Light traffic approximations in many-server queues

1992 ◽  
Vol 24 (01) ◽  
pp. 202-218 ◽  
Author(s):  
D. J. Daley ◽  
T. Rolski
Keyword(s):  
Waiting Time ◽  
Arrival Process ◽  
Light Traffic ◽  
Traffic Conditions ◽  
Poisson Arrival ◽  
Many Server Queues ◽  
Special Case ◽  
Poisson Arrival Process

This paper complements two previous studies (Daley and Rolski (1984), (1991)) by investigating limit properties of the waiting time in k-server queues with renewal arrival process under ‘light traffic' conditions. Formulae for the limits of the probability of waiting and the waiting time moments are derived for the two approaches of dilation and thinning of the arrival process. Asmussen's (1991) approach to light traffic limits applies to the cases considered, of which the Poisson arrival process (i.e. M/G/k) is a special case and for which formulae are given.

Light traffic approximations in many-server queues

1992 ◽  
Vol 24 (1) ◽  
pp. 202-218 ◽  
Author(s):  
D. J. Daley ◽  
T. Rolski
Keyword(s):  
Waiting Time ◽  
Arrival Process ◽  
Light Traffic ◽  
Traffic Conditions ◽  
Poisson Arrival ◽  
Many Server Queues ◽  
Special Case ◽  
Poisson Arrival Process

This paper complements two previous studies (Daley and Rolski (1984), (1991)) by investigating limit properties of the waiting time in k-server queues with renewal arrival process under ‘light traffic' conditions. Formulae for the limits of the probability of waiting and the waiting time moments are derived for the two approaches of dilation and thinning of the arrival process. Asmussen's (1991) approach to light traffic limits applies to the cases considered, of which the Poisson arrival process (i.e. M/G/k) is a special case and for which formulae are given.

The departure process from a queueing system

1976 ◽  
Vol 80 (2) ◽  
pp. 283-285 ◽  
Author(s):  
F. P. Kelly
Keyword(s):  
Queueing System ◽  
Arrival Process ◽  
Single Server ◽  
Departure Process ◽  
Common Distribution ◽  
Poisson Arrival ◽  
Queue Discipline ◽  
Common Distribution Function ◽  
Image Position ◽  
Poisson Arrival Process

Consider a single-server queueing system with a Poisson arrival process at rate λ and positive service requirements independently distributed with common distribution function B(z) and finite expectationwhere βλ < 1, i.e. an M/G/1 system. When the queue discipline is first come first served, or last come first served without pre-emption, the stationary departure process is Poisson if and only if G = M (i.e. B(z) = 1 − exp (−z/β)); see (8), (4) and (2). In this paper it is shown that when the queue discipline is last come first served with pre-emption the stationary departure process is Poisson whatever the form of B(z). The method used is adapted from the approach of Takács (10) and Shanbhag and Tambouratzis (9).

A time-varying Poisson arrival process generator

SIMULATION ◽  
1984 ◽  
Vol 43 (4) ◽  
pp. 193-195 ◽  
Author(s):  
Robert W. Klein ◽  
Stephen D. Roberts
Keyword(s):  
Arrival Process ◽  
Time Varying ◽  
Poisson Arrival ◽  
Poisson Arrival Process

The mean waiting time of a GI/G/1 queue in light traffic via random thinning

1995 ◽  
Vol 32 (01) ◽  
pp. 256-266
Author(s):  
Soracha Nananukul ◽  
Wei-Bo Gong
Keyword(s):  
Waiting Time ◽  
Padé Approximation ◽  
Radius Of Convergence ◽  
Arrival Process ◽  
Maclaurin Series ◽  
Light Traffic ◽  
Numerical Examples ◽  
Mean Waiting Time ◽  
The Mean ◽  
Random Thinning

In this paper, we derive the MacLaurin series of the mean waiting time in light traffic for a GI/G/1 queue. The light traffic is defined by random thinning of the arrival process. The MacLaurin series is derived with respect to the admission probability, and we prove that it has a positive radius of convergence. In the numerical examples, we use the MacLaurin series to approximate the mean waiting time beyond light traffic by means of Padé approximation.

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

1969 ◽  
Vol 6 (3) ◽  
pp. 565-572 ◽  
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.

Two sufficient properties for the insensitivity of a class of queueing models

1991 ◽  
Vol 28 (03) ◽  
pp. 664-672
Author(s):  
Moshe Haviv
Keyword(s):  
Arrival Process ◽  
Queueing Models ◽  
Sufficient Condition ◽  
Poisson Arrival ◽  
System 2 ◽  
Poisson Arrival Process

For indivisible strong work-conserving queueing models with a Poisson arrival process, each of the following two properties is a sufficient condition for insensitivity. (1) The completed workload of a job receiving service is independent of the number of jobs in the system. (2) Independently of the completed workloads of the jobs in the system, they all are equally likely to be in service. For models which additionally belong to the class described by two families of parameters each of these properties is also necessary for insensitivity.

The mean waiting time of a GI/G/1 queue in light traffic via random thinning

1995 ◽  
Vol 32 (1) ◽  
pp. 256-266 ◽  
Author(s):  
Soracha Nananukul ◽  
Wei-Bo Gong
Keyword(s):  
Waiting Time ◽  
Padé Approximation ◽  
Radius Of Convergence ◽  
Arrival Process ◽  
Maclaurin Series ◽  
Light Traffic ◽  
Numerical Examples ◽  
Mean Waiting Time ◽  
The Mean ◽  
Random Thinning

In this paper, we derive the MacLaurin series of the mean waiting time in light traffic for a GI/G/1 queue. The light traffic is defined by random thinning of the arrival process. The MacLaurin series is derived with respect to the admission probability, and we prove that it has a positive radius of convergence. In the numerical examples, we use the MacLaurin series to approximate the mean waiting time beyond light traffic by means of Padé approximation.

The asymptotic behavior o queues with time-varying arrival rates

1984 ◽  
Vol 21 (01) ◽  
pp. 143-156 ◽  
Author(s):  
Daniel P. Heyman ◽  
Ward Whitt
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Stability ◽  
Limit Theorems ◽  
Arrival Process ◽  
Periodic Case ◽  
Time Varying ◽  
Poisson Arrival ◽  
Poisson Arrival Process

This paper discusses the asymptotic behavior of the Mt/G/c queue having a Poisson arrival process with a general deterministic intensity. Since traditional equilibrium does not always exist, other notions of asymptotic stability are introduced and investigated. For the periodic case, limit theorems are proved complementing Harrison and Lemoine (1977) and Lemoine (1981).

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