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A load balancing system in the many-server heavy-traffic asymptotics

Abstract

We study a load balancing system in the many-server heavy-traffic regime. We consider a system with N servers, where jobs arrive to the system according to a Poisson process and have an exponentially distributed size with mean 1. We parametrize the arrival rate so that the arrival rate per server is \(1-N^{-\alpha }\), where \(\alpha >0\) is a parameter that represents how fast the load grows with respect to the number of servers. The many-server heavy-traffic regime corresponds to the limit as \(N\rightarrow \infty \), and subsumes several regimes, such as the Halfin–Whitt regime (\(\alpha =1/2\)), the NDS regime (\(\alpha =1\)), as \(\alpha \downarrow 0\) it approximates mean field and as \(\alpha \rightarrow \infty \) it approximates the classical heavy-traffic regime. Most of the prior work focuses on regimes with \(\alpha \in [0,1]\). In this paper, we focus on the case when \(\alpha >1\) and the routing algorithm is power-of-d choices with \(d=\lceil cN^\beta \rceil \) for some constants \(c>0\) and \(\beta \ge 0\). We prove that \(\alpha +\beta >3\) is sufficient to observe that the average queue length scaled by \(N^{1-\alpha }\) converges to an exponential random variable. In other words, if \(\alpha +\beta >3\), the scaled average queue length behaves similarly to the classical heavy-traffic regime. In particular, this result implies that if d is constant, we require \(\alpha >3\) and if routing occurs according to JSQ we require \(\alpha >2\). We provide two proofs to our result: one based on the Transform method introduced in Hurtado-Lange and Maguluri (Stoch Syst 10(4):275–309, 2020) and one based on Stein’s method. In the second proof, we also compute the rate of convergence in Wasserstein’s distance. In both cases, we additionally compute the rate of convergence in expected value. All of our proofs are powered by state space collapse.

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References

  1. Adan, I., van Houtum, G.J., van der Wal, J.: Upper and lower bounds for the waiting time in the symmetric shortest queue system. Ann. Oper. Res. 48(2), 197–217 (1994)

    Article  Google Scholar 

  2. Atar, R.: A diffusion regime with nondegenerate slowdown. Oper. Res. 60(2), 490–500 (2012)

    Article  Google Scholar 

  3. Badonnel, R., Burgess, M.: Dynamic pull-based load balancing for autonomic servers. In: NOMS 2008-2008 IEEE Network Operations and Management Symposium, pp. 751–754. IEEE (2008)

  4. Banerjee, S., Mukherjee, D.: Join-the-shortest queue diffusion limit in Halfin–Whitt regime: tail asymptotics and scaling of extrema. Ann. Appl. Probab. 29(2), 1262–1309 (2019)

    Article  Google Scholar 

  5. Banerjee, S., Mukherjee, D.: Join-the-shortest queue diffusion limit in Halfin–Whitt regime: sensitivity on the heavy-traffic parameter. Ann. Appl. Probab. 30(1), 80–144 (2020)

    Article  Google Scholar 

  6. Bramson, M.: State space collapse with application to heavy-traffic limits for multiclass queueing networks. Queueing Syst. Theory Appl., 89 – 148 (1998)

  7. Braverman, A.: Steady-state analysis of the join-the-shortest-queue model in the Halfin–Whitt regime. Math. Oper. Res. (2020)

  8. Braverman, A., Dai, J.: Stein’s method for steady-state diffusion approximations of M/Ph/n+ M systems. Ann. Appl. Probab. 27(1), 550–581 (2017)

    Article  Google Scholar 

  9. Braverman, A., Dai, J., Feng, J.: Stein’s method for steady-state diffusion approximations: an introduction through the Erlang-A and Erlang-C models. Stoch. Syst. 6(2), 301–366 (2017)

    Article  Google Scholar 

  10. Braverman, A., Dai, J., Miyazawa, M.: Heavy traffic approximation for the stationary distribution of a Generalized Jackson Network: The BAR approach. Stoch. Syst. 7(1), 143–196 (2017)

    Article  Google Scholar 

  11. Dai, J.: Steady-state approximations: achievement lecture. In: Abstracts of the 2018 ACM International Conference on Measurement and Modeling of Computer Systems, pp. 1–1. ACM (2018)

  12. Dai, J., Lin, W.: Asymptotic optimality of maximum pressure policies in stochastic processing networks. Ann. Appl. Probab. 18(6), 2239–2299 (2008)

    Article  Google Scholar 

  13. Dai, J., Tezcan, T.: State space collapse in many-server diffusion limits of parallel server systems. Math. Oper. Res. 36(2), 271–320 (2011)

    Article  Google Scholar 

  14. Ephremides, A., Varaiya, P., Walrand, J.: A simple dynamic routing problem. IEEE Trans. Autom. Control 25(4), 690–693 (1980)

    Article  Google Scholar 

  15. Eryilmaz, A., Srikant, R.: Asymptotically tight steady-state queue length bounds implied by drift conditions. Queueing Syst. 72(3–4), 311–359 (2012)

    Article  Google Scholar 

  16. Eschenfeldt, P., Gamarnik, D.: Join the Shortest Queue with many servers. The heavy-traffic asymptotics. Math. Oper. Res. 43(3), 867–886 (2018)

    Article  Google Scholar 

  17. Foschini, G., Salz, J.: A basic dynamic routing problem and diffusion. IEEE Trans. Commun. 26(3), 320–327 (1978)

    Article  Google Scholar 

  18. Foss, S., Stolyar, A.L.: Large-scale join-idle-queue system with general service times. J. Appl. Probab. 995–1007 (2017)

  19. Gamarnik, D., Zeevi, A.: Validity of heavy traffic steady-state approximations in Generalized Jackson Networks. Ann. Appl. Probab. 56–90 (2006)

  20. Gibbs, A.L., Su, F.E.: On choosing and bounding probability metrics. Int. Stat. Rev. 70(3), 419–435 (2002)

    Article  Google Scholar 

  21. Gupta, V., Walton, N.: Load balancing in the nondegenerate slowdown regime. Oper. Res. 67(1), 281–294 (2019)

    Article  Google Scholar 

  22. Gurvich, I.: Diffusion models and steady-state approximations for exponentially ergodic Markovian queues. Ann. Appl. Probab. 24(6), 2527–2559 (2014)

