On a property of a refusals stream

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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Single-server queueing systems with uniformly limited queueing time

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700025325 ◽

1964 ◽

Vol 4 (4) ◽

pp. 489-505 ◽

Cited By ~ 41

Author(s):

D. J. Daley

Keyword(s):

Integral Equation ◽

Waiting Time ◽

Time Distribution ◽

Busy Period ◽

Queueing System ◽

Queueing Systems ◽

Single Server ◽

Waiting Time Distribution ◽

Loss Distribution ◽

Queueing Time

SummaryThe paper considers the queueing system GI/G/1 with a type of customer impatience, namely, that the total queueing-time is uniformly limited. Using Lindiley's approach [10], an integral equation for the limiting waiting- time distribution is derived, and this is solved explicitly for M/G/1 using an expansion of the Pollaczek-Khintchine formula. It is also solved, in principle for Ej/G/l, and explicitly for Ej/Ek/l. A duality noted between GIA(x)/GB(x)/l and GIB(x)/GA(x)/l relates solutions for GI/Ek/l to Ek/G/l. Finally the equation for the busy period in GI/G/l is derived and related to the no-customer-loss distribution and dual distributions.

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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 the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals

Journal of Applied Probability ◽

10.2307/3212173 ◽

1971 ◽

Vol 8 (3) ◽

pp. 494-507 ◽

Cited By ~ 34

Author(s):

E. K. Kyprianou

Keyword(s):

Waiting Time ◽

Busy Period ◽

Queueing System ◽

Random Variable ◽

Single Server ◽

Virtual Waiting Time ◽

Waiting Time Process ◽

Poisson Arrivals ◽

Image Position ◽

Quasi Stationary Distribution

We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt, and the service time has distribution dB(t), 0 < t < ∞. Let W(t) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable denote the first busy period initiated by a waiting time u at time t = 0.

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

Journal of Applied Probability ◽

10.1017/s0021900200026607 ◽

1969 ◽

Vol 6 (03) ◽

pp. 550-564 ◽

Cited By ~ 6

Author(s):

D. J. Daley

Keyword(s):

Waiting Time ◽

Queueing Theory ◽

Busy Period ◽

Queueing System ◽

Road Traffic ◽

Waiting Times ◽

Random Variable ◽

Single Server ◽

Single Server Queue ◽

Server Queue

A quantity of particular interest in the study of (road) traffic jams is the total waiting time X of all vehicles involved in a given hold-up (Gaver (1969): see note following (2.3) below and the first paragraph of Section 5). With certain assumptions on the process this random variable X is the same as the sum of waiting times of customers in a busy period of a GI/G/1 queueing system, and it is the object of this paper and its sequel to study the random variable in the queueing theory context.

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

Journal of Applied Probability ◽

10.2307/3212101 ◽

1969 ◽

Vol 6 (3) ◽

pp. 550-564 ◽

Cited By ~ 13

Author(s):

D. J. Daley

Keyword(s):

Waiting Time ◽

Queueing Theory ◽

Busy Period ◽

Queueing System ◽

Road Traffic ◽

Waiting Times ◽

Random Variable ◽

Single Server ◽

Single Server Queue ◽

Server Queue

A quantity of particular interest in the study of (road) traffic jams is the total waiting time X of all vehicles involved in a given hold-up (Gaver (1969): see note following (2.3) below and the first paragraph of Section 5). With certain assumptions on the process this random variable X is the same as the sum of waiting times of customers in a busy period of a GI/G/1 queueing system, and it is the object of this paper and its sequel to study the random variable in the queueing theory context.

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The Busy Period in Relation to the Single-Server Queueing System with General Independent Arrivals and Erlangian Service-Time

Journal of the Royal Statistical Society Series B (Methodological) ◽

10.1111/j.2517-6161.1960.tb00355.x ◽

1960 ◽

Vol 22 (1) ◽

pp. 89-96 ◽

Cited By ~ 2

Author(s):

B. W. Conolly

Keyword(s):

Service Time ◽

Busy Period ◽

Queueing System ◽

Single Server

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On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals

Journal of Applied Probability ◽

10.1017/s0021900200035580 ◽

1971 ◽

Vol 8 (03) ◽

pp. 494-507 ◽

Cited By ~ 4

Author(s):

E. K. Kyprianou

Keyword(s):

Waiting Time ◽

Busy Period ◽

Queueing System ◽

Random Variable ◽

Single Server ◽

Virtual Waiting Time ◽

Waiting Time Process ◽

Poisson Arrivals ◽

Image Position ◽

Quasi Stationary Distribution

We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt, and the service time has distribution dB(t), 0 < t < ∞. Let W(t) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable denote the first busy period initiated by a waiting time u at time t = 0.

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Queues with semi-Markovian arrivals

Journal of Applied Probability ◽

10.2307/3212030 ◽

1967 ◽

Vol 4 (2) ◽

pp. 365-379 ◽

Cited By ~ 56

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 queueing systems by retrials

Journal of Applied Probability ◽

10.1017/s0021900200023524 ◽

1983 ◽

Vol 20 (02) ◽

pp. 380-389 ◽

Cited By ~ 1

Author(s):

Vidyadhar G. Kulkarni

Keyword(s):

General Result ◽

Service Time ◽

Queueing Systems ◽

Single Server ◽

Expected Number ◽

Waiting Room ◽

Second Moments ◽

Different Types ◽

General Distributions ◽

Arbitrary Service Time

A general result for queueing systems with retrials is presented. This result relates the expected total number of retrials conducted by an arbitrary customer to the expected total number of retrials that take place during an arbitrary service time. This result is used in the analysis of a special system where two types of customer arrive in an independent Poisson fashion at a single-server service station with no waiting room. The service times of the two types of customer have independent general distributions with finite second moments. When the incoming customer finds the server busy he immediately leaves and tries his luck again after an exponential amount of time. The retrial rates are different for different types of customers. Expressions are derived for the expected number of retrial customers of each type.

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