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.

INFINITE-SERVER QUEUES WITH BATCH ARRIVALS AND DEPENDENT SERVICE TIMES

2012 ◽  
Vol 26 (2) ◽  
pp. 197-220 ◽  
Author(s):  
Guodong Pang ◽  
Ward Whitt
Keyword(s):  
Large Scale ◽  
Heavy Traffic ◽  
Arrival Rate ◽  
Arrival Process ◽  
Regularity Conditions ◽  
Time Varying ◽  
Performance Impact ◽  
Batch Arrivals ◽  
Infinite Server ◽  
Service Times

Motivated by large-scale service systems, we consider an infinite-server queue with batch arrivals, where the service times are dependent within each batch. We allow the arrival rate of batches to be time varying as well as constant. As regularity conditions, we require that the batch sizes be i.i.d. and independent of the arrival process of batches, and we require that the service times within different batches be independent. We exploit a recently established heavy-traffic limit for the number of busy servers to determine the performance impact of the dependence among the service times. The number of busy servers is approximately a Gaussian process. The dependence among the service times does not affect the mean number of busy servers, but it does affect the variance of the number of busy servers. Our approximations quantify the performance impact upon the variance. We conduct simulations to evaluate the heavy-traffic approximations for the stationary model and the model with a time-varying arrival rate. In the simulation experiments, we use the Marshall–Olkin multivariate exponential distribution to model dependent exponential service times within a batch. We also introduce a class of Marshall–Olkin multivariate hyperexponential distributions to model dependent hyper-exponential service times within a batch.

scholarly journals Linear stochastic fluid networks

1999 ◽  
Vol 36 (01) ◽  
pp. 244-260 ◽  
Author(s):  
Offer Kella ◽  
Ward Whitt
Keyword(s):  
Infinite Server ◽  
Open Networks ◽  
Fluid Network ◽  
Stochastic Transition ◽  
Fluid Networks ◽  
Batch Sizes ◽  
Arrival Processes ◽  
Flow Through ◽  
Infinite Server Queues ◽  
Stochastic Fluid

We introduce open stochastic fluid networks that can be regarded as continuous analogues or fluid limits of open networks of infinite-server queues. Random exogenous input may come to any of the queues. At each queue, a c.d.f.-valued stochastic process governs the proportion of the input processed by a given time after arrival. The routeing may be deterministic (a specified sequence of successive queue visits) or proportional, i.e. a stochastic transition matrix may govern the proportion of the output routed from one queue to another. This stochastic fluid network with deterministic c.d.f.s governing processing at the queues arises as the limit of normalized networks of infinite-server queues with batch arrival processes where the batch sizes grow. In this limit, one can think of each particle having an evolution through the network, depending on its time and place of arrival, but otherwise independent of all other particles. A key property associated with this independence is the linearity: the workload associated with a superposition of inputs, each possibly having its own pattern of flow through the network, is simply the sum of the component workloads. As with infinite-server queueing models, the tractability makes the linear stochastic fluid network a natural candidate for approximations.

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.

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