    Article  Google Scholar 

  23. Hajek, B.: Random Processes for Engineers. Cambridge University Press (2015)

  24. Halfin, S., Whitt, W.: Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29(3), 567–588 (1981)

    Article  Google Scholar 

  25. Harrison, J.: Brownian models of queueing networks with heterogeneous customer populations. In: Stochastic Differential Systems, Stochastic Control Theory and Applications, pp. 147–186. Springer (1988)

  26. Harrison, J.: Heavy traffic analysis of a system with parallel servers: asymptotic optimality of discrete review policies. Ann. App. Probab. 822–848 (1998)

  27. Harrison, J., López, M.: Heavy traffic resource pooling in parallel-server systems. Queueing Syst. 339–368 (1999)

  28. Hurtado-Lange, D., Maguluri, S.T.: Heavy-traffic analysis of queueing systems with no complete resource pooling. arXiv preprint arXiv:1904.10096 (2019)

  29. Hurtado-Lange, D., Maguluri, S.T.: Throughput and delay optimality of power-of-d choices in inhomogeneous load balancing systems. arXiv preprint arXiv:2004.00538 (2020)

  30. Hurtado-Lange, D., Maguluri, S.T.: Transform methods for heavy-traffic analysis. Stoch. Syst. 10(4), 275–309 (2020)

    Article  Google Scholar 

  31. Kang, W., Kelly, F., Lee, N., Williams, R.: State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy. Ann. Appl. Probab. 1719–1780 (2009)

  32. Liu, X., Ying, L.: In: On universal scaling of distributed queues under load balancing (2019). ArXiv preprint arXiv:1912.11904

  33. Liu, X., Ying, L.: A simple steady-state analysis of load balancing algorithms in the sub-Halfin–Whitt regime. ACM SIGMETRICS Perform. Eval. Rev. 46(2), 15–17 (2019)

    Article  Google Scholar 

  34. Lu, Y., Xie, Q., Kliot, G., Geller, A., Larus, J., Greenberg, A.: Join-Idle-Queue: a novel load balancing algorithm for dynamically scalable web services. Perform. Eval. 68(11), 1056–1071 (2011)

    Article  Google Scholar 

  35. Maguluri, S.T., Burle, S., Srikant, R.: Optimal heavy-traffic queue length scaling in an incompletely saturated switch. Queueing Syst. 88(3–4), 279–309 (2018)

    Article  Google Scholar 

  36. Maguluri, S.T., Srikant, R.: Heavy traffic queue length behavior in a switch under the MaxWeight algorithm. Stoch. Syst. 6(1), 211–250 (2016). https://doi.org/10.1214/15-SSY193

    Article  Google Scholar 

  37. Maguluri, S.T., Srikant, R., Ying, L.: Heavy traffic optimal resource allocation algorithms for cloud computing clusters. Perform. Eval. 81, 20–39 (2014)

    Article  Google Scholar 

  38. Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: theory of majorization and its applications, vol. 143. Springer (1979)

  39. Mitzenmacher, M.: Load balancing and density dependent jump Markov processes. In: FOCS, p. 213. IEEE (1996)

  40. Mitzenmacher, M.: The power of two choices in randomized load balancing. IEEE Trans. Parallel Distrib. Syst. 12(10), 1094–1104 (2001)

    Article  Google Scholar 

  41. Miyazawa, M.: Diffusion approximation for stationary analysis of queues and their networks: a review. J. Oper. Res. Soc. Jpn. 58(1), 104–148 (2015)

    Google Scholar 

  42. Mukherjee, D., Borst, S.C., Van Leeuwaarden, J.S., Whiting, P.A.: Universality of power-of-d load balancing in many-server systems. Stoch. Syst. 8(4), 265–292 (2018)

    Article  Google Scholar 

  43. Mukherjee, D., Borst, S.C., Van Leeuwaarden, J.S., Whiting, P.A., et al.: Universality of load balancing schemes on the diffusion scale. J. Appl. Probab. 53(4), 1111–1124 (2016)

    Article  Google Scholar 

  44. Ross, N.: Fundamentals of Stein’s method. Probab. Surv. 8, 210–293 (2011)

  45. Shah, D., Wischik, D.: Switched networks with maximum weight policies: fluid approximation and multiplicative state space collapse. Ann. Appl. Probab. 22(1), 70–127 (2012)

    Google Scholar 

  46. Stein, C.: A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Probability Theory. The Regents of the University of California (1972)

  47. Stolyar, A.: MaxWeight scheduling in a generalized switch: state space collapse and workload minimization in heavy traffic. Ann. Appl. Probab., 1–53 (2004)

  48. Stolyar, A.: Tightness of stationary distributions of a flexible-server system in the Halfin–Whitt asymptotic regime. Stoch. Syst. 5(2), 239–267 (2015)

    Article  Google Scholar 

  49. Stolyar, A.: Pull-based load distribution among heterogeneous parallel servers: the case of multiple routers. Queueing Syst. 85(1–2), 31–65 (2017)

    Article  Google Scholar 

  50. van der Boor, M., Borst, S., van Leeuwaarden, J., Mukherjee, D.: Scalable load balancing in networked systems: a survey of recent advances. arXiv preprint arXiv:1806.05444 (2018)

  51. Vvedenskaya, N., Dobrushin, R., Karpelevich, F.: Queueing system with selection of the shortest of two queues: an asymptotic approach. Probl. Inf. Transm. 32(1), 15–27 (1996)

    Google Scholar 

  52. Wang, C.H., Maguluri, S.T., Javidi, T.: Heavy traffic queue length behavior in switches with reconfiguration delay. In: INFOCOM 2017-IEEE Conference on Computer Communications, IEEE, pp. 1–9. IEEE (2017)

  53. Wang, W., Maguluri, S.T., Srikant, R., Ying, L.: Heavy-traffic insensitive bounds for weighted proportionally fair bandwidth sharing policies. Math. Oper. Res. (2022)

