Some general results for many server queues

Pollaczek's theory for the many server queue is generalized and extended. Pollaczek (1961) found the distribution of the actual waiting times in the model G/G/s as a solution of a set of integral equations. We give a somewhat more general set of integral equations from which the joint distribution of the actual waiting time and some other random variables may be found. With this joint distribution we can obtain distributions of a number of characteristic quantities, such as the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. For a wide class of many server queues the formal expressions may lead to explicit results.

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On the many server queue with exponential service times

Advances in Applied Probability ◽

10.2307/1425970 ◽

1973 ◽

Vol 5 (1) ◽

pp. 170-182 ◽

Cited By ~ 21

Author(s):

J. H. A. De Smit

Keyword(s):

Waiting Time ◽

Queue Length ◽

Busy Period ◽

Virtual Waiting Time ◽

Server Queue ◽

Poisson Arrivals ◽

Service Times ◽

The Many ◽

Special Case ◽

Busy Cycle

The general theory for the many server queue due to Pollaczek (1961) and generalized by the author (de Smit (1973)) is applied to the system with exponential service times. In this way many explicit results are obtained for the distributions of characteristic quantities, such as the actual waiting time, the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. Most of these results are new, even for the special case of Poisson arrivals.

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On the many server queue with exponential service times

Advances in Applied Probability ◽

10.1017/s000186780003901x ◽

1973 ◽

Vol 5 (01) ◽

pp. 170-182 ◽

Cited By ~ 6

Author(s):

J. H. A. De Smit

Keyword(s):

Waiting Time ◽

Queue Length ◽

Busy Period ◽

Virtual Waiting Time ◽

Server Queue ◽

Poisson Arrivals ◽

Service Times ◽

The Many ◽

Special Case ◽

Busy Cycle

The general theory for the many server queue due to Pollaczek (1961) and generalized by the author (de Smit (1973)) is applied to the system with exponential service times. In this way many explicit results are obtained for the distributions of characteristic quantities, such as the actual waiting time, the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. Most of these results are new, even for the special case of Poisson arrivals.

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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 steady-state solution of the M/K2/m queue

Advances in Applied Probability ◽

10.1017/s0001867800035503 ◽

1980 ◽

Vol 12 (03) ◽

pp. 799-823

Author(s):

Per Hokstad

Keyword(s):

Steady State ◽

Queue Length ◽

Simple Formula ◽

Length Distribution ◽

Steady State Solution ◽

Queue Length Distribution ◽

Server Queue ◽

The Mean ◽

Mean Queue Length ◽

The Many

The many-server queue with service time having rational Laplace transform of order 2 is considered. An expression for the asymptotic queue-length distribution is obtained. A relatively simple formula for the mean queue length is also found. A few numerical results on the mean queue length and on the probability of having to wait are given for the case of three servers. Some approximations for these quantities are also considered.

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The virtual waiting time of the GI/G/1 queue in heavy traffic

Advances in Applied Probability ◽

10.2307/1426170 ◽

1971 ◽

Vol 3 (2) ◽

pp. 249-268 ◽

Cited By ~ 16

Author(s):

E. Kyprianou

Keyword(s):

Markov Chains ◽

Waiting Time ◽

Limit Theorems ◽

Queueing System ◽

Heavy Traffic ◽

Waiting Times ◽

Double Sequence ◽

Traffic Intensity ◽

Virtual Waiting Time

Investigations in the theory of heavy traffic were initiated by Kingman ([5], [6] and [7]) in an effort to obtain approximations for stable queues. He considered the Markov chains {Wni} of a sequence {Qi} of stable GI/G/1 queues, where Wni is the waiting time of the nth customer in the ith queueing system, and by making use of Spitzer's identity obtained limit theorems as first n → ∞ and then ρi ↑ 1 as i → ∞. Here &rHi is the traffic intensity of the ith queueing system. After Kingman the theory of heavy traffic was developed by a number of Russians mainly. Prohorov [10] considered the double sequence of waiting times {Wni} and obtained limit theorems in the three cases when n1/2(ρi-1) approaches (i) - ∞, (ii) -δ and (iii) 0 as n → ∞ and i → ∞ simultaneously. The case (i) includes the result of Kingman. Viskov [12] also studied the double sequence {Wni} and obtained limits in the two cases when n1/2(ρi − 1) approaches + δ and + ∞ as n → ∞ and i → ∞ simultaneously.

