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.

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.

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.

Some explicit formulas and computational methods for infinite-server queues with phase-type arrivals

1980 ◽  
Vol 17 (2) ◽  
pp. 498-514 ◽  
Author(s):  
V. Ramaswami ◽  
Marcel F. Neuts
Keyword(s):  
Queue Length ◽  
Arrival Process ◽  
Infinite Server ◽  
Highly Sensitive ◽  
Phase Type ◽  
Random Variability ◽  
Infinite Server Queues ◽  
Linear Differential ◽  
Mean Queue Length ◽  
Transient And Steady State

This paper discusses infinite-server queues with phase-type input. The problems of obtaining the transient and steady-state distributions and moments of the queue length are reduced to the solution of certain well-behaved systems of linear differential equations. Sample computations, provided with as many as ten phases, show that although (even the time-dependent) mean queue length is very insensitive to substantial random variability in the arrival process, the higher moments of the queue length are highly sensitive. These examples indicate that considerable caution should be exercised in using robustness results for such stochastic models.

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

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.

scholarly journals On a multilevel controlled bulk queueing system MX/Gr,R/1

1992 ◽  
Vol 5 (3) ◽  
pp. 237-260 ◽  
Author(s):  
Lev Abolnikov ◽  
Jewgeni H. Dshalalow
Keyword(s):  
Stationary Distribution ◽  
Queue Length ◽  
Transition Probability ◽  
Queueing Systems ◽  
Transition Probability Matrix ◽  
Queueing Process ◽  
Poisson Input ◽  
Service Delay ◽  
Associated Functions ◽  
Semi Markov Process

The authors introduce and study a class of bulk queueing systems with a compound Poisson input modulated by a semi-Markov process, multilevel control service time and a queue length dependent service delay discipline. According to this discipline, the server immediately starts the next service act if the queue length is not less than r; in this case all available units, or R (capacity of the server) of them, whichever is less, are taken for service. Otherwise, the server delays the service act until the number of units in the queue reaches or exceeds level r.The authors establish a necessary and sufficient criterion for the ergodicity of the embedded queueing process in terms of generating functions of the entries of the corresponding transition probability matrix and of the roots of a certain associated functions in the unit disc of the complex plane. The stationary distribution of this process is found by means of the results of a preliminary analysis of some auxiliary random processes which arise in the “first passage problem” of the queueing process over level r. The stationary distribution of the queueing process with continuous time parameter is obtained by using semi-regenerative techniques. The results enable the authors to introduce and analyze some functionals of the input and output processes via ergodic theorems. A number of different examples (including an optimization problem) illustrate the general methods developed in the article.

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