  54. Weber, R.: On the optimal assignment of customers to parallel servers. J. Appl. Probab. 15(2), 406–413 (1978)

    Article  Google Scholar 

  55. Weng, W., Wang, W.: Dispatching parallel jobs to achieve zero queuing delay. arXiv preprint arXiv:2004.02081 (2020)

  56. Wheeden, R.L.: Measure and Integral: An Introduction to Real Analysis, vol. 308. CRC Press (2015)

  57. Williams, R.: Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse. Queueing Syst. Theory Appl. pp. 27 – 88 (1998)

  58. Williams, R.: On dynamic scheduling of a parallel server system with complete resource pooling. Fields Inst. Commun. 28(49–71), 5–1 (2000)

    Google Scholar 

  59. Winston, W.: Optimality of the shortest line discipline. J. Appl. Probab. 14(1), 181–189 (1977). https://doi.org/10.1017/S0021900200104772

    Article  Google Scholar 

  60. Ying, L.: On the approximation error of mean-field models. In: Proceedings of the 2016 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Science, SIGMETRICS ’16, pp. 285–297. ACM, New York, NY, USA (2016). https://doi.org/10.1145/2896377.2901463

  61. Ying, L.: Stein’s method for mean field approximations in light and heavy traffic regimes. Proc. ACM Meas. Anal. Comput. Syst. 1(1), 12:1–12:27 (2017). https://doi.org/10.1145/3084449

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Acknowledgements

We would like to thank the anonymous reviewers for carefully checking the correctness of our arguments and their meaningful feedback to improve the presentation of our paper.

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Correspondence to Daniela Hurtado-Lange.

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Appendices

Appendix

Details of proofs using Transform method

Proof of Lemma 2

Proof

(of Lemma 2) We omit the dependence on N and t of the variables, for ease of exposition. By definition of indicator function, for any \(i\in [N]\) we have

$$\begin{aligned} \mathbbm {1}_{\left\{ q_i=0 \right\} }\exp \left( {\scriptstyle \theta N^{-\alpha }q_\Sigma }\right)&= \mathbbm {1}_{\left\{ q_i=0 \right\} }\exp \left( {\scriptstyle -\theta N^{1-\alpha }q_i}\right) \exp \left( {\scriptstyle \theta N^{-\alpha }q_\Sigma }\right) \\&{\mathop {=}\limits ^{(a)}} \mathbbm {1}_{\left\{ q_i=0 \right\} } + \mathbbm {1}_{\left\{ q_i=0 \right\} } \left( \exp \left( {\scriptstyle -\theta N^{1-\alpha }q_{\perp i}}\right) -1\right) , \end{aligned}$$

where \(q_{\perp i}\) is the \(i{}^{\text {th}}\) component of \({\varvec{q}}_\perp \). Here, (a) holds by definition of \({\varvec{q}}_\perp \) according to (2), and after adding and subtracting \(\mathbbm {1}_{\left\{ q_i=0 \right\} }\). Then, recalling the definition of \(\phi ({\varvec{q}},N)\) and reorganizing terms we obtain

$$\begin{aligned} \phi ({\varvec{q}},N)&{\mathop {=}\limits ^{\triangle }}\left( \exp \left( {\scriptstyle \theta N^{1-\alpha }q_\Sigma }\right) - 1\right) \left( \sum _{i=1}^N \mathbbm {1}_{\left\{ q_i=0 \right\} }\right) \\&= \sum _{i=1}^N \mathbbm {1}_{\left\{ q_i=0 \right\} } \left( \exp \left( {\scriptstyle -\theta N^{1-\alpha }q_{\perp i}}\right) -1\right) . \end{aligned}$$

We now compute the desired bound. We have

$$\begin{aligned} \left| {\mathbb {E}}\left[ \phi ({\overline{{\varvec{q}}}},N) \right] \right|&{\mathop {\le }\limits ^{(a)}} {\mathbb {E}}\left[ \sum _{i=1}^N \mathbbm {1}_{\left\{ {\overline{q}}_i=0 \right\} } \left| \exp \left( {\scriptstyle -\theta N^{1-\alpha } {\overline{q}}_{\perp i}}\right) -1 \right| \right] \nonumber \\&{\mathop {\le }\limits ^{(b)}} |\theta | N^{1-\alpha } {\mathbb {E}}\left[ \sum _{i=1}^N \mathbbm {1}_{\left\{ {\overline{q}}_i=0 \right\} }|{\overline{q}}_{\perp i}| \exp \left( {\scriptstyle |\theta | N^{1-\alpha }|{\overline{q}}_{\perp i}|}\right) \right] \nonumber \\&{\mathop {\le }\limits ^{(c)}} |\theta | N^{1-\alpha } {\mathbb {E}}\left[ \sum _{i=1}^N \mathbbm {1}_{\left\{ {\overline{q}}_i=0 \right\} } \right] ^{1-\frac{1}{r}} {\mathbb {E}}\left[ \sum _{i=1}^N |{\overline{q}}_{\perp i}|^r \exp \left( {\scriptstyle |\theta | N^{1-\alpha } r |{\overline{q}}_{\perp i}|}\right) \right] ^{\frac{1}{r}} \nonumber \\&{\mathop {=}\limits ^{(d)}} |\theta | N^{(1-\alpha )\left( 2-\frac{1}{r}\right) } {\mathbb {E}}\left[ \sum _{i=1}^N |{\overline{q}}_{\perp i}|^r \exp \left( {\scriptstyle |\theta | N^{1-\alpha } r |{\overline{q}}_{\perp i}|}\right) \right] ^{\frac{1}{r}}, \end{aligned}$$
(25)

where \(r>1\). Here, (a) holds by triangle inequality; (b) holds because \(|\exp \left( {\scriptstyle x}\right) -1|\le |x|\exp \left( {\scriptstyle |x|}\right) \) for all \(x\in {\mathbb {R}}\); (c) hods by Hölder’s inequality for the vectors \({\varvec{X}}\) and \({\varvec{Y}}\) with elements \(X_i=\mathbbm {1}_{\left\{ {\overline{q}}_i=0 \right\} }\) and \(Y_i=|{\overline{q}}_{\perp i}| \exp \left( {\scriptstyle |\theta | N^{1-\alpha } |{\overline{q}}_{\perp i}|}\right) \) for \(i\in [N]\), and noticing that \(X_i^r = X_i\) because it is an indicator function; and (d) holds by Lemma 1.

