconnects these points as it utilizes the remaining probe rate. The combination of (λ, R values for which the pull strategy is outperforme by the rate-base push strategy with T =, by the rate-base push strategy with T an by the maxpush strategy, respectively, is shown in Figure 0. The pull strategy is still the most effective for larger loas λ, however, for a large range of (λ, R values the elay of the pull strategy can be reuce using a rate-base push strategy with T > or a max-push strategy. VII. FINITE VERSUS INFINITE SYSTEM MODEL Similar to [8] for the rate-base strategies with T =, we can efine a family of ensity epenent Markov chains [9] to escribe the behavior of the stochastic finite systems with N noes for both the rate-base pull/push an max-push strategy. In case of the max-push strategy convergence towars the infinite system moel over finite time scales follows from Kurtz s well-known theorem [9] an the convergence extens to the stationary regime as we showe that the set of ODEs given by (7-(9 has a unique global attractor in E T, ue to a result by Benaïm []. For the rate-base pull/push strategy with T we can rely on the following generalization of Kurtz s theorem [3, Theorem 3.3]: Theorem 9 (Kurtz. Consier a family of ensity epenent CTMCs, with F Lipschitz. Let lim N X (N (0 = x a.s. an let φ t ( x be the unique solution to the initial value problem tx(t = F (x(t with x(0 = x. Consier the path {φ t ( x, t T } for some fixe T 0 an assume that there exists a neighborhoo K aroun this path satisfying l sup β l (x <, ( x K then l L lim sup X (N (t φ t ( x = 0 a.s. N t T For the rate-base pull/push strategy, the above conition ( correspons to showing that there exists an environment K such that i sup x K(x i x i+ <. Such an environment can be shown to exist by repeating the argument for T = from [4, Theorem 7]. To show that the convergence extens to the stationary regime, we can make use of a theorem by Benaïm an Le Bouec [5] as in the T = case, where the require proof for the tightness of the measures can be proven as in [4]. VIII. CONCLUSION AND FUTURE WORK In this paper we introuce a new class of rate-base pull an push strategies that can match any preefine maximum allowe probe rate R. This class relie on a threshol parameter T such that jobs can only be exchange between ile noes an noes with a queue length exceeing T, where the class of strategies introuce in [8] correspons to setting T =. We erive a close form expression for the mean elay of this new class of strategies in a homogeneous network with Poisson arrivals an exponential job urations uner the socalle infinite system moel.
8 We showe that setting T = is optimal for the pull strategies consiere, while for the push strategy setting T > may reuce the mean elay for some values of (λ, R, i.e., for larger λ an smaller R values. We further introuce the max-push strategy, which utilizes the remaining probe rate capacity in case R > λ T + /( λ T, erive a close form expression for its mean elay an (numerically ientifie the (λ, R region where it outperforms the pull strategy. We prove that the infinite system moels of both the ratebase strategies with T > an the max-push strategy, are the proper limit processes of the finite stochastic systems with N noes as N tens to infinity over any finite timescale. Moreover, the convergence was shown to exten to the stationary regime by proving that the ODEs have a global attractor. We valiate these theoretical results by simulation, an have shown that the infinite moel is an accurate approximation for finite systems of moerate size. The current results can be extene in a number of ways: networks with finite queues can be consiere (this actually makes some of the technical issues less involve, the assumptions on the arrival an service time istribution coul be relaxe (which makes the analysis more challenging or heterogeneous networks coul be stuie. The class of