The shortest queue problem

Halfin, Shlomo

doi:10.1017/s0021900200108101

Cited by 19 publications

(20 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Early work on the JSQ model appeared in the late 50's and early 60's [22,30], followed by a number of papers in the 70's-90's [12,13,23,25,42]. This body of literature first studied the JSQ model with two servers, and later considered heavy-traffic asymptotics in the setting where the number of servers n is fixed, and λ → 1; see [10] for an itemized description of the aforementioned works.…”

Section: Literature Review and Contributionsmentioning

confidence: 99%

Steady-State Analysis of the Join-the-Shortest-Queue Model in the Halfin–Whitt Regime

Braverman

2020

Mathematics of OR

View full text Add to dashboard Cite

This paper studies the steady-state properties of the Join the Shortest Queue model in the Halfin-Whitt regime. We focus on the process tracking the number of idle servers, and the number of servers with non-empty buffers. Recently, [10] proved that a scaled version of this process converges, over finite time intervals, to a two-dimensional diffusion limit as the number of servers goes to infinity. In this paper we prove that the diffusion limit is exponentially ergodic, and that the diffusion scaled sequence of the steady-state number of idle servers and non-empty buffers is tight. Combined with the process-level convergence proved in [10], our results imply convergence of steady-state distributions. The methodology used is the generator expansion framework based on Stein's method, also referred to as the drift-based fluid limit Lyapunov function approach in [36]. One technical contribution to the framework is to show how it can be used as a general tool to establish exponential ergodicity. arXiv:1801.05121v2 [math.PR]

show abstract

Section: Literature Review and Contributionsmentioning

confidence: 99%

Steady-State Analysis of the Join-the-Shortest-Queue Model in the Halfin–Whitt Regime

Braverman

2020

Mathematics of OR

View full text Add to dashboard Cite

show abstract

“…This is, in fact, a tighter upper bound and has a simpler form than the one presented in [21], which is valid only for 1 2 ≤ ρ < 1 (see Figure 4). To prove this inequality we need an equivalent form of Lemma 1 for the infinite capacity case, which is…”

Section: 3mentioning

confidence: 77%

Stationary analysis of the shortest queue problem

2017

View full text Add to dashboard Cite

Abstract. A simple analytical solution is proposed for the stationary loss system of two parallel queues with finite capacity K, in which new customers join the shortest queue, or one of the two with equal probability if their lengths are equal. The arrival process is Poisson, service times at each queue have exponential distributions with the same parameter, and both queues have equal capacity. Using standard generating function arguments, a simple expression for the blocking probability is derived, which as far as we know is original. Using coupling arguments and explicit formulas, comparisons with related loss systems are then provided. Bounds are similarly obtained for the average total number of customers, with the stationary distribution explicitly determined on {K, . . . , 2K}, and elsewhere upper bounded. Furthermore, from the balance equations, all stationary probabilities are obtained as explicit combinations of their values at states (0, k) for 0 ≤ k ≤ K. These expressions extend to the infinite capacity and asymmetric cases, i.e., when the queues have different service rates. For the initial symmetric finite capacity model, the stationary probabilities of states (0, k) can be obtained recursively from the blocking probability. In the other cases, they are implicitly determined through a functional equation that characterizes their generating function. The whole approach shows that the stationary distribution of the infinite capacity symmetric process is the limit of the corresponding finite capacity distributions. For the infinite capacity symmetric model, we provide an elementary proof of a result by Cohen which gives the solution of the functional equation in terms of an infinite product with explicit zeroes and poles.

show abstract

“…They thus obtain the generating function explicitly and obtain from it various asymptotic results for the tails of the joint probability distribution. Previously some partial asymptotic results for this problem were obtained by Kingman [18], and elementary bounds are discussed by Halfin [16]. Boxma and Cohen [8] studied both the symmetric and asymmetric models and reduce the problem of determining the bivariate generating function to a standard boundary value problem.…”

Section: Introductionmentioning

confidence: 99%

On the Shortest Queue Version of the Erlang Loss Model

Yao

Knessl

2008

Stud Appl Math

View full text Add to dashboard Cite

We consider two parallel M/M/N/N queues. Thus there are N servers in each queue and no waiting line(s). The network is fed by a single Poisson arrival stream of rate λ, and the 2N servers are identical exponential servers working at rate µ. A new arrival is routed to the queue with the smaller number of occupied servers. If both have the same occupancy then the arrival is routed randomly, with the probability of joining either queue being 1/2. This model may be viewed as the shortest queue version of the classic Erlang loss model. If all 2N servers are occupied further arrivals are turned away and lost. We let ρ = λ/µ and a = N /ρ = N µ/λ. We study this model both numerically and asymptotically. For the latter we consider heavily loaded systems (ρ → ∞) with a comparably large number of servers (N → ∞ with a = O(1)). We obtain asymptotic approximations to the joint steady state distribution of finding m servers occupied in the first queue and n in the second. We also consider the marginal distribution of the number of occupied servers in the second queue, as well as some conditional distributions. We show that aspects of the solution are much different according as a > 1/2, a ≈ 1/2, 1/4 < a < 1/2, a ≈ 1/4 or 0 < a < 1/4. The asymptotic approximations are shown to be quite accurate numerically.

show abstract

The shortest queue problem

Cited by 19 publications

References 6 publications

Steady-State Analysis of the Join-the-Shortest-Queue Model in the Halfin–Whitt Regime

Steady-State Analysis of the Join-the-Shortest-Queue Model in the Halfin–Whitt Regime

Stationary analysis of the shortest queue problem

On the Shortest Queue Version of the Erlang Loss Model

Contact Info

Product

Resources

About