Now we bound the expectation in (25) using properties of norms, Cauchy-Schwarz inequality and SSC. For \(r\ge 2\) we have

$$\begin{aligned}&{\mathbb {E}}\left[ \sum _{i=1}^N |{\overline{q}}_{\perp i}|^r \exp \left( {\scriptstyle |\theta | N^{1-\alpha } r |{\overline{q}}_{\perp i}|}\right) \right] ^{\frac{1}{r}} \nonumber \\&{\mathop {\le }\limits ^{(a)}} {\mathbb {E}}\left[ \left\| {\overline{{\varvec{q}}}}_\perp \right\| ^r_r \exp \left( {\scriptstyle |\theta | N^{1-\alpha } r \Vert {\overline{{\varvec{q}}}}_\perp \Vert }\right) \right] ^{\frac{1}{r}} \nonumber \\&{\mathop {\le }\limits ^{(b)}} {\mathbb {E}}\left[ \left\| {\overline{{\varvec{q}}}}_\perp \right\| ^r \exp \left( {\scriptstyle |\theta | N^{1-\alpha } r \Vert {\overline{{\varvec{q}}}}_\perp \Vert }\right) \right] ^{\frac{1}{r}} \nonumber \\&{\mathop {\le }\limits ^{(c)}} {\mathbb {E}}\left[ \Vert {\overline{{\varvec{q}}}}_\perp \Vert ^{2r} \right] ^{\frac{1}{2r}} {\mathbb {E}}\left[ \exp \left( {\scriptstyle |\theta | N^{1-\alpha } 2r \Vert {\overline{{\varvec{q}}}}_\perp \Vert }\right) \right] ^{\frac{1}{2r}}, \nonumber \\ \end{aligned}$$
(26)

where (a) holds using that \(|{\overline{q}}_{\perp i}|\le \Vert {\overline{{\varvec{q}}}}_\perp \Vert \) in the exponent and by definition of the r-norm; (b) holds because the r-norm is smaller than the Euclidean norm for all \(r\ge 2\); and (c) holds by Cauchy-Schwarz inequality.

Now we bound each of the terms in (26) using SSC. From Proposition 1, recall that for every positive integer k we have

$$\begin{aligned} {\mathbb {E}}\left[ \left\| {\overline{{\varvec{q}}}}_\perp \right\| ^k \right] ^{\frac{1}{k}} \le {\overline{C}}k \left( \dfrac{N^2}{d-1}\right) , \end{aligned}$$

and for every \(\theta ^*\) satisfying \(|\theta ^*| < \frac{1}{2}\log \left( 1 + \tfrac{\lambda _0(d-1)}{2N^2}\right) \) we have

$$\begin{aligned} {\mathbb {E}}\left[ \exp \left( {\scriptstyle \theta ^* \left\| {\overline{{\varvec{q}}}}_\perp \right\| }\right) \right] \le \dfrac{\lambda _0(d-1)\exp \left( {\scriptstyle \tfrac{2\theta ^*N^2}{\lambda _0(d-1)}}\right) }{\lambda _0(d-1) + 2N^2\left( 1- \exp \left( {\scriptstyle 2\theta ^*}\right) \right) }. \end{aligned}$$

Using these results in (26) with \(k=2r\) and \(\theta ^*=2|\theta |rN^{1-\alpha }\), we obtain

$$\begin{aligned}&{\mathbb {E}}\left[ \sum _{i=1}^N |{\overline{q}}_{\perp i}|^r \exp \left( {\scriptstyle |\theta | N^{1-\alpha } r |{\overline{q}}_{\perp i}|}\right) \right] ^{\frac{1}{r}} \\&\le 2{\overline{C}}\lambda _0 \left( \dfrac{ rN^2 \exp \left( {\scriptstyle \tfrac{4|\theta | r N^{3-\alpha }}{\lambda _0(d-1)}}\right) }{\lambda _0(d-1) + 2N^2\left( 1-\exp \left( {\scriptstyle 4|\theta | r N^{1-\alpha }}\right) \right) }\right) . \end{aligned}$$

Using this result in (25), we obtain

$$\begin{aligned} \left| {\mathbb {E}}\left[ \phi \left( {\overline{{\varvec{q}}}},N\right) \right] \right|&\le 2{\overline{C}}\lambda _0 |\theta | \left( \dfrac{rN^{(1-\alpha )\left( 1-\frac{1}{r}\right) } N^2 \exp \left( {\scriptstyle \tfrac{4|\theta | r N^{3-\alpha }}{\lambda _0(d-1)}}\right) }{\lambda _0(d-1) + 2N^2\left( 1 - \exp \left( {\scriptstyle 4|\theta | r N^{1-\alpha }}\right) \right) } \right) . \end{aligned}$$

Since this upper bound holds for every \(r\ge 2\), we minimize the bound with respect to r and we obtain that \(r = \lceil \alpha -1\rceil \lceil \log (N)\rceil \) gives the tightest bound. Replacing this value we obtain the result. \(\square \)

Proof of Lemma 3

In this proof we use the definition of drift and we reorganize terms appropriately.

