scholarly journals The asymptotic variance of departures in critically loaded queues

2011 ◽  
Vol 43 (1) ◽  
pp. 243-263 ◽  
Author(s):  
A. Al Hanbali ◽  
M. Mandjes ◽  
Y. Nazarathy ◽  
W. Whitt
Keyword(s):  
Explicit Expression ◽  
Counting Process ◽  
Asymptotic Variance ◽  
Arrival Rate ◽  
Finite Capacity ◽  
System Load ◽  
Multiserver Queues ◽  
Service Patterns ◽  
Coefficients Of Variation ◽  
Service Times

We consider the asymptotic variance of the departure counting processD(t) of the GI/G/1 queue;D(t) denotes the number of departures up to timet. We focus on the case where the system load ϱ equals 1, and prove that the asymptotic variance rate satisfies limt→∞varD(t) /t= λ(1–2/π)(ca2+cs2), where λ is the arrival rate, andca2andcs2are squared coefficients of variation of the interarrival and service times, respectively. As a consequence, the departures variability has a remarkable singularity in the case in which ϱ equals 1, in line with the BRAVO (balancing reduces asymptotic variance of outputs) effect which was previously encountered in finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multiserver queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue, we present an explicit expression of the variance ofD(t) for anyt.

scholarly journals The asymptotic variance of departures in critically loaded queues

2011 ◽  
Vol 43 (01) ◽  
pp. 243-263 ◽  
Author(s):  
A. Al Hanbali ◽  
M. Mandjes ◽  
Y. Nazarathy ◽  
W. Whitt
Keyword(s):  
Explicit Expression ◽  
Counting Process ◽  
Asymptotic Variance ◽  
Arrival Rate ◽  
Finite Capacity ◽  
System Load ◽  
Multiserver Queues ◽  
Service Patterns ◽  
Coefficients Of Variation ◽  
Service Times

We consider the asymptotic variance of the departure counting process D(t) of the GI/G/1 queue; D(t) denotes the number of departures up to time t. We focus on the case where the system load ϱ equals 1, and prove that the asymptotic variance rate satisfies lim t→∞varD(t) / t = λ(1–2/π)(c a 2 + c s 2), where λ is the arrival rate, and c a 2 and c s 2 are squared coefficients of variation of the interarrival and service times, respectively. As a consequence, the departures variability has a remarkable singularity in the case in which ϱ equals 1, in line with the BRAVO (balancing reduces asymptotic variance of outputs) effect which was previously encountered in finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multiserver queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue, we present an explicit expression of the variance of D(t) for any t.

scholarly journals Simple and Yet Efficient Estimators for Markovian Multiserver Queues

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Emilio Suyama ◽  
Roberto C. Quinino ◽  
Frederico R. B. Cruz
Keyword(s):  
Maximum Likelihood ◽  
Arrival Rate ◽  
Research Area ◽  
Service Rate ◽  
Traffic Intensity ◽  
Moment Estimator ◽  
Likelihood Estimator ◽  
Future Developments ◽  
Multiserver Queues

Estimators for the parameters of the Markovian multiserver queues are presented, from samples that are the number of clients in the system at arbitrary points and their sojourn times. As estimation in queues is a recognizably difficult inferential problem, this study focuses on the estimators for the arrival rate, the service rate, and the ratio of these two rates, which is known as the traffic intensity. Simulations are performed to verify the quality of the estimations for sample sizes up to 400. This research also relates notable new insights, for example, that the maximum likelihood estimator for the traffic intensity is equivalent to its moment estimator. Some limitations of the results are presented along with a detailed numerical example and topics for future developments in this research area.

The convexity of a general performance measure for multiserver queues

1987 ◽  
Vol 24 (03) ◽  
pp. 725-736 ◽  
Author(s):  
Arie Harel ◽  
Paul Zipkin
Keyword(s):  
Queueing Systems ◽  
Arrival Rate ◽  
Performance Measure ◽  
Service Rate ◽  
Service Time Distribution ◽  
Multiserver Queues ◽  
General Performance ◽  
Production Models ◽  
Convexity Properties ◽  
Special Case

This paper examines a general performance measure for queueing systems. This criterion reflects both the mean and the variance of sojourn times; the standard deviation is a special case. The measure plays a key role in certain production models, and it should be useful in a variety of other applications. We focus here on convexity properties of an approximation of the measure for the M/G/c queue. For c ≧ 2 we show that this quantity is convex in the arrival rate. Assuming the service rate acts as a scale factor in the service-time distribution, the measure is convex in the service rate also.

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 Asymptotic Analysis of a Storage Allocation Model with Finite Capacity: Joint Distribution

2016 ◽  
Vol 2016 ◽  
pp. 1-56 ◽  
Author(s):  
Eunju Sohn ◽  
Charles Knessl
Keyword(s):  
Steady State ◽  
Joint Distribution ◽  
Arrival Rate ◽  
Kolmogorov Equation ◽  
Finite Capacity ◽  
Storage Allocation ◽  
Allocation Model ◽  
Singular Perturbation Analysis ◽  
Poisson Arrival ◽  
Time Period

We consider a storage allocation model with a finite number of storage spaces. There aremprimary spaces andRsecondary spaces. All of them are numbered and ranked. Customers arrive according to a Poisson process and occupy a space for an exponentially distributed time period, and a new arrival takes the lowest ranked available space. We letN1andN2denote the numbers of occupied primary and secondary spaces and study the joint distributionProb[N1=k,N2=r]in the steady state. The joint process(N1,N2)behaves as a random walk in a lattice rectangle. We study the problem asymptotically as the Poisson arrival rateλbecomes large, and the storage capacitiesmandRare scaled to be commensurably large. We use a singular perturbation analysis to approximate the forward Kolmogorov equation(s) satisfied by the joint distribution.

Overflow processes for multiserver queues with finite waiting room

1984 ◽  
Vol 16 (1) ◽  
pp. 8-8
Author(s):  
Jos H. A. De Smit
Keyword(s):  
Renewal Process ◽  
Numerical Results ◽  
Markov Renewal Process ◽  
Waiting Room ◽  
Multiserver Queues ◽  
Phase Type ◽  
Service Times ◽  
Overflow Process ◽  
Explicit Expressions

The overflow process of the multiserver queue with phase-type service times and finite waiting room is a Markov renewal process. The solution for this process is obtained. If the service times are exponential the overflow process reduces to a renewal process. For the latter case explicit expressions and numerical results are given.

scholarly journals Discrete-time multiserver queues with geometric service times

2004 ◽  
Vol 31 (1) ◽  
pp. 81-99 ◽  
Author(s):  
Peixia Gao ◽  
Sabine Wittevrongel ◽  
Herwig Bruneel
Keyword(s):  
Discrete Time ◽  
Multiserver Queues ◽  
Service Times
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