The MX/G/1 queue with queue length dependent service times

It is shown that, in a multiserver queue with interarrival and service-time distributions of phase type (PH/PH/c), the waiting-time distributionW(x) has an asymptotically exponential tail, i.e., 1 –W(x) ∽Ke–ckx. The parameter k is the unique positive number satisfyingT*(ck)S*(–k) = 1, whereT*(s) andS*(s) are the Laplace–Stieltjes transforms of the interarrival and the service-time distributions. It is also shown that the queue-length distribution has an asymptotically geometric tail with the rate of decay η =T*(ck). The proofs of these results are based on the matrix-geometric form of the state probabilities of the system in the steady state.The equation for k shows interesting relations between single- and multiserver queues in the rates of decay of the tails of the waiting-time and the queue-length distributions.The parameters k and η can be easily computed by solving an algebraic equation. The multiplicative constantKis not so easy to compute. In order to obtain its numerical value we have to solve the balance equations or estimate it from simulation.

Download Full-text

GeoX1, GeoX2/D/c HOL Priority Queueing System with Random Order Selection within Each Priority Class

Probability in the Engineering and Informational Sciences ◽

10.1017/s026996480000509x ◽

1998 ◽

Vol 12 (1) ◽

pp. 125-139 ◽

Cited By ~ 6

Author(s):

Bong Dae Choi ◽

Yutae Lee ◽

Doo Il Choi

Keyword(s):

Queue Length ◽

Time Distribution ◽

Queueing System ◽

Length Distribution ◽

Random Order ◽

Order Selection ◽

Waiting Time Distribution ◽

Queue Length Distribution ◽

Packet Switch ◽

Priority Queueing

We model the virtual contention queue in an ATM nonblocking packet switch with capacity c and input queues by a Geox1, Geox2/D/c Head-of-Line priority queueing system with Random Order Selection within each class and find the joint queue length distribution and the waiting time distribution for each class.

Download Full-text

Asymptotic exponentiality of the tail of the waiting-time distribution in a Ph/Ph/C queue

Advances in Applied Probability ◽

10.2307/1426788 ◽

1981 ◽

Vol 13 (3) ◽

pp. 619-630 ◽

Cited By ~ 50

Author(s):

Yukio Takahashi

Keyword(s):

Waiting Time ◽

Service Time ◽

Queue Length ◽

Time Distribution ◽

Length Distribution ◽

Waiting Time Distribution ◽

Queue Length Distribution ◽

Multiserver Queues ◽

The Matrix ◽

Rates Of Decay

It is shown that, in a multiserver queue with interarrival and service-time distributions of phase type (PH/PH/c), the waiting-time distribution W(x) has an asymptotically exponential tail, i.e., 1 – W(x) ∽ Ke–ckx. The parameter k is the unique positive number satisfying T*(ck) S*(–k) = 1, where T*(s) and S*(s) are the Laplace–Stieltjes transforms of the interarrival and the service-time distributions. It is also shown that the queue-length distribution has an asymptotically geometric tail with the rate of decay η = T*(ck). The proofs of these results are based on the matrix-geometric form of the state probabilities of the system in the steady state.The equation for k shows interesting relations between single- and multiserver queues in the rates of decay of the tails of the waiting-time and the queue-length distributions.The parameters k and η can be easily computed by solving an algebraic equation. The multiplicative constant K is not so easy to compute. In order to obtain its numerical value we have to solve the balance equations or estimate it from simulation.

Download Full-text

A non-exponential queueing system with independent arrivals and batch servicing

Journal of Applied Probability ◽

10.1017/s0021900200038857 ◽

1990 ◽

Vol 27 (02) ◽

pp. 401-408

Author(s):

Nico M. Van Dijk ◽

Eric Smeitink

Keyword(s):

Queue Length ◽

Queueing System ◽

Length Distribution ◽

Service Systems ◽

Batch Service ◽

Repair System ◽

Theoretical Interest ◽

Form Expression ◽

Queue Length Distribution ◽

Service Times

We study a queueing system with a finite number of input sources. Jobs are individually generated by a source but wait to be served in batches, during which the input of that source is stopped. The service speed of a server depends on the mode of other sources and thus includes interdependencies. The input and service times are allowed to be generally distributed. A classical example is a machine repair system where the machines are subject to shocks causing cumulative damage. A product-form expression is obtained for the steady state joint queue length distribution and shown to be insensitive (i.e. to depend on only mean input and service times). The result is of both practical and theoretical interest as an extension of more standard batch service systems.

Download Full-text

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.

Download Full-text

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.

