Exponential spectra as a tool for the study of server-systems with several classes of customers

For a single-server system having several Poisson streams of customers with exponentially distributed service times, busy period densities, waiting time densities, and idle state probabilities are completely monotone. The exponential spectra for such densities are of importance for understanding the transient behavior of such systems. Algorithms are given for the computation of such spectra. Applications to heavy traffic situations and priority systems are also discussed.

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On a property of a refusals stream

Journal of Applied Probability ◽

10.1017/s0021900200101469 ◽

1997 ◽

Vol 34 (03) ◽

pp. 800-805 ◽

Cited By ~ 5

Author(s):

Vyacheslav M. Abramov

Keyword(s):

Waiting Time ◽

Service Time ◽

Busy Period ◽

Elementary Proof ◽

Queueing System ◽

Queueing Systems ◽

Single Server ◽

Wide Family

This paper consists of two parts. The first part provides a more elementary proof of the asymptotic theorem of the refusals stream for an M/GI/1/n queueing system discussed in Abramov (1991a). The central property of the refusals stream discussed in the second part of this paper is that, if the expectations of interarrival and service time of an M/GI/1/n queueing system are equal to each other, then the expectation of the number of refusals during a busy period is equal to 1. This property is extended for a wide family of single-server queueing systems with refusals including, for example, queueing systems with bounded waiting time.

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A single server tandem queue

Journal of Applied Probability ◽

10.2307/3211840 ◽

1971 ◽

Vol 8 (1) ◽

pp. 95-109 ◽

Cited By ~ 16

Author(s):

Sreekantan S. Nair

Keyword(s):

Waiting Time ◽

Queue Length ◽

Busy Period ◽

Single Server ◽

Queueing Model ◽

Tandem Queue ◽

Limiting Behavior ◽

Virtual Waiting Time ◽

Priority Model

Avi-Itzhak, Maxwell and Miller (1965) studied a queueing model with a single server serving two service units with alternating priority. Their model explored the possibility of having the alternating priority model treated in this paper with a single server serving alternately between two service units in tandem.Here we study the distribution of busy period, virtual waiting time and queue length and their limiting behavior.

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The stable M/G/1 queue in heavy traffic and its covariance function

Advances in Applied Probability ◽

10.1017/s0001867800043184 ◽

1977 ◽

Vol 9 (01) ◽

pp. 169-186 ◽

Cited By ~ 6

Author(s):

Teunis J. Ott

Keyword(s):

Brownian Motion ◽

Waiting Time ◽

Covariance Function ◽

Busy Period ◽

Stationary Process ◽

Heavy Traffic ◽

Time Process ◽

Period Distribution ◽

Virtual Waiting Time ◽

Waiting Time Process

Let X(t) be the virtual waiting-time process of a stable M/G/1 queue. Let R(t) be the covariance function of the stationary process X(t), B(t) the busy-period distribution of X(t); and let E(t) = P{X(t) = 0|X(0) = 0}. For X(t) some heavy-traffic results are given, among which are limiting expressions for R(t) and its derivatives and for B(t) and E(t). These results are used to find the covariance function of stationary Brownian motion on [0, ∞).

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Queues with semi-Markovian arrivals

Journal of Applied Probability ◽

10.1017/s0021900200032113 ◽

1967 ◽

Vol 4 (02) ◽

pp. 365-379 ◽

Cited By ~ 17

Author(s):

Erhan Çinlar

Keyword(s):

Distribution Function ◽

Markov Chain ◽

Finite Number ◽

Waiting Time ◽

Busy Period ◽

Queueing System ◽

Interarrival Time ◽

Single Server ◽

Queue Size

A queueing system with a single server is considered. There are a finite number of types of customers, and the types of successive arrivals form a Markov chain. Further, the nth interarrival time has a distribution function which may depend on the types of the nth and the n–1th arrivals. The queue size, waiting time, and busy period processes are investigated. Both transient and limiting results are given.

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On two stationary distributions for the stable GI/G/1 queue

Journal of Applied Probability ◽

10.1017/s0021900200118303 ◽

1974 ◽

Vol 11 (04) ◽

pp. 849-852 ◽

Cited By ~ 4

Author(s):

Austin J. Lemoine

Keyword(s):

Waiting Time ◽

Queue Length ◽

Distribution Functions ◽

Single Server ◽

Stationary Distributions ◽

Virtual Waiting Time ◽

Server Systems ◽

Stationary Queue Length

This paper provides simple proofs of two standard results for the stable GI/G/1 queue on the structure of the distribution functions of the stationary virtual waiting time and the stationary queue-length Our argument is applicable to more general single server systems than the queue GI/G/1.

