On the busy period distribution of the M/G/2 queueing system

1989 ◽  
Vol 26 (4) ◽  
pp. 858-865 ◽  
Author(s):  
Douglas P. Wiens
Keyword(s):  
Linear Combination ◽  
Busy Period ◽  
Queueing System ◽  
Rate Parameter ◽  
Period Distribution ◽  
Special Cases ◽  
Service Times ◽  
Arbitrary Linear Combination

Equations are derived for the distribution of the busy period of the GI/G/2 queue. The equations are analyzed for the M/G/2 queue, assuming that the service times have a density which is an arbitrary linear combination, with respect to both the number of stages and the rate parameter, of Erlang densities. The coefficients may be negative. Special cases and examples are studied.

On the busy period distribution of the M/G/2 queueing system

1989 ◽  
Vol 26 (04) ◽  
pp. 858-865 ◽  
Author(s):  
Douglas P. Wiens
Keyword(s):  
Linear Combination ◽  
Busy Period ◽  
Queueing System ◽  
Rate Parameter ◽  
Period Distribution ◽  
Special Cases ◽  
Service Times ◽  
Arbitrary Linear Combination

Equations are derived for the distribution of the busy period of the GI/G/2 queue. The equations are analyzed for the M/G/2 queue, assuming that the service times have a density which is an arbitrary linear combination, with respect to both the number of stages and the rate parameter, of Erlang densities. The coefficients may be negative. Special cases and examples are studied.

scholarly journals Single server queues with modified service mechanisms

1962 ◽  
Vol 2 (4) ◽  
pp. 499-507 ◽  
Author(s):  
G. F. Yeo
Keyword(s):  
Time Distribution ◽  
Busy Period ◽  
Queueing System ◽  
Probability Generating Function ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Period Distribution ◽  
Time Dependent Problem ◽  
Stationary Waiting Time ◽  
Special Case

SummaryThis paper considers a generalisation of the queueing system M/G/I, where customers arriving at empty and non-empty queues have different service time distributions. The characteristic function (c.f.) of the stationary waiting time distribution and the probability generating function (p.g.f.) of the queue size are obtained. The busy period distribution is found; the results are generalised to an Erlangian inter-arrival distribution; the time-dependent problem is considered, and finally a special case of server absenteeism is discussed.

Tandem Conveyor Queues

1989 ◽  
Vol 3 (4) ◽  
pp. 517-536
Author(s):  
F. Baccelli ◽  
E.G. Coffman ◽  
E.N. Gilbert
Keyword(s):  
Boundary Value Problem ◽  
Joint Distribution ◽  
Queueing System ◽  
The Other ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary ◽  
Special Cases ◽  
A Cell ◽  
Service Times ◽  
Input Position

This paper analyzes a queueing system in which a constant-speed conveyor brings new items for service and carries away served items. The conveyor is a sequence of cells each able to hold at most one item. At each integer time, a new cell appears at the queue's input position. This cell holds an item requiring service with probability a, holds a passerby requiring no service with probability b, and is empty with probability (1– a – b). Service times are integers synchronized with the arrival of cells at the input, and they are geometrically distributed with parameter μ. Items requiring service are placed in an unbounded queue to await service. Served items are put in a second unbounded queue to await replacement on the conveyor in cells at the input position. Two models are considered. In one, a served item can only be placed into a cell that was empty on arrival; in the other, the served item can be placed into a cell that was either empty or contained an item requiring service (in the latter case unloading and loading at the input position can take place in the same time unit). The stationary joint distribution of the numbers of items in the two queues is studied for both models. It is verified that, in general, this distribution does not have a product form. Explicit results are worked out for special cases, e.g., when b = 0, and when all service times are one time unit (μ = 1). It is shown how the analysis of the general problem can be reduced to the solution of a Riemann boundary-value problem.

An Uncertain Single-Server Queueing Model

2021 ◽  
pp. 2150001
Author(s):  
Kai Yao
Keyword(s):  
Queueing Theory ◽  
Busy Period ◽  
Queueing System ◽  
Single Server ◽  
Queueing Model ◽  
Uncertainty Distribution ◽  
Uncertain Variables ◽  
Service Efficiency ◽  
Service Times ◽  
Interarrival Times

In the queueing theory, the interarrival times between customers and the service times for customers are usually regarded as random variables. This paper considers human uncertainty in a queueing system, and proposes an uncertain queueing model in which the interarrival times and the service times are regarded as uncertain variables. The busyness index is derived analytically which indicates the service efficiency of a queueing system. Besides, the uncertainty distribution of the busy period is obtained.

