On the waiting time distribution in a generalized GI/G/1 queueing system

1971 ◽  
Vol 8 (2) ◽  
pp. 241-251 ◽  
Author(s):  
Jacqueline Loris-Teghem
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Queueing System ◽  
Random Variables ◽  
Markovian Process ◽  
Random Time ◽  
Arbitrary Distribution ◽  
Waiting Time Distribution ◽  
Transient Behaviour ◽  
Set Up

The model considered in this paper describes a queueing system in which the station is dismantled at the end of a busy period and re-established on arrival of a new customer, in such a way that the closing-down process consists of N1 phases of random duration and that a customer 𝒞n who arrives while the station is being closed down must wait a random time idn(i = 1, ···, N1) if the ith phase is going on at the arrival instant. (For each fixed index i, the random variables idn are identically distributed.) A customer 𝒞n arriving when the closing-down of the station is already accomplished has to wait a random time (N1 + 1)dn corresponding to the set up time of the station. Besides, a customer 𝒞n who arrives when the station is busy has to wait an additional random time 0dn. We thus have (N1 + 2) types of “delay” (additional waiting time). Similarly, we consider (N2 + 2) types of service time and (N3 + 2) probabilities of joining the queue. This may be formulated as a model with (N + 2) types of triplets (delay, service time, probability of joining the queue). We consider the general case where the random variables defining the model all have an arbitrary distribution.The process {wn}, where wn denotes the waiting time of customer 𝒞n if he joins the queue at all, is not necessarily Markovian, so that we first study (by algebraic considerations) the transient behaviour of a Markovian process {vn} related to {wn}, and then derive the distribution of the variables wn.

On the waiting time distribution in a generalized GI/G/1 queueing system

1971 ◽  
Vol 8 (02) ◽  
pp. 241-251 ◽  
Author(s):  
Jacqueline Loris-Teghem
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Queueing System ◽  
Random Variables ◽  
Markovian Process ◽  
Random Time ◽  
Arbitrary Distribution ◽  
Waiting Time Distribution ◽  
Transient Behaviour ◽  
Set Up

The model considered in this paper describes a queueing system in which the station is dismantled at the end of a busy period and re-established on arrival of a new customer, in such a way that the closing-down process consists of N 1 phases of random duration and that a customer 𝒞 n who arrives while the station is being closed down must wait a random time idn (i = 1, ···, N 1) if the ith phase is going on at the arrival instant. (For each fixed index i, the random variables idn are identically distributed.) A customer 𝒞 n arriving when the closing-down of the station is already accomplished has to wait a random time (N 1 + 1) dn corresponding to the set up time of the station. Besides, a customer 𝒞 n who arrives when the station is busy has to wait an additional random time 0 dn. We thus have (N 1 + 2) types of “delay” (additional waiting time). Similarly, we consider (N 2 + 2) types of service time and (N 3 + 2) probabilities of joining the queue. This may be formulated as a model with (N + 2) types of triplets (delay, service time, probability of joining the queue). We consider the general case where the random variables defining the model all have an arbitrary distribution. The process {wn }, where wn denotes the waiting time of customer 𝒞 n if he joins the queue at all, is not necessarily Markovian, so that we first study (by algebraic considerations) the transient behaviour of a Markovian process {vn } related to {wn }, and then derive the distribution of the variables wn.

On the waiting time distribution in a generalized queueing system with uniformly bounded sojourn times

1972 ◽  
Vol 9 (3) ◽  
pp. 642-649 ◽  
Author(s):  
Jacqueline Loris-Teghem
Keyword(s):  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Queueing System ◽  
Waiting Time Distribution ◽  
Stieltjes Transform ◽  
Sojourn Times ◽  
Stieltjes Transforms ◽  
Uniformly Bounded

A generalized queueing system with (N + 2) types of triplets (delay, service time, probability of joining the queue) and with uniformly bounded sojourn times is considered. An expression for the generating function of the Laplace-Stieltjes transforms of the waiting time distributions is derived analytically, in a case where some of the random variables defining the model have a rational Laplace-Stieltjes transform.The standard Kl/Km/1 queueing system with uniformly bounded sojourn times is considered in particular.

On the waiting time distribution in a generalized queueing system with uniformly bounded sojourn times

1972 ◽  
Vol 9 (03) ◽  
pp. 642-649
Author(s):  
Jacqueline Loris-Teghem
Keyword(s):  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Queueing System ◽  
Waiting Time Distribution ◽  
Stieltjes Transform ◽  
Sojourn Times ◽  
Stieltjes Transforms ◽  
Uniformly Bounded

A generalized queueing system with (N+ 2) types of triplets (delay, service time, probability of joining the queue) and with uniformly bounded sojourn times is considered. An expression for the generating function of the Laplace-Stieltjes transforms of the waiting time distributions is derived analytically, in a case where some of the random variables defining the model have a rational Laplace-Stieltjes transform.The standardKl/Km/1 queueing system with uniformly bounded sojourn times is considered in particular.