Proof

(of Lemma 3) We have:

$$\begin{aligned}&\Delta V_\parallel ({\varvec{q}}) \\&= \lambda N \sum _{i=1}^N \dfrac{\left( {\begin{array}{c}N-i\\ d-1\end{array}}\right) }{\left( {\begin{array}{c}N\\ d\end{array}}\right) } \left( \left\| \left( {\varvec{q}}+{\varvec{e}}^{(\psi _{{\varvec{q}}}(i))}\right) _\parallel \right\| ^2 - \left\| {\varvec{q}}_\parallel \right\| ^2 \right) + \sum _{i=1}^N \left( 1-\mathbbm {1}_{\left\{ q_i=0 \right\} }\right) \\&\quad \times \left( \left\| \left( {\varvec{q}}-{\varvec{e}}^{(i)}\right) _\parallel \right\| ^2 - \left\| {\varvec{q}}_\parallel \right\| ^2 \right) \\&{\mathop {=}\limits ^{(a)}} \lambda \sum _{i=1}^N \dfrac{\left( {\begin{array}{c}N-i\\ d-1\end{array}}\right) }{\left( {\begin{array}{c}N\\ d\end{array}}\right) } \left( 1 + 2\sum _{j=1}^N q_j\right) + \frac{1}{N} \sum _{i=1}^N \left( 1-\mathbbm {1}_{\left\{ q_i=0 \right\} }\right) \left( 1-2\sum _{j=1}^N q_j \right) , \end{aligned}$$

where (a) holds by the definition of \({\varvec{x}}_\parallel \) given a vector \({\varvec{x}}\) in (2), and computing the norms. This completes the proof. \(\square \)

Details of the Proof of Proposition 1

Proof of Lemma 6

In the proof of Lemma 6, we use the bound (18) to compute an upper bound on the moment generating function of \(Z({\overline{X}})\).

Proof

(of Lemma 6) First observe that \(Z({\overline{X}})\ge 0\) by assumption of Lemma 5. Then,

$$\begin{aligned} \exp \left( {\scriptstyle \theta Z({\overline{X}})}\right) \le \exp \left( {\scriptstyle |\theta | Z({\overline{X}})}\right) . \end{aligned}$$

We compute an upper bound for \({\mathbb {E}}\left[ \exp \left( {\scriptstyle |\theta | Z({\overline{X}})}\right) \right] \). Let \(F_Z(x)\) be the cumulative distribution function of \(Z({\overline{X}})\). Then,

$$\begin{aligned}&{\mathbb {E}}\left[ \exp \left( {\scriptstyle |\theta | Z({\overline{X}})}\right) \right] \\&= \int _0^\infty \exp \left( {\scriptstyle |\theta | x}\right) \,\mathrm{d}F_Z(x) \\&{\mathop {=}\limits ^{(a)}} \left[ -\exp \left( {\scriptstyle |\theta | x}\right) {\mathbb {P}}\left( Z({\overline{X}})>x\right) \right] ^\infty _0 + |\theta |\int _0^\infty \exp \left( {\scriptstyle |\theta | x}\right) {\mathbb {P}}\left( Z({\overline{X}})>x\right) \,\mathrm{d}x \\&= {\mathbb {P}}\left( Z({\overline{X}})>0\right) + |\theta | \int _0^B \exp \left( {\scriptstyle |\theta | x}\right) {\mathbb {P}}\left( Z({\overline{X}})>x\right) \,\mathrm{d}x \\ {}&\quad + |\theta | \int _B^\infty \exp \left( {\scriptstyle |\theta | x}\right) {\mathbb {P}}\left( Z({\overline{X}})>x\right) \,\mathrm{d}x \\&{\mathop {\le }\limits ^{(b)}} \exp \left( {\scriptstyle |\theta | B}\right) + \sum _{j=0}^\infty \int _{B+2\nu _{\max }j}^{B+2\nu _{\max }(j+1)} |\theta | \exp \left( {\scriptstyle |\theta | x}\right) {\mathbb {P}}\left( Z({\overline{X}})>x\right) \, \mathrm{d}x \\&{\mathop {\le }\limits ^{(c)}} \exp \left( {\scriptstyle |\theta | B}\right) + \sum _{j=0}^\infty \int _{B+2\nu _{\max }j}^{B+2\nu _{\max }(j+1)} |\theta | \exp \left( {\scriptstyle |\theta | x}\right) {\mathbb {P}}\left( Z({\overline{X}})> B+2\nu _{\max }j\right) \,\mathrm{d}x \\&{\mathop {\le }\limits ^{(d)}} \exp \left( {\scriptstyle |\theta | B}\right) + \exp \left( {\scriptstyle |\theta | B}\right) \left( \exp \left( {\scriptstyle 2|\theta |\nu _{\max }}\right) -1 \right) \left( \dfrac{G_{\max }\nu _{\max }}{G_{\max }\nu _{\max }+\gamma } \right) \\ {}&\quad \times \sum _{j=0}^\infty \left( \dfrac{G_{\max }\nu _{\max }\exp \left( {\scriptstyle 2|\theta |\nu _{\max }}\right) }{G_{\max }\nu _{\max }+\gamma } \right) ^j \\&{\mathop {=}\limits ^{(e)}} \dfrac{\exp \left( {\scriptstyle |\theta | B}\right) \gamma }{\gamma + G_{\max }\nu _{\max }(1-\exp \left( {\scriptstyle 2\nu _{\max }|\theta |}\right) )} \end{aligned}$$

where (a) holds integrating by parts; (b) holds because probabilities are upper bounded by 1, solving \(\int _0^B \exp \left( {\scriptstyle |\theta | x}\right) \,dx\), and breaking the last integral into intervals; (c) holds because \(f(x)=1-F_Z(x)={\mathbb {P}}\left( Z({\overline{X}})>x\right) \) is a nonincreasing function; (d) holds by (18) and solving the integral; and (e) holds after solving the geometric summation and reorganizing terms, because \(|\theta |<\frac{1}{2\nu _{\max }}\log \left( 1+\tfrac{\gamma }{G_{\max }\nu _{\max }}\right) \) by assumption and, hence, the geometric sum converges. \(\square \)

Proof of Lemma 7

In this proof we use the definition of drift and properties of concave functions.

