Heavy-traffic fluid limits for periodic infinite-server queues

2016 ◽  
Vol 84 (1-2) ◽  
pp. 111-143 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Heavy Traffic ◽  
Fluid Limits ◽  
Infinite Server ◽  
Infinite Server Queues

A new view of the heavy-traffic limit theorem for infinite-server queues

1991 ◽  
Vol 23 (01) ◽  
pp. 188-209 ◽  
Author(s):  
Peter W. Glynn ◽  
Ward Whitt
Keyword(s):  
Heavy Traffic ◽  
Rates Of Convergence ◽  
Infinite Server ◽  
Open Networks ◽  
Heavy Traffic Limit ◽  
Many Server Queues ◽  
Finite Set ◽  
Deterministic Service ◽  
Infinite Server Queues ◽  
Service Times

This paper presents a new approach for obtaining heavy-traffic limits for infinite-server queues and open networks of infinite-server queues. The key observation is that infinite-server queues having deterministic service times can easily be analyzed in terms of the arrival counting process. A variant of the same idea applies when the service times take values in a finite set, so this is the key assumption. In addition to new proofs of established results, the paper contains several new results, including limits for the work-in-system process, limits for steady-state distributions, limits for open networks with general customer routes, and rates of convergence. The relatively tractable Gaussian limits are promising approximations for many-server queues and open networks of such queues, possibly with finite waiting rooms.

A new view of the heavy-traffic limit theorem for infinite-server queues

1991 ◽  
Vol 23 (1) ◽  
pp. 188-209 ◽  
Author(s):  
Peter W. Glynn ◽  
Ward Whitt
Keyword(s):  
Heavy Traffic ◽  
Rates Of Convergence ◽  
Infinite Server ◽  
Open Networks ◽  
Heavy Traffic Limit ◽  
Many Server Queues ◽  
Finite Set ◽  
Deterministic Service ◽  
Infinite Server Queues ◽  
Service Times

This paper presents a new approach for obtaining heavy-traffic limits for infinite-server queues and open networks of infinite-server queues. The key observation is that infinite-server queues having deterministic service times can easily be analyzed in terms of the arrival counting process. A variant of the same idea applies when the service times take values in a finite set, so this is the key assumption. In addition to new proofs of established results, the paper contains several new results, including limits for the work-in-system process, limits for steady-state distributions, limits for open networks with general customer routes, and rates of convergence. The relatively tractable Gaussian limits are promising approximations for many-server queues and open networks of such queues, possibly with finite waiting rooms.

scholarly journals Two-parameter heavy-traffic limits for infinite-server queues

2010 ◽  
Vol 65 (4) ◽  
pp. 325-364 ◽  
Author(s):  
Guodong Pang ◽  
Ward Whitt
Keyword(s):  
Heavy Traffic ◽  
Infinite Server ◽  
Infinite Server Queues ◽  
Two Parameter

On joint queue-length characteristics in infinite-server tandem queues with heavy traffic

1987 ◽  
Vol 19 (02) ◽  
pp. 474-486 ◽  
Author(s):  
Volker Schmidt
Keyword(s):  
Stationary Distribution ◽  
Simple Proof ◽  
Shot Noise ◽  
Queue Length ◽  
Heavy Traffic ◽  
Asymptotic Formulas ◽  
Tandem Queues ◽  
Infinite Server ◽  
Poisson Input ◽  
Infinite Server Queues

For m infinite-server queues with Poisson input which are connected in a series, a simple proof is given of a formula derived in [3] for the generating function of the joint customer-stationary distribution of the successive numbers of customers a randomly chosen customer finds at his arrival epochs at two queues of the system. In this connection, a shot-noise representation of the queue-length characteristics under consideration is used. Moreover, using this representation, corresonding asymptotic formulas are derived for infinite-server tandem queues with general high-density renewal input.

scholarly journals Infinite-server queues with Hawkes input

2018 ◽  
Vol 55 (3) ◽  
pp. 920-943 ◽  
Author(s):  
D. T. Koops ◽  
M. Saxena ◽  
O. J. Boxma ◽  
M. Mandjes
Keyword(s):  
Heavy Traffic ◽  
Probability Generating Function ◽  
Arrival Process ◽  
Hawkes Process ◽  
Asymptotic Results ◽  
Recursive Procedure ◽  
Infinite Server ◽  
Infinite Server Queues ◽  
Heavy Tailed ◽  
Number Of Customers

Abstract In this paper we study the number of customers in infinite-server queues with a self-exciting (Hawkes) arrival process. Initially we assume that service requirements are exponentially distributed and that the Hawkes arrival process is of a Markovian nature. We obtain a system of differential equations that characterizes the joint distribution of the arrival intensity and the number of customers. Moreover, we provide a recursive procedure that explicitly identifies (transient and stationary) moments. Subsequently, we allow for non-Markovian Hawkes arrival processes and nonexponential service times. By viewing the Hawkes process as a branching process, we find that the probability generating function of the number of customers in the system can be expressed in terms of the solution of a fixed-point equation. We also include various asymptotic results: we derive the tail of the distribution of the number of customers for the case that the intensity jumps of the Hawkes process are heavy tailed, and we consider a heavy-traffic regime. We conclude by discussing how our results can be used computationally and by verifying the numerical results via simulations.

On joint queue-length characteristics in infinite-server tandem queues with heavy traffic

1987 ◽  
Vol 19 (2) ◽  
pp. 474-486 ◽  
Author(s):  
Volker Schmidt
Keyword(s):  
Stationary Distribution ◽  
Simple Proof ◽  
Shot Noise ◽  
Queue Length ◽  
Heavy Traffic ◽  
Asymptotic Formulas ◽  
Tandem Queues ◽  
Infinite Server ◽  
Poisson Input ◽  
Infinite Server Queues

For m infinite-server queues with Poisson input which are connected in a series, a simple proof is given of a formula derived in [3] for the generating function of the joint customer-stationary distribution of the successive numbers of customers a randomly chosen customer finds at his arrival epochs at two queues of the system. In this connection, a shot-noise representation of the queue-length characteristics under consideration is used. Moreover, using this representation, corresonding asymptotic formulas are derived for infinite-server tandem queues with general high-density renewal input.

scholarly journals Perfect sampling for infinite server and loss systems

2015 ◽  
Vol 47 (03) ◽  
pp. 761-786 ◽  
Author(s):  
Jose Blanchet ◽  
Jing Dong
Keyword(s):  
Point Processes ◽  
Marked Point ◽  
Heavy Traffic ◽  
Arrival Rate ◽  
Marked Point Processes ◽  
Perfect Sampling ◽  
Running Time ◽  
Infinite Server ◽  
Loss Systems ◽  
Server System

We present the first class of perfect sampling (also known as exact simulation) algorithms for the steady-state distribution of non-Markovian loss systems. We use a variation of dominated coupling from the past. We first simulate a stationary infinite server system backwards in time and analyze the running time in heavy traffic. In particular, we are able to simulate stationary renewal marked point processes in unbounded regions. We then use the infinite server system as an upper bound process to simulate the loss system. The running time analysis of our perfect sampling algorithm for loss systems is performed in the quality-driven (QD) and the quality-and-efficiency-driven regimes. In both cases, we show that our algorithm achieves subexponential complexity as both the number of servers and the arrival rate increase. Moreover, in the QD regime, our algorithm achieves a nearly optimal rate of complexity.

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