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The total waiting time in a busy period of a stable single-server queue, II

Journal of Applied Probability ◽

10.2307/3212102 ◽

1969 ◽

Vol 6 (3) ◽

pp. 565-572 ◽

Cited By ~ 15

Author(s):

D. J. Daley ◽

D. R. Jacobs

Keyword(s):

Waiting Time ◽

Busy Period ◽

Bessel Functions ◽

Arrival Process ◽

Single Server ◽

Single Server Queue ◽

Common Distribution ◽

Poisson Arrival ◽

Server Queue ◽

Common Distribution Function

This paper is a continuation of Daley (1969), referred to as (I), whose notation and numbering is continued here. We shall indicate various approaches to the study of the total waiting time in a busy period2 of a stable single-server queue with a Poisson arrival process at rate λ, and service times independently distributed with common distribution function (d.f.) B(·). Let X'i denote3 the total waiting time in a busy period which starts at an epoch when there are i (≧ 1) customers in the system (to be precise, the service of one customer is just starting and the remaining i − 1 customers are waiting for service). We shall find the first two moments of X'i, prove its asymptotic normality for i → ∞ when B(·) has finite second moment, and exhibit the Laplace-Stieltjes transform of X'i in M/M/1 as the ratio of two Bessel functions.

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On Mx/(G1G2)/1/G(BS)/Vs vacation queue with two types of general heterogeneous service

Journal of Applied Mathematics and Decision Sciences ◽

10.1155/jamds.2005.123 ◽

2005 ◽

Vol 2005 (3) ◽

pp. 123-135 ◽

Cited By ~ 11

Author(s):

Kailash C. Madan ◽

Z. R. Al-Rawi ◽

Amjad D. Al-Nasser

Keyword(s):

Steady State ◽

Performance Measures ◽

Waiting Time ◽

Busy Period ◽

Single Server ◽

Queue Size ◽

Single Vacation ◽

The Mean ◽

Expected Waiting Time ◽

Heterogeneous Service

We analyze a batch arrival queue with a single server providing two kinds of general heterogeneous service. Just before his service starts, a customer may choose one of the services and as soon as a service (of any kind) gets completed, the server may take a vacation or may continue staying in the system. The vacation times are assumed to be general and the server vacations are based on Bernoulli schedules under a single vacation policy. We obtain explicit queue size distribution at a random epoch as well as at a departure epoch and also the mean busy period of the server under the steady state. In addition, some important performance measures such as the expected queue size and the expected waiting time of a customer are obtained. Further, some interesting particular cases are also discussed.

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On queues in which customers are served in random order

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100036239 ◽

1962 ◽

Vol 58 (1) ◽

pp. 79-91 ◽

Cited By ~ 40

Author(s):

J. F. C. Kingman

Keyword(s):

Steady State ◽

Exponential Decay ◽

Waiting Time ◽

Time Distribution ◽

Busy Period ◽

Asymptotic Form ◽

Heavy Traffic ◽

Formal Solution ◽

Random Order ◽

Waiting Time Distribution

ABSTRACTThe queue M |G| l is considered in the case in which customers are served in random order. A formal solution is obtained for the waiting time distribution in the steady state, and is used to consider the exponential decay of the distribution. The moments of the waiting time are examined, and the asymptotic form of the distribution in heavy traffic is found. Finally, the problem is related to those of the busy period and the approach to the steady state.

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Exponential approximation of waiting time and queue size for queues in heavy traffic

Advances in Applied Probability ◽

10.2307/1427606 ◽

1990 ◽

Vol 22 (1) ◽

pp. 230-240 ◽

Cited By ~ 10

Author(s):

Władysław Szczotka

Keyword(s):

Waiting Time ◽

Wide Class ◽

Time Distribution ◽

Heavy Traffic ◽

Single Server ◽

Queue Size ◽

Exponential Approximation ◽

Waiting Time Distribution ◽

Stationary Waiting Time ◽

Stationary Representation

An exponential approximation for the stationary waiting time distribution and the stationary queue size distribution for single-server queues in heavy traffic is given for a wide class of queues. This class contains for example not only queues for which the generic sequence, i.e. the sequence of service times and interarrival times, is stationary but also such queues for which the generic sequence is asymptotically stationary in some sense. The conditions ensuring the exponential approximation of the characteristics considered in heavy traffic are expressed in terms of the invariance principle for the stationary representation of the generic sequence and its first two moments.

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