Proof

(of Lemma 7) First observe that if g(x) is a differentiable concave function on \({\mathbb {R}}_+\), we have that for any \(x,y\in {\mathbb {R}}_+\)

$$\begin{aligned} g(x)-g(y)\le g'(y)(x-y). \end{aligned}$$
(27)

Now, observe that \(W_\perp ({\varvec{q}})=\left\| {\varvec{q}}_\perp \right\| =\sqrt{\left\| {\varvec{q}}_\perp \right\| ^2}\) and \(g(x)=\sqrt{x}\) is a concave function. Therefore, by definition of drift in Definition 1, and the generator matrix in (1), we have

$$\begin{aligned}&\Delta W_\perp ({\varvec{q}}) \\&= \lambda N\sum _{i=1}^N \dfrac{\left( {\begin{array}{c}N-i\\ d-1\end{array}}\right) }{\left( {\begin{array}{c}N\\ d\end{array}}\right) } \left( W_\perp \left( {\varvec{q}}+{\varvec{e}}^{(\psi _{{\varvec{q}}}(i))}\right) - W_\perp \left( {\varvec{q}}\right) \right) \\ {}&\quad + \sum _{i=1}^N \left( 1-\mathbbm {1}_{\left\{ q_i=0 \right\} } \right) \left( W_\perp \left( {\varvec{q}}-{\varvec{e}}^{(i)}\right) - W_\perp ({\varvec{q}}) \right) \\&{\mathop {\le }\limits ^{(a)}} \lambda N \sum _{i=1}^N \dfrac{\left( {\begin{array}{c}N-i\\ d-1\end{array}}\right) }{\left( {\begin{array}{c}N\\ d\end{array}}\right) } \left( \dfrac{\left\| \left( {\varvec{q}}+{\varvec{e}}^{(\psi _{{\varvec{q}}}(i))} \right) _\perp \right\| ^2 - \left\| {\varvec{q}}_\perp \right\| ^2}{2\left\| {\varvec{q}}_\perp \right\| } \right) \\&\quad + \sum _{i=1}^N \left( 1-\mathbbm {1}_{\left\{ q_i=0 \right\} }\right) \left( \dfrac{\left\| \left( {\varvec{q}}- {\varvec{e}}^{(i)}\right) _\perp \right\| ^2 - \left\| {\varvec{q}}_\perp \right\| ^2}{2\left\| {\varvec{q}}_\perp \right\| } \right) \\&{\mathop {=}\limits ^{(b)}} \dfrac{\lambda N}{2\left\| {\varvec{q}}_\perp \right\| }\sum _{i=1}^N \dfrac{\left( {\begin{array}{c}N-i\\ d-1\end{array}}\right) }{\left( {\begin{array}{c}N\\ d\end{array}}\right) }\left( V\left( {\varvec{q}}+{\varvec{e}}^{(\psi _{{\varvec{q}}}(i))}\right) - V({\varvec{q}}) - \left( V_\parallel \left( {\varvec{q}}+{\varvec{e}}^{(\psi _{{\varvec{q}}}(i))}\right) - V_\parallel ({\varvec{q}}) \right) \right) \\&\quad + \sum _{i=1}^N \left( \dfrac{1-\mathbbm {1}_{\left\{ q_i=0 \right\} }}{2\left\| {\varvec{q}}_\perp \right\| }\right) \left( V\left( {\varvec{q}}-{\varvec{e}}^{(i)}\right) - V({\varvec{q}}) - \left( V_\parallel \left( {\varvec{q}}-{\varvec{e}}^{(i)}\right) - V_\parallel ({\varvec{q}}) \right) \right) \\&{\mathop {=}\limits ^{(c)}} \dfrac{1}{2\left\| {\varvec{q}}_\perp \right\| }\left( \Delta V({\varvec{q}}) - \Delta V_{\parallel }({\varvec{q}}) \right) \end{aligned}$$

where (a) holds by (27) applied in the first and the second term in the following way. In the first term we use \(x=\left\| \left( {\varvec{q}}+{\varvec{e}}^{(\psi _{{\varvec{q}}}(i))} \right) _\perp \right\| ^2\) and \(y=\left\| {\varvec{q}}_\perp \right\| ^2\), and in the second term we use \(x=\left\| \left( {\varvec{q}}- {\varvec{e}}^{(i)}\right) _\perp \right\| ^2\) and \(y=\left\| {\varvec{q}}_\perp \right\| ^2\). Equality (b) holds by the definition of \(V(\cdot )\) and \(V_\parallel (\cdot )\) in (20) and because for any vector \({\varvec{x}}\in {\mathbb {R}}^N\), we have \(\left\| {\varvec{x}}_\perp \right\| ^2=\left\| {\varvec{x}}\right\| ^2 - \left\| {\varvec{x}}_\parallel \right\| ^2\); and (c) holds by reorganizing terms and by definition of drift. \(\square \)

Proof of Lemma 8

In this proof we use properties of the order statistics \(q_{(i)}\) for \(i\in [N]\). Recall that \(q_{(i)}\) represents the \(i{}^{\text {th}}\) shortest element of \({\varvec{q}}\), with ties broken by the minimum index.

Proof

(of Lemma 8) We have

$$\begin{aligned}&\Delta V({\varvec{q}}) \nonumber \\&= \lambda N \sum _{i=1}^N \dfrac{\left( {\begin{array}{c}N-i\\ d-1\end{array}}\right) }{\left( {\begin{array}{c}N\\ d\end{array}}\right) } \left( \left\| {\varvec{q}}+ {\varvec{e}}^{(\psi _{{\varvec{q}}}(i))}\right\| ^2 - \left\| {\varvec{q}}\right\| ^2 \right) \nonumber \\ {}&\quad + \sum _{i=1}^N \left( 1-\mathbbm {1}_{\left\{ q_i=0 \right\} }\right) \left( \left\| {\varvec{q}}-{\varvec{e}}^{(i)} \right\| ^2 - \left\| {\varvec{q}}\right\| ^2\right) \nonumber \\&{\mathop {=}\limits ^{(a)}} \lambda N \sum _{i=1}^N \dfrac{\left( {\begin{array}{c}N-i\\ d-1\end{array}}\right) }{\left( {\begin{array}{c}N\\ d\end{array}}\right) } \left( 1+2 q_{(i)}\right) + \sum _{i=1}^N (1-\mathbbm {1}_{\left\{ q_i=0 \right\} }) \left( 1-2q_i \right) \nonumber \\&{\mathop {\le }\limits ^{(b)}} N(\lambda +1) - 2(1-\lambda ) \sum _{i=1}^N q_i + 2\lambda \sum _{i=1}^N \left( \dfrac{N \left( {\begin{array}{c}N-i\\ d-1\end{array}}\right) }{\left( {\begin{array}{c}N\\ d\end{array}}\right) }-1\right) q_{(i)}, \end{aligned}$$
(28)

where (a) holds because, by definition of \(\psi _{{\varvec{q}}}(i)\), we have \(q_{\psi _{{\varvec{q}}}(i)}=q_{(i)}\); and (b) holds because \(\mathbbm {1}_{\left\{ q_i=0 \right\} }q_i=0\) for all \(i\in [N]\), because \(\sum _{i=1}^N \mathbbm {1}_{\left\{ q_i=0 \right\} }\ge 0\) and reorganizing terms.