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The probability of large queue lengths and waiting times in a heterogeneous multiserver queue II: Positive recurrence and logarithmic limits

Advances in Applied Probability ◽

10.2307/1427839 ◽

1995 ◽

Vol 27 (2) ◽

pp. 567-583 ◽

Cited By ~ 13

Author(s):

John S. Sadowsky

Keyword(s):

Waiting Time ◽

Queue Length ◽

Waiting Times ◽

General Setting ◽

Batch Arrival ◽

Positive Recurrence ◽

Waiting Time Distributions ◽

Harris Recurrence ◽

Stationary Queue Length ◽

Queue Lengths

We continue our investigation of the batch arrival-heterogeneous multiserver queue begun in Part I. In a general setting we prove the positive Harris recurrence of the system, and with no additional conditions we prove logarithmic tail limits for the stationary queue length and waiting time distributions.

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The steady-state solution of the M/K2/m queue

Advances in Applied Probability ◽

10.2307/1426432 ◽

1980 ◽

Vol 12 (3) ◽

pp. 799-823 ◽

Cited By ~ 17

Author(s):

Per Hokstad

Keyword(s):

Steady State ◽

Queue Length ◽

Simple Formula ◽

Length Distribution ◽

Steady State Solution ◽

Queue Length Distribution ◽

Server Queue ◽

The Mean ◽

Mean Queue Length ◽

The Many

The many-server queue with service time having rational Laplace transform of order 2 is considered. An expression for the asymptotic queue-length distribution is obtained. A relatively simple formula for the mean queue length is also found. A few numerical results on the mean queue length and on the probability of having to wait are given for the case of three servers. Some approximations for these quantities are also considered.

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The G/M/m queue with finite waiting room

Journal of Applied Probability ◽

10.2307/3212728 ◽

1975 ◽

Vol 12 (4) ◽

pp. 779-792 ◽

Cited By ~ 17

Author(s):

Per Hokstad

Keyword(s):

Asymptotic Distribution ◽

Waiting Time ◽

Joint Distribution ◽

Transient Solution ◽

Waiting Room ◽

Waiting Time Distributions ◽

Virtual Waiting Time ◽

The Moment ◽

Number Of Customers

The G/M/m queue with only s waiting places is studied. We start by studying the joint distribution of the number of customers present at time t and the time elapsing until the next arrival after t. This gives the asymptotic distribution of the number of customers at the moment of an arrival and at an arbitrary moment. Then waiting time and virtual waiting time distributions are easily obtained. For the G/M/1 queue also the transient solution is given. Finally the case s = ∞ is considered.

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Laws of Little in an open queueing network

Nonlinear Analysis Modelling and Control ◽

10.15388/na.17.3.14059 ◽

2012 ◽

Vol 17 (3) ◽

pp. 327-342 ◽

Cited By ~ 1

Author(s):

Saulius Minkevičius ◽

Stasys Steišūnas

Keyword(s):

Waiting Time ◽

Queueing Theory ◽

Queue Length ◽

Heavy Traffic ◽

Queueing Network ◽

Fluid Limit ◽

Fluid Approximation ◽

Large Numbers ◽

Virtual Waiting Time ◽

Open Queueing Network

The object of this research in the queueing theory is theorems about the functional strong laws of large numbers (FSLLN) under the conditions of heavy traffic in an open queueing network (OQN). The FSLLN is known as a fluid limit or fluid approximation. In this paper, FSLLN are proved for the values of important probabilistic characteristics of the OQN investigated as well as the virtual waiting time of a customer and the queue length of customers. As applications of the proved theorems laws of Little in OQN are presented.

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