scholarly journals Sensitivity of mean-field fluctuations in Erlang loss models with randomized routing

2021 ◽  
Vol 58 (2) ◽  
pp. 428-448
Author(s):  
Thirupathaiah Vasantam ◽  
Ravi R. Mazumdar
Keyword(s):  
Mean Field ◽  
Blocking Probabilities ◽  
Field Fluctuations ◽  
Mean And Variance ◽  
Join The Shortest Queue ◽  
Shortest Queue ◽  
Erlang Loss ◽  
Randomized Policies ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

AbstractIn this paper we study a large system of N servers, each with capacity to process at most C simultaneous jobs; an incoming job is routed to a server if it has the lowest occupancy amongst d (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of d) policies. Under the assumption that jobs arrive according to a Poisson process with rate $N\lambda^{(N)}$ where $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}\,}$ , $\sigma\in\mathbb{R}_+$ and $\beta\in\mathbb{R}$ , we establish functional central limit theorems for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein–Uhlenbeck process whose mean and variance depend on the mean field of the considered model. Using this, we obtain approximations to the blocking probabilities for large N, where we can precisely estimate the accuracy of first-order approximations.

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

2020 ◽  
Vol 45 (3) ◽  
pp. 1069-1103
Author(s):  
Anton Braverman
Keyword(s):  
Steady State ◽  
Diffusion Limit ◽  
Fluid Limit ◽  
Time Intervals ◽  
Dimensional Diffusion ◽  
Queue Model ◽  
Join The Shortest Queue ◽  
Shortest Queue ◽  
Process Level ◽  
General Tool

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 nonempty buffers. Recently, Eschenfeldt and Gamarnik 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 nonempty buffers is tight. Combined with the process-level convergence proved by Eschenfeldt and Gamarnik, 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 Stolyar. One technical contribution to the framework is to show how it can be used as a general tool to establish exponential ergodicity.

Queues in transportation systems, II: An independently-dispatched system

1974 ◽  
Vol 11 (01) ◽  
pp. 145-158 ◽  
Author(s):  
Michael A. Crane
Keyword(s):  
Limit Theorems ◽  
Transportation Systems ◽  
Linear Network ◽  
Random Functions ◽  
Arrival Processes ◽  
Vehicle Fleet ◽  
Vehicle Capacity ◽  
Service Groups ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

We consider a transportation system consisting of a linear network of N + 1 terminals served by S vehicles of fixed capacity. Customers arrive stochastically at terminal i, 1 ≦ i ≦ N, seeking transportation to some terminal j, 0 ≦ j ≦ i − 1, and are served as empty units of vehicle capacity become available at i. The vehicle fleet is partitioned into N service groups, with vehicles in the ith group stopping at terminals i, i − 1,···,0. Travel times between terminals and idle times at terminals are stochastic and are independent of the customer arrival processes. Functional central limit theorems are proved for random functions induced by processes of interest, including customer queue size processes. The results are of most interest in cases where the system is unstable. This occurs whenever, at some terminal, the rate of customer arrivals is at least as great as the rate at which vehicle capacity is made available.

Weak convergence and first passage times

1975 ◽  
Vol 12 (02) ◽  
pp. 324-332
Author(s):  
Allan Gut
Keyword(s):  
First Passage Time ◽  
Domain Of Attraction ◽  
Passage Time ◽  
Partial Sums ◽  
First Passage ◽  
Stable Law ◽  
Common Distribution Function ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit ◽  
Passage Times

Let Sn, n = 1, 2, ‥, denote the partial sums of i.i.d. random variables with the common distribution function F and positive, finite mean. Let N(c) = min [k; Sk > c‥kp ], c ≥ 0, 0 ≤ p < 1. Under the assumption that F belongs to the domain of attraction of a stable law with index α, 1 < α ≤ 2, functional central limit theorems for the first passage time process N(nt), 0 ≤ t ≤ 1, when n → ∞, are derived in the function space D[0,1].

Heavy traffic limit theorems for a queueing system in which customers join the shortest line

1989 ◽  
Vol 21 (02) ◽  
pp. 451-469 ◽  
Author(s):  
Zhang Hanqin ◽  
Wang Rongxin
Keyword(s):  
Stochastic Processes ◽  
Central Limit ◽  
Limit Theorems ◽  
Queueing System ◽  
Heavy Traffic ◽  
Traffic Intensity ◽  
Heavy Traffic Limit ◽  
Functional Central Limit Theorems ◽  
Service Channels ◽  
Functional Central Limit

The queueing system considered in this paper consists of r independent arrival channels and s independent service channels, where, as usual, the arrival and service channels are independent. In the queueing system, each server of the system has his own queue and arriving customers join the shortest line in the system. We give functional central limit theorems for the stochastic processes characterizing this system after appropriately scaling and translating the processes in traffic intensity ρ > 1.

Blocking probabilities for large multirate erlang loss systems

1993 ◽  
Vol 25 (04) ◽  
pp. 997-1009 ◽  
Author(s):  
Pawel Gazdzicki ◽  
Ioannis Lambadaris ◽  
Ravi R. Mazumdar
Keyword(s):  
Large Deviations ◽  
Unified Framework ◽  
Blocking Probabilities ◽  
Asymptotic Expressions ◽  
Loss Systems ◽  
Heavy Loads ◽  
Erlang Loss

This paper is concerned with the computation of asymptotic blocking probabilities for a generalized Erlangian system which results when M independent Poisson streams of traffic with rates access a trunk group of C circuits with traffic from stream k requiring Ak circuits which are simultaneously held and released after a time which is randomly distributed with unit mean and independent of earlier arrivals and holding times. A call from stream k is lost if on arrival less than Ak circuits are available. Although exact expressions for the blocking probabilities are known, their computation is unwieldy for even moderate-sized switches. It is shown that as the size of the switch increases in that both the traffic rates and trunk capacity are scaled together, simple asymptotic expressions for the blocking probabilities are obtained. In particular the expression is different for light, moderate and heavy loads. The approach is via exponential centering and large deviations and provides a unified framework for the analysis.

Sign in / Sign up

Export Citation Format

Share Document