The last step of the proof is to show that

$$\begin{aligned} \sum _{i=1}^N \left( \dfrac{N \left( {\begin{array}{c}N-i\\ d-1\end{array}}\right) }{\left( {\begin{array}{c}N\\ d\end{array}}\right) }-1\right) q_{(i)} \le -\left( \dfrac{d-1}{N}\right) \left\| {\varvec{q}}_\perp \right\| , \end{aligned}$$
(29)

which we do at the end of this section. Using the bound (29), we obtain the result. \(\square \)

In the proof of (29), we use properties of the order statistics and majorization. Specifically, we use the following lemma, which is proved in [38, Section 16.A.2.a].

Lemma 10

Consider three vectors \({\varvec{a}},{\varvec{b}},{\varvec{x}}\in {\mathbb {R}}^N\). The inequality

$$\begin{aligned} \sum _{i=1}^N a_i x_{(i)}\le \sum _{i=1}^N b_i x_{(i)} \end{aligned}$$

holds if and only if

  1. (C1)

    The total sum satisfies

    $$\begin{aligned} \sum _{i=1}^N a_i = \sum _{i=1}^N b_i. \end{aligned}$$
  2. (C2)

    For every \(k\in [N]\), the partial sums satisfy

    $$\begin{aligned} \sum _{i=k}^N a_i \le \sum _{i=k}^N b_i. \end{aligned}$$

Now we show (29).

Proof

(of (29)) For each \(i\in [N]\) define

$$\begin{aligned} \eta _i {\mathop {=}\limits ^{\triangle }}\dfrac{ N \left( {\begin{array}{c}N-i\\ d-1\end{array}}\right) }{\left( {\begin{array}{c}N\\ d\end{array}}\right) }, \end{aligned}$$
(30)

and observe that \(\eta _i=0\) for \(i\ge N-d+1\). Then,

$$\begin{aligned}&\sum _{i=1}^N \left( \dfrac{N \left( {\begin{array}{c}N-i\\ d-1\end{array}}\right) }{\left( {\begin{array}{c}N\\ d\end{array}}\right) }-1\right) q_{(i)} = \sum _{i=1}^N \left( \eta _i -1\right) q_{(i)}. \end{aligned}$$

Observe that \(\eta _1=d\). Then,

$$\begin{aligned}&\sum _{i=1}^N \left( \eta _i-1\right) q_{(i)}\nonumber \\&= (d-1)q_{(1)} + \sum _{i=2}^N \left( \eta _i - 1\right) q_{(i)} \nonumber \\&{\mathop {=}\limits ^{(a)}} \left( \dfrac{d-1}{N}\right) \sum _{i=1}^N \left( q_{(1)}-q_i\right) + \sum _{i=1}^N \left( \eta _i - \dfrac{N-d+1}{N}\right) q_{(i)} - (d-1)q_{(1)} , \end{aligned}$$
(31)

where (a) holds after reorganizing terms. We bound each of the terms of (31). For the first term we have

$$\begin{aligned} \left( \dfrac{d-1}{N}\right) \sum _{i=1}^N \left( q_{(1)}-q_i \right)&{\mathop {=}\limits ^{(a)}} -\left( \dfrac{d-1}{N}\right) \sum _{i=1}^N \left| q_i-q_{(1)}\right| \\&= -\left( \dfrac{d-1}{N}\right) \left\| {\varvec{q}}-q_{(1)}{\varvec{1}}\right\| _1 \\&{\mathop {\le }\limits ^{(b)}} -\left( \dfrac{d-1}{N}\right) \left\| {\varvec{q}}- q_{(1)}{\varvec{1}}\right\| \\&{\mathop {\le }\limits ^{(c)}} -\left( \dfrac{d-1}{N}\right) \left\| {\varvec{q}}_\perp \right\| , \end{aligned}$$

where (a) holds because \(q_{(1)}=\min _{i\in [N]}q_i\); (b) holds because norm-1 upper bounds the Euclidean norm; and (c) holds because, by definition of projection, the function \(g(x)=\left\| {\varvec{q}}-x{\varvec{1}}\right\| \) is minimized at \(x=\dfrac{1}{N}\sum _{i=1}^N q_i\), which equals the elements of \({\varvec{q}}_\parallel \). Then, the inequality holds by definition of \({\varvec{q}}_\perp {\mathop {=}\limits ^{\triangle }}{\varvec{q}}-{\varvec{q}}_\parallel \).

Now we only need to show that

$$\begin{aligned} \sum _{i=1}^N \eta _i q_{(i)} - (d-1)q_{(1)}\le \left( \dfrac{N-d+1}{N}\right) \sum _{i=1}^N q_{(i)}. \end{aligned}$$

We use Lemma 10 with \({\varvec{a}}\) and \({\varvec{b}}\) defined as follows:

$$\begin{aligned} a_1&{\mathop {=}\limits ^{\triangle }}\eta _1 - (d-1) = 1,\quad a_i{\mathop {=}\limits ^{\triangle }}\eta _i = \dfrac{N\left( {\begin{array}{c}N-i\\ d-1\end{array}}\right) }{\left( {\begin{array}{c}N\\ d\end{array}}\right) }\quad \forall i\in [N], i\ge 2\\ b_i&{\mathop {=}\limits ^{\triangle }}\dfrac{N-d+1}{N} \quad \forall i\in [N]. \end{aligned}$$

We first show that condition (C1) is satisfied. To do so, we compute the sum of the elements of \({\varvec{a}}\) and \({\varvec{b}}\). For the vector \({\varvec{a}}\) we obtain

$$\begin{aligned} \sum _{i=1}^N a_i&= 1 + \dfrac{N}{\left( {\begin{array}{c}N\\ d\end{array}}\right) } \sum _{i=2}^N \left( {\begin{array}{c}N-i\\ d-1\end{array}}\right) {\mathop {=}\limits ^{(a)}} 1 + (N-1) \dfrac{\left( {\begin{array}{c}N-2\\ d-1\end{array}}\right) }{\left( {\begin{array}{c}N-1\\ d-1\end{array}}\right) } {\mathop {=}\limits ^{(b)}} N-d+1, \end{aligned}$$

where (a) holds after solving the summation; and (b) holds after simplifying the last term.