Download Full-text

An N-policy discrete-time Geo/G/1 queue with modified multiple server vacations and Bernoulli feedback*

RAIRO - Operations Research ◽

10.1051/ro/2017027 ◽

2019 ◽

Vol 53 (2) ◽

pp. 367-387

Author(s):

Shaojun Lan ◽

Yinghui Tang

Keyword(s):

Discrete Time ◽

Queue Length ◽

Length Distribution ◽

Probability Generating Function ◽

Renewal Theory ◽

Function Technique ◽

Single Server ◽

Bernoulli Feedback ◽

Queue Length Distribution ◽

Special Cases

This paper deals with a single-server discrete-time Geo/G/1 queueing model with Bernoulli feedback and N-policy where the server leaves for modified multiple vacations once the system becomes empty. Applying the law of probability decomposition, the renewal theory and the probability generating function technique, we explicitly derive the transient queue length distribution as well as the recursive expressions of the steady-state queue length distribution. Especially, some corresponding results under special cases are directly obtained. Furthermore, some numerical results are provided for illustrative purposes. Finally, a cost optimization problem is numerically analyzed under a given cost structure.

Download Full-text

Complementary generating functions for the MX/GI/1/k and GI/My/1/K queues and their application to the comparison of loss probabilities

Journal of Applied Probability ◽

10.2307/3214551 ◽

1990 ◽

Vol 27 (3) ◽

pp. 684-692 ◽

Cited By ~ 15

Author(s):

Masakiyo Miyazawa

Keyword(s):

Waiting Time ◽

Queue Length ◽

Length Distribution ◽

Direct Proof ◽

Interarrival Time ◽

Convex Order ◽

Queue Length Distribution ◽

Virtual Waiting Time ◽

Loss Probabilities ◽

Waiting Time Process

A direct proof is presented for the fact that the stationary system queue length distribution just after the service completion epochs in the Mx/GI/1/k queue is given by the truncation of a measure on Z+ = {0, 1, ·· ·}. The related truncation formulas are well known for the case of the traffic intensity ρ < 1 and for the virtual waiting time process in M/GI/1 with a limited waiting time (Cohen (1982) and Takács (1974)). By the duality of GI/MY/1/k to Mx/GI/1/k + 1, we get a similar result for the system queue length distribution just before the arrival of a customer in GI/MY/1/k. We apply those results to prove that the loss probabilities of Mx/GI/1/k and GI/MY/1/k are increasing for the convex order of the service time and interarrival time distributions, respectively, if their means are fixed.

Download Full-text

The N/G/1 queue and its detailed analysis

Advances in Applied Probability ◽

10.1017/s0001867800033474 ◽

1980 ◽

Vol 12 (01) ◽

pp. 222-261 ◽

Cited By ~ 26

Author(s):

V. Ramaswami

Keyword(s):

Poisson Process ◽

Waiting Time ◽

Queue Length ◽

Renewal Theory ◽

Single Server ◽

Virtual Waiting Time ◽

Special Cases ◽

Complex Models ◽

Markov Modulated ◽

Markov Renewal Theory

We discuss a single-server queue whose input is the versatile Markovian point process recently introduced by Neuts [22] herein to be called the N-process. Special cases of the N-process discussed earlier in the literature include a number of complex models such as the Markov-modulated Poisson process, the superposition of a Poisson process and a phase-type renewal process, etc. This queueing model has great appeal in its applicability to real world situations especially such as those involving inhibition or stimulation of arrivals by certain renewals. The paper presents formulas in forms which are computationally tractable and provides a unified treatment of many models which were discussed earlier by several authors and which turn out to be special cases. Among the topics discussed are busy-period characteristics, queue-length distributions, moments of the queue length and virtual waiting time. We draw particular attention to our generalization of the Pollaczek–Khinchin formula for the Laplace–Stieltjes transform of the virtual waiting time of the M/G/1 queue to the present model and the resulting Volterra system of integral equations. The analysis presented here serves as an example of the power of Markov renewal theory.

Download Full-text

Analytical solution of finite capacity M/D/1 queues

Journal of Applied Probability ◽

10.1017/s0021900200018258 ◽

2000 ◽

Vol 37 (04) ◽

pp. 1092-1098

Author(s):

Olivier Brun ◽

Jean-Marie Garcia

Keyword(s):

Queue Length ◽

Length Distribution ◽

Point Of View ◽

Queueing Model ◽

Finite Capacity ◽

Computational Point ◽

Queue Length Distribution ◽

The Mean ◽

Mean Queue Length ◽

Number Of Customers

Although the M/D/1/N queueing model is well solved from a computational point of view, there is no known analytical expression of the queue length distribution. In this paper, we derive closed-form formulae for the distribution of the number of customers in the system in the finite-capacity M/D/1 queue. We also give an explicit solution for the mean queue length and the average waiting time.

Download Full-text