An approximation for the busy period of the M/G/1 queue using a diffusion model

1974 ◽  
Vol 11 (1) ◽  
pp. 159-169 ◽  
Author(s):  
D. P. Heyman
Keyword(s):  
Density Function ◽  
Diffusion Model ◽  
Gamma Distribution ◽  
Service Time ◽  
Busy Period ◽  
Exact Density ◽  
Mean And Variance ◽  
Negative Exponential

A diffusion model for the M/G/1 queue due to D. P. Gaver is used to obtain an approximation for the density function of the busy period. The approximation has the same mean and variance as the exact density function, and can be given explicitly when the service time is constant, or has a negative exponential or gamma distribution, or is a mixture of these types.

An approximation for the busy period of the M/G/1 queue using a diffusion model

1974 ◽  
Vol 11 (01) ◽  
pp. 159-169 ◽  
Author(s):  
D. P. Heyman
Keyword(s):  
Density Function ◽  
Diffusion Model ◽  
Gamma Distribution ◽  
Service Time ◽  
Busy Period ◽  
Exact Density ◽  
Mean And Variance ◽  
Negative Exponential

A diffusion model for the M/G/1 queue due to D. P. Gaver is used to obtain an approximation for the density function of the busy period. The approximation has the same mean and variance as the exact density function, and can be given explicitly when the service time is constant, or has a negative exponential or gamma distribution, or is a mixture of these types.

On a property of a refusals stream

1997 ◽  
Vol 34 (03) ◽  
pp. 800-805 ◽  
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.

A Correlated Queue

1969 ◽  
Vol 6 (01) ◽  
pp. 122-136 ◽  
Author(s):  
B.W. Conolly ◽  
N. Hadidi
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Busy Period ◽  
The State ◽  
Queueing Model ◽  
Time Process ◽  
Waiting Time Process ◽  
The Subject ◽  
Service Pattern ◽  
Arrival Pattern

A “correlated queue” is defined to be a queueing model in which the arrival pattern influences the service pattern or vice versa. A particular model of this nature is considered in this paper. It is such that the service time of a customer is directly proportional to the interval between his own arrival and that of his predecessor. The initial busy period, state and output processes are analyzed in detail. For completeness, a sketch is also given of the analysis of the waiting time process which forms the subject of another paper. The results are used in the analysis of the state and output processes.

Busy period in GIX/G/∞

1996 ◽  
Vol 33 (3) ◽  
pp. 815-829 ◽  
Author(s):  
Liming Liu ◽  
Ding-Hua Shi
Keyword(s):  
Service Time ◽  
Busy Period ◽  
Random Variables ◽  
Batch Service ◽  
Idle Period ◽  
Special Cases ◽  
Infinite Server Queues ◽  
General Relations ◽  
Number Of Customers ◽  
Busy Cycle

Busy period problems in infinite server queues are studied systematically, starting from the batch service time. General relations are given for the lengths of the busy cycle, busy period and idle period, and for the number of customers served in a busy period. These relations show that the idle period is the most difficult while the busy cycle is the simplest of the four random variables. Renewal arguments are used to derive explicit results for both general and special cases.

On the integral of the workload process of the single server queue

2003 ◽  
Vol 40 (01) ◽  
pp. 200-225 ◽  
Author(s):  
A. A. Borovkov ◽  
O. J. Boxma ◽  
Z. Palmowski
Keyword(s):  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Single Server ◽  
Service Time Distribution ◽  
Whittaker Functions ◽  
Single Server Queue ◽  
Stieltjes Transform ◽  
Server Queue ◽  
Sequential Approximation

This paper is devoted to a study of the integral of the workload process of the single server queue, in particular during one busy period. Firstly, we find asymptotics of the area 𝒜 swept under the workload process W(t) during the busy period when the service time distribution has a regularly varying tail. We also investigate the case of a light-tailed service time distribution. Secondly, we consider the problem of obtaining an explicit expression for the distribution of 𝒜. In the general GI/G/1 case, we use a sequential approximation to find the Laplace—Stieltjes transform of 𝒜. In the M/M/1 case, this transform is obtained explicitly in terms of Whittaker functions. Thirdly, we consider moments of 𝒜 in the GI/G/1 queue. Finally, we show asymptotic normality of .

Some Duality Results in the Theory of Queues

1969 ◽  
Vol 6 (01) ◽  
pp. 99-121 ◽  
Author(s):  
Irwin Greenberg
Keyword(s):  
Service Time ◽  
Busy Period ◽  
Original System ◽  
Dual System ◽  
Duality Results ◽  
System A ◽  
Duality Relations

When the interarrival and service time distributions of a queue are interchanged a new queue is obtained which can be considered as the dual of the original. Another dual system, a dam, can also be associated with the original queue. Events defined for the original system can be transformed into events defined for the duals and conversely, and hence, probabilities obtained for one system can be extended to the others. In this paper several duality relations are derived, with particular emphasis on results pertaining to a single busy period. Examples are given, most of which refer to the M/G/1 – G/M/l queues.

The general N server finite queue

1971 ◽  
Vol 8 (04) ◽  
pp. 828-834 ◽  
Author(s):  
Asha Seth Kapadia
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Queueing System ◽  
List Type ◽  
Type Number ◽  
Stochastic Nature ◽  
Mean Waiting Time ◽  
Queue Discipline ◽  
Mean And Variance ◽  
The Mean

Kingman (1962) studied the effect of queue discipline on the mean and variance of the waiting time. He made no assumptions regarding the stochastic nature of the input and the service distributions, except that the input and service processes are independent of each other. When the following two conditions hold: (a) no server sits idle while there are customers waiting to be served; (b) the busy period is finite with probability one (i.e., the queue empties infinitely often with probability one); he has shown that the mean waiting time is independent of the queue discipline and the variance of the waiting time is a minimum when the customers are served in order of their arrival. Conditions (a) and (b) will henceforward be called Kingman conditions and a queueing system satisfying Kingman conditions will be referred to in the text as a Kingman queue.

On the busy period of the modified GI/G/1 queue

1973 ◽  
Vol 10 (01) ◽  
pp. 192-197 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Laplace Transform ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Service Time Distribution ◽  
Duality Results ◽  
The Laplace Transform ◽  
Period Duration

Proceeding from duality results for the GI/G/1 queue, this paper obtains the probability of the number served in a busy period of aGI/G/1 system where customers initiating a busy period have a different service time distribution from other customers. Using duality arguments for processes with interchangeable increments, the Laplace transform of the busy period duration is found for a modified GI/M/1 queue.

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