For the vector \({\varvec{b}}\) we obtain

$$\begin{aligned} \sum _{i=1}^N b_i&= \sum _{i=1}^N \dfrac{N-d+1}{N} = N-d+1, \end{aligned}$$

where the last equality holds because the general term of the summation does not depend on the index i. Hence, condition (C1) is satisfied.

To prove condition (C2), we consider three cases: (i) \(k\ge N-d+2\), (ii) \(2\le k\le N-d+1\), and (iii) \(k=1\). First observe that in case (iii) the inequality trivially holds after proving (C1). Now we prove the other two cases.

We start with case (i). Since \(k\ge N-d+2\), we have \(\left( {\begin{array}{c}N-k\\ d-1\end{array}}\right) =0\) for all k. Additionally, \(b_i\ge 0\) for all \(i\in [N]\) by definition. Therefore, condition (C2) is satisfied for \(k\ge N-d+2\).

For case (i) we compute the partial sums. We obtain

$$\begin{aligned} \sum _{i=k}^N a_i&= \dfrac{N}{\left( {\begin{array}{c}N\\ d\end{array}}\right) } \sum _{i=k}^N \left( {\begin{array}{c}N-i\\ d-1\end{array}}\right) \nonumber \\&{\mathop {=}\limits ^{(a)}} \dfrac{N}{\left( {\begin{array}{c}N\\ d\end{array}}\right) } \left( \dfrac{N+1-k}{d}\right) \left( {\begin{array}{c}N-k\\ d-1\end{array}}\right) \nonumber \\&{\mathop {=}\limits ^{(b)}} (N+1-k)\dfrac{\left( {\begin{array}{c}N-k\\ d-1\end{array}}\right) }{\left( {\begin{array}{c}N-1\\ d-1\end{array}}\right) } \nonumber \\&{\mathop {=}\limits ^{(c)}} (N+1-k)\dfrac{\left( {\begin{array}{c}N-2\\ d-1\end{array}}\right) }{\left( {\begin{array}{c}N-1\\ d-1\end{array}}\right) } \nonumber \\&{\mathop {=}\limits ^{(d)}} (N+1-k)\left( \dfrac{N-d}{N-1}\right) \end{aligned}$$
(32)

where (a) holds after solving the summation; (b) holds after reorganizing terms; (c) holds because \(k\ge 2\). Then, it suffices to show that

$$\begin{aligned} (32) \le \sum _{i=k}^N b_i = \dfrac{(N-k+1)(N-d+1)}{N}, \end{aligned}$$

which is satisfied if and only if

$$\begin{aligned} \dfrac{N-d}{N-1}\le \dfrac{N-d+1}{N}. \end{aligned}$$
(33)

Reorganizing terms in (33) we see that the condition is equivalent to \(d\ge 1\), which holds by assumption. This completes the proof. \(\square \)

Proof of Lemma 9

The goal of this section is to compute a lower bound on \(\Delta V_\parallel ({\varvec{q}})\). We use Lemma 3 (where we computed \(\Delta V_\parallel ({\varvec{q}})\)), properties of the Euclidean norm and of indicator functions.

Proof

(of Lemma 9) From Lemma 3 we have

$$\begin{aligned} \Delta V_\parallel ({\varvec{q}})&= \lambda \sum _{i=1}^N \dfrac{\left( {\begin{array}{c}N-i\\ d-1\end{array}}\right) }{\left( {\begin{array}{c}N\\ d\end{array}}\right) } \left( 1 + 2\sum _{j=1}^N q_j\right) + \frac{1}{N} \sum _{i=1}^N \left( 1-\mathbbm {1}_{\left\{ q_i=0 \right\} }\right) \left( 1-2\sum _{j=1}^N q_j \right) \nonumber \\&{\mathop {=}\limits ^{(a)}} \lambda - 2(1-\lambda ) \sum _{i=1}^N q_i + \dfrac{1}{N}\sum _{i=1}^N \left( 1-\mathbbm {1}_{\left\{ q_i=0 \right\} }\right) \nonumber \\ {}&\quad + \dfrac{2}{N}\left( \sum _{i=1}^N \mathbbm {1}_{\left\{ q_i=0 \right\} }\right) \left( \sum _{i=1}^N q_i \right) \nonumber \\&{\mathop {\ge }\limits ^{(b)}} -2(1-\lambda )\sum _{i=1}^N q_i, \end{aligned}$$
(34)

where (a) holds after reorganizing terms; and (b) holds because \(\lambda \ge 0\), \(1-\mathbbm {1}_{\left\{ q_i=0 \right\} }\ge 0\) for all \(i\in [N]\), and \(\left( \sum _{i=1}^N \mathbbm {1}_{\left\{ q_i=0 \right\} }\right) \left( \sum _{i=1}^N q_i \right) \ge 0\) since every term is nonnegative. This completes the proof. \(\square \)

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Hurtado-Lange, D., Maguluri, S.T. A load balancing system in the many-server heavy-traffic asymptotics. Queueing Syst 101, 353–391 (2022). https://doi.org/10.1007/s11134-022-09847-7

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  • DOI: https://doi.org/10.1007/s11134-022-09847-7

Keywords

  • Many-server heavy-traffic
  • Load balancing system
  • Stein’s method
  • Transform method
  • State space collapse
  • Join the shortest queue
  • Power-of-d choices
  • Drift method
  • Lyapunov drift

Mathematics Subject Classification

  • 60K25
  • 68M20
  • 90B22
  • 60H99