The Brownian approximation for rate-control throttles and the G/G/1/C queue

Berger, Arno; Whitt, Ward

doi:10.1007/bf01797279

Cited by 38 publications

(22 citation statements)

References 55 publications

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“…Let N(p, cz, ca, m, K) denote the number in system in the GI/G/m/K model with K extra waiting spaces at an arbitrary time in steady state. The proposed approximation is then P (N(p, cz, Yao andBuzacott ( 1985a, 1985b) and Berger and Whitt ( 1992) for related work.…”

Section: Finite Bujandersmentioning

confidence: 99%

APPROXIMATIONS FOR THE GI/G/m QUEUE

Whitt

1993

Production and Operations Management

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Queueing models can usefully represent production systems experiencing congestion due to irregular flows, but exact analyses of these queueing models can be difficult. Thus it is natural to seek relatively simple approximations that are suitably accurate for engineering purposes. Here approximations for a basic queueing model are developed and evaluated. The model is the GI/G/m queue, which has m identical servers in parallel, unlimited waiting room, and the first‐come first‐served queue discipline, with service and interarrival times coming from independent sequences of independent and identically distributed random variables with general distributions. The approximations depend on the general interarrival‐time and service‐time distributions only through their first two moments. The main focus is on the expected waiting time and the probability of having to wait before beginning service, but approximations are also developed for other congestion measures, including the entire distributions of waiting time, queue‐length and number in system. These relatively simple approximations are useful supplements to algorithms for computing the exact values that have been developed in recent years. The simple approximations can serve as starting points for developing approximations for more complicated systems for which exact solutions are not yet available. These approximations are especially useful for incorporating GI/G/m models in larger models, such as queueing networks, wherein the approximations can be components of rapid modeling tools.

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Section: Finite Bujandersmentioning

confidence: 99%

APPROXIMATIONS FOR THE GI/G/m QUEUE

Whitt

1993

Production and Operations Management

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297

148

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“…Hence we now briefly discuss the standard M/M/1 queue. Sensitivity of the general single-server queue, G/G/1/C, to the key parameters was studied previously via the Brownian heavy-traffic approximation in Section 9 of Berger and Whitt (1992).…”

Section: Proposition 3 (Elasticities In the Ed Limiting Regime)mentioning

confidence: 99%

“…Earlier papers that focus on many-server heavy-traffic scaling are Erlang (1924), Jagerman (1974) and Halfin and Whitt (1981). Sensitivity of performance in the G/G/1/C model was studied via derivatives of the Brownian heavy-traffic approximation in Section 9 of Berger and Whitt (1992).…”

Section: Introductionmentioning

confidence: 99%

Sensitivity of Performance in the Erlang-A Queueing Model to Changes in the Model Parameters

Whitt

2006

Operations Research

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This paper studies the M/M/s + M queue, i.e., the M/M/s queue with customer abandonment, also called the Erlang-A model, having independent and identically distributed customer abandon times with an exponential distribution (the +M ), focusing on the case in which the arrival rate and the number of servers are large. The goal is to better understand the sensitivity of performance to changes in the model parameters: the arrival rate, the service rate, the number of servers, and the abandonment rate. Elasticities are used to show the percentage change of a performance measure caused by a small percentage change in a parameter.Elasticities are calculated using an exact numerical algorithm and simple finite-difference approximations. Insight is gained by applying fluid and diffusion approximations. The analysis shows that performance is quite sensitive to small percentage changes in the arrival rate or the service rate, but relatively insensitive to small percentage changes in the abandonment rate.

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“…Williams [28], Equation (6); Berger and Whitt [6], Equation (29)). Figure 2 shows a contour plot of c 2 U (λ, µ).…”

Section: Then P = E(d)/(e(u ) + E(d))mentioning

confidence: 99%

Two perishable inventory systems with one-way substitution

2016

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Motivated by the ABO issue of the blood banks system, in which the portions stored have constant shelf life, we consider two subsystems of perishable inventory. The two Perishable Inventory Subsystems -PIS A and PIS B, are correlated to each other through a so-called one-way substitution of demands. Specifically, the input streams and the demand streams applied to each subsystem are four Poisson processes which are independent of one another. However, if the shelf of PIS A (blood of type O) is empty of items an arriving demand of type A is unsatisfied, since demand of type A cannot be satisfied by an item of type B (blood portions of type AB), but if the shelf of PIS B is empty of items an arriving demand of type B is applied to PIS A, since demands of type B can be satisfied by both types. Such a one-way substitution of the issuing policy generates for PIS A a modulated Poisson demand process operating in a two-state non-Markovian environment. The performance analysis of PIS B is known from previous work. Hence, in this study we focus on the marginal performance analysis of PIS A. Based on a fluid formulation and a Markovian approximation for the one-way substitution demands process, we develop a unified approach to efficiently and accurately approximate the performance of PIS A. The effectiveness of the approach is investigated by extensive numerical experiments.

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The Brownian approximation for rate-control throttles and the G/G/1/C queue

Cited by 38 publications

References 55 publications

APPROXIMATIONS FOR THE GI/G/m QUEUE

APPROXIMATIONS FOR THE GI/G/m QUEUE

Sensitivity of Performance in the Erlang-A Queueing Model to Changes in the Model Parameters

Two perishable inventory systems with one-way substitution

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