On the service time distribution and the waiting time process of a potentially infinite capacity queueing system

1969 ◽  
Vol 6 (3) ◽  
pp. 594-603 ◽  
Author(s):  
Nasser Hadidi
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Queueing System ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Poisson Stream ◽  
Waiting Time Process ◽  
The Subject ◽  
Effective Service

In [1] the authors dealt with a particular queueing system in which arrivals occurred in a Poisson stream and the probability differential of a service completion was μσn when the queue contained n customers. Much of the theory could not be carried out further analytically for a general σn, which is a purely n-dependent quantity. To carry the analysis further to the extent of finding the “effective” service time and the waiting time distribution when σn is a linear function of n, (which is considered to be rather general and sufficient for practical purposes), constitutes the subject matter of this paper.

On the service time distribution and the waiting time process of a potentially infinite capacity queueing system

1969 ◽  
Vol 6 (03) ◽  
pp. 594-603 ◽  
Author(s):  
Nasser Hadidi
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Queueing System ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Poisson Stream ◽  
Waiting Time Process ◽  
The Subject ◽  
Effective Service

In [1] the authors dealt with a particular queueing system in which arrivals occurred in a Poisson stream and the probability differential of a service completion was μσn when the queue contained n customers. Much of the theory could not be carried out further analytically for a general σn , which is a purely n-dependent quantity. To carry the analysis further to the extent of finding the “effective” service time and the waiting time distribution when σn is a linear function of n, (which is considered to be rather general and sufficient for practical purposes), constitutes the subject matter of this paper.

Analysis of the M x/G/1 queue by N-policy and multiple vacations

1994 ◽  
Vol 31 (02) ◽  
pp. 476-496
Author(s):  
Ho Woo Lee ◽  
Soon Seok Lee ◽  
Jeong Ok Park ◽  
K. C. Chae
Keyword(s):  
Performance Measures ◽  
Queue Length ◽  
Time Distribution ◽  
Queueing System ◽  
Random Variables ◽  
System Size ◽  
The Other ◽  
Waiting Time Distribution ◽  
Multiple Vacations ◽  
Linear Cost

We consider an Mx /G/1 queueing system with N-policy and multiple vacations. As soon as the system empties, the server leaves for a vacation of random length V. When he returns, if the queue length is greater than or equal to a predetermined value N(threshold), the server immediately begins to serve the customers. If he finds less than N customers, he leaves for another vacation and so on until he finally finds at least N customers. We obtain the system size distribution and show that the system size decomposes into three random variables one of which is the system size of ordinary Mx /G/1 queue. The interpretation of the other random variables will be provided. We also derive the queue waiting time distribution and other performance measures. Finally we derive a condition under which the optimal stationary operating policy is achieved under a linear cost structure.

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.

Characterizations of the Exponential Distribution via the Blocking Time in a Queueing System with an Unreliable Server

1998 ◽  
Vol 35 (1) ◽  
pp. 236-239 ◽  
Author(s):  
Jian-Lun Xu
Keyword(s):  
Exponential Distribution ◽  
Service Time ◽  
Coefficient Of Variation ◽  
Queueing System ◽  
Random Variables ◽  
Unreliable Server ◽  
Failure Times ◽  
Server Failure

The characterization of the exponential distribution via the coefficient of the variation of the blocking time in a queueing system with an unreliable server, as given by Lin (1993), is improved by substantially weakening the conditions. Based on the coefficient of variation of certain random variables, including the blocking time, the normal service time and the minimum of the normal service and the server failure times, two new characterizations of the exponential distribution are obtained.

A queue with Markov-dependent service times

1968 ◽  
Vol 5 (02) ◽  
pp. 461-466
Author(s):  
Gerold Pestalozzi
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Queueing System ◽  
Single Channel ◽  
Probability Generating Function ◽  
Average Waiting Time ◽  
Poisson Input ◽  
The Mean

A queueing system is considered where each item has a property associated with it, and where the service time interposed between two items depends on the properties of both of these items. The steady state of a single-channel queue of this type, with Poisson input, is investigated. It is shown how the probability generating function of the number of items waiting can be found. Easily applied approximations are given for the mean number of items waiting and for the average waiting time.

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