strategies consiere in the paper can also be generalize. For instance, it shoul be possible to incorporate another parameter B, such that any noe with a queue length below B is allowe to accept push requests instea of only the ile noes. REFERENCES [] M. Mitzenmacher, The power of two choices in ranomize loa balancing, IEEE Trans. Parallel Distrib. Syst., vol., pp. 094 04, October 00. [] Y. Lu, Q. Xie, G. Kliot, A. Geller, J. R. Larus, an A. Greenberg, Joinile-queue: A novel loa balancing algorithm for ynamically scalable web services, Perform. Eval., vol. 68, pp. 056 07, 0. [3] D. Eager, E. Lazowska, an J. Zahorjan, A comparison of receiverinitiate an sener-initiate aaptive loa sharing, Perform. Eval., vol. 6, no., pp. 53 68, 986. [4] R. Mirchananey, D. Towsley, an J. Stankovic, Analysis of the effects of elays on loa sharing, IEEE Trans. Comput., vol. 38, no., pp. 53 55, 989. [5] M. Mitzenmacher, Analyses of loa stealing moels base on families of ifferential equations, Theory of Computing Systems, vol. 34, pp. 77 98, 00. [6] N. Gast an B. Gaujal, A mean fiel moel of work stealing in largescale systems, SIGMETRICS Perform. Eval. Rev., vol. 38, no., pp. 3 4, 00. [7] R. Mirchananey, D. Towsley, an J. A. Stankovic, Aaptive loa sharing in heterogeneous istribute systems, J. Parallel Distrib. Comput., vol. 9, no. 4, pp. 33 346, 990. [8] W. Minnebo an B. Van Hout, Pull versus push mechanism in large istribute networks: Close form results, in Proc. of the 4-th International Teletraffic Congress, Krakau (Polan, 0. [9] T. Kurtz, Approximation of population processes. Society for Inustrial an Applie Mathematics, 98. [0] S. Ethier an T. Kurtz, Markov processes: characterization an convergence. Wiley, 986. [] K. Deimling, Orinary Differential Equations in Banach spaces. Lect. Notes in Math. 596, 977. [] M. Benaïm, Recursive algorithms, urn processes an chaining number of chain recurrent sets, Ergoic Theory an Dynamical Systems, vol. 8, pp. 53 87, 998. [3] M. Mitzenmacher, The power of two choices in ranomize loa balancing, Ph.D. issertation, University of California, Berkeley, 996. [4] W. Minnebo an B. Van Hout, A fair comparison of pull an push strategies in large istribute networks, to appear in IEEE/ACM Transactions on Networking, 03. [5] M. Benaïm an J. Le Bouec, On mean fiel convergence an stationary regime, CoRR, vol. abs/.570, Nov 4 0. [6] J. Walker, Dynamical Systems an Evolution Equations. Theory an Applications. Plenum Press, New York, 980. APPENDIX A PROOF OF THEOREM 3 We start by proving the following Lemma: Lemma. Let x(t be the unique solution of the set of ODEs given by ( to (3 with x(0 E. The L -istance to the unique fixe point i x i(t π i oes not increase as a function of t. Proof: Define ɛ i (t = x i (t π i, for i, such that Φ(t = i ɛ i(t represents the L -istance. As t x i(t = t ɛ i(t an π is a fixe point of ( to (3, we fin t ɛ (t = ɛ (t( + λ rɛ (t( π T + + ɛ T + (t + rɛ T + (t( π + ɛ (t, ( t ɛ i(t = λɛ i (t ( + λɛ i (t + ɛ i+ (t, (3 for i T, an t ɛ i(t = λ(ɛ i (t ɛ i (t + rɛ (t( π i π i+ (ɛ i (t ɛ i+ (t( rɛ (t + r( π, (4 for i > T. Assume for now that ɛ i (t 0 for all i such that Φ(t is properly efine as t t Φ(t = i:ɛ i(t>0 t ɛ i(t i:ɛ i(t<0 t ɛ i(t. If ɛ i (t has the same sign for all i, one fins that t Φ(t = ɛ (t by summing ( to (4, we will show that this inequality also hols in general. Let I = {i, i,...}, with i < i <..., be the set of inices where ɛ i (t changes sign, that is, ɛ i (t an ɛ i (t have a ifferent sign if an only if i I. Assume ɛ (t < 0 an let I k = {i I : i T + }, I m = {i I : i > T + }, By means of ( to (4, we fin that if ɛ i (t an ɛ i (t iffer in sign, tφ(t contains an extra term given by for i =,..., T +, sign(ɛ i (t(λɛ i (t ɛ i (t, sign(ɛ i (t[λɛ i (t+rɛ (t π i ɛ i (t( rɛ (t+r( π ], for i > T +. Further, if ɛ (t an ɛ T + (t iffer in sign, Φ(t contains an extra term given by t sign(ɛ T + (tr[ɛ (t( π T + + ɛ T + (t ɛ T + (t( π ].
9 This implies that for ɛ (t 0 t Φ(t = ɛ (t +α }{{} + sign(ɛ i (t{λɛ i (t ɛ i (t( + r( π } }{{} i I m + sign(ɛ i (t{rɛ (t(ɛ i (t + π i } i I m + i I k sign(ɛ i (t{λɛ i (t ɛ i (t} }{{} where α is equal to rɛ (t(ɛ T + (t + π T + ɛ T + (tr( π. }{{} if ɛ T + (t > 0 an zero otherwise. Hence, t Φ(t ɛ (t provie that sign(ɛ i (t(ɛ i (t + π i = sign(ɛ i (tx i (t 0, i I m i I m if ɛ T + (t 0 an x T + (t + i I m sign(ɛ i (tx i (t 0, if ɛ T + (t > 0. Let I m = {i 0, i,...}. In case ɛ T + (t 0, the sign(ɛ in (t is equal to for n even an for n o. Hence, the conition reuces to (x ik (t x ik+ (t 0, k 0 which hols as x i (t x j (t for i < j. Similarly, if ɛ T + (t > 0, the sign(ɛ in (t is equal to for n even an for n o. Hence, the conition reuces to (x T + (t x i0 (t + k 0(x ik+ x ik+ 0, which again hols as x i (t x j (t for i < j. Hence, t Φ(t ɛ (t if ɛ (t 0. A similar argument can be use for ɛ (t 0. Finally, the technical issue of efining tφ(t in case ɛ i (t = 0 for some i an t = t 0, can be resolve as in [4]. The above lemma shows that the L -istance to the fixe point oes not increase along any trajectory x(t in E, an can only remain the same whenever x (t = π an there are no sign changes in the ɛ i (t s. Lemma. The only trajectory x(t of the ODEs given by ( to (3 with x(0 E for which the L -istance oes not ecrease is given by x(t = π for all t. Proof: From the proof of Lemma, we know that x (t = π = λ for all t, whenever the L -istance oes not ecrease. Equation ( therefore implies that x T + (t = λ x (t r( λ such a trajectory. Hence, if x (t = π + c, then x T + (t = λ π (t r( λ c r( λ = π c T + r( λ. Hence, ɛ (t = x (t π an ɛ T + (t = x T + (t π T + iffer in sign unless x (t = π an x T + (t = π T +. The fact that x(t = π on such a trajectory now follows from ( to (3. We now recall La Salle s invariance principle for Banach spaces, where a (positively invariant subset of K E of an ODE efine on E is such that x(t K for all t provie that x(t is the unique solution of the ODE with x(0 K. Theorem 0 ( [6]. Let V (x be a continuous real value function from E to R with t V (x = lim sup t 0 + t (V (x(t V (x 0, where x(t is the unique solution of an ODE with x(0 = x. Let K = {x E tv (x = 0} an let M be the largest (positively invariant subset of K. If x(t is precompact (i.e., remains in a compact set for x(0 E, then lim ist(x(t, M = 0, t where ist(x, M represents the Banach istance between the point x E an the set M E. Using La Salle s invariance principle, Theorem 3 can be proven analogously to [4, Theorem 3] with V (x equal to the L -istance. APPENDIX B PROOF OF THEOREM 8 We start by proving the following Lemma: Lemma 3. Let x(t be the unique solution of the ODEs given by (7 to (9 with x(0 E T. The L -istance to the unique fixe point i x i(t π i oes not increase as a function of t. Proof: Using the same efinitions as in Appenix A, we fin t ɛ (t = λ(ɛ T (t ɛ (t (ɛ (t ɛ (t + rɛ T (t( (ɛ (t + π r π T ɛ (t, (5 t ɛ i(t = λɛ i (t ( + λɛ i (t + ɛ i+ (t, (6 for < i < T, an t ɛ T (t = λ(ɛ T (t ɛ T (t + rɛ (t π T ɛ T (t( + r( (ɛ (t + π. (7 Assume for now that ɛ i (t 0 for all i such that tφ(t is properly efine. If ɛ i (t has the same sign for all i, one fins that t Φ(t = ɛ (t by summing (5 to (7, we will show that this inequality also hols in general. If ɛ i (t an ɛ i (t iffer in sign, tφ(t changes by sign(ɛ i (t(λɛ i (t ɛ i (t, on
0 for i =,..., T, while a ifference in sign between the terms ɛ (t an ɛ T (t creates a term of the form sign(ɛ T (t{λɛ T (t + rɛ (t π T ɛ T (tr( (ɛ (t + π }. Assume ɛ (t 0, then we fin t Φ(t = ɛ (t +α + {sign(ɛ i (t(λɛ i (t ɛ i (t} }{{}}{{} i I (8 where α is equal to α = λɛ T (t rɛ T (t( (ɛ (t + π + r π T ɛ (t. }{{}}{{}}{{} (9 if ɛ T (t > 0 an α = 0 otherwise. Hence, t Φ(t ɛ (t. A similar argument can be use for ɛ (t > 0 by reversing all the signs. As in Appenix A, the technical issue of efining t Φ(t in case ɛ i (t = 0 for some i an t = t 0 is resolve by relying on the upper right-han erivative (as in [, Theorem 3]. The above lemma shows that the L -istance to the fixe point oes not increase along any trajectory x(t in E T, an only remains the same whenever x (t = π (as ɛ (t = 0 in such a case. Lemma 4. The only trajectory x(t of the ODE given by (7 to (9 with x(0 E T for which the L -istance oes not ecrease is given by x(t = π for all t. Proof: If x (t = π = λ for all t, then (7 implies that x T (t = λ x(t λ+r( λ an the proof procees as in Lemma. Using La Salle s invariance principle for Banach spaces as given by Theorem 0, we can now prove theorem 8: Proof of Theorem 8: We rely on La Salle s invariance principle for Banach spaces by setting V (x equal to the L - istance to the fixe point, i.e., V (x = T i= x i π i. Lemma 3 implies that tv (x 0, while Lemma 4 shows that M = { π} is a singleton. Hence, π is a global attractor since E T itself is a compact set an all trajectory are containe within E T by efinition.