Distribution of the busy period in a controllable M/M/2 queue operating under the triadic (0, K, N, M) policy

1990 ◽  
Vol 27 (02) ◽  
pp. 425-432
Author(s):  
Hahn-Kyou Rhee ◽  
B. D. Sivazlian
Keyword(s):  
Steady State ◽  
Busy Period ◽  
Queueing System ◽  
Service Completion ◽  
Special Cases ◽  
Service Stations ◽  
Gambler’S Ruin ◽  
Gambler's Ruin Problem ◽  
Number Of Customers ◽  
System Operating

We consider an M/M/2 queueing system with removable service stations operating under steady-state conditions. We assume that the number of operating service stations can be adjusted at customers' arrival or service completion epochs depending on the number of customers in the system. The objective of this paper is to obtain the distribution of the busy period using the theory of the gambler's ruin problem. As special cases, the distributions of the busy periods for the ordinary M/M/2 queueing system, the M/M/1 queueing system operating under the N policy and the ordinary M/M/1 queueing system are obtained.

scholarly journals A new approach to the distribution of the duration of the busy period for a G/G/l queueing system

1990 ◽  
Vol 48 (1) ◽  
pp. 89-100 ◽  
Author(s):  
Wolfgang Stadje
Keyword(s):  
Integral Equations ◽  
Joint Distribution ◽  
Busy Period ◽  
Queueing System ◽  
Laplace Transforms ◽  
Theoretic Approach ◽  
New Approach ◽  
Standard Methods ◽  
Period Distribution ◽  
Number Of Customers

AbstractFor a G/G/l queueing system let Xt be the number of customers present at time t and Yt(Zt) be the time elapsed since the last arrival of a customer (the last completion of a service) at time t. Let τl be the time until the number of customers in the sustem is reduced from j to j – l, given that X0 = j ≧ l, Y0 = y, Z0 = z. For the joint distribution of τl and Yτl and the Laplace transforms of the τl intergral equations are derived. Under slight conditions these integral equations have unique solutions which can be determined by standard methods. Our results offer a method for calculating the busy period distribution which is completely different from the usual fluctuatuion theoretic approach.

Analysis of the busy period for the M/M/c queue: an algorithmic approach

2001 ◽  
Vol 38 (01) ◽  
pp. 209-222
Author(s):  
J. R. Artalejo ◽  
M. J. Lopez-Herrero
Keyword(s):  
Busy Period ◽  
Queueing System ◽  
Linear Equations ◽  
Time Interval ◽  
System Of Linear Equations ◽  
Recursive Procedure ◽  
Algorithmic Approach ◽  
Period Distribution ◽  
Algorithmic Analysis ◽  
Number Of Customers

This paper presents an algorithmic analysis of the busy period for the M/M/c queueing system. By setting the busy period equal to the time interval during which at least one server is busy, we develop a first step analysis which gives the Laplace-Stieltjes transform of the busy period as the solution of a finite system of linear equations. This approach is useful in obtaining a suitable recursive procedure for computing the moments of the length of a busy period and the number of customers served during it. The maximum entropy formalism is then used to analyse what is the influence of a given set of moments on the distribution of the busy period and to estimate the true busy period distribution. Our study supplements a recent work of Daley and Servi (1998) and other studies where the busy period of a multiserver queue follows a different definition, i.e., a busy period is the time interval during which all servers are engaged.

Distribution of the busy period in a controllable M/M/2 queue operating under the triadic (0, K, N, M) policy

1990 ◽  
Vol 27 (2) ◽  
pp. 425-432 ◽  
Author(s):  
Hahn-Kyou Rhee ◽  
B. D. Sivazlian
Keyword(s):  
Steady State ◽  
Busy Period ◽  
Queueing System ◽  
Service Completion ◽  
Special Cases ◽  
Service Stations ◽  
Gambler’S Ruin ◽  
Gambler's Ruin Problem ◽  
Number Of Customers ◽  
System Operating

We consider an M/M/2 queueing system with removable service stations operating under steady-state conditions. We assume that the number of operating service stations can be adjusted at customers' arrival or service completion epochs depending on the number of customers in the system. The objective of this paper is to obtain the distribution of the busy period using the theory of the gambler's ruin problem. As special cases, the distributions of the busy periods for the ordinary M/M/2 queueing system, the M/M/1 queueing system operating under the N policy and the ordinary M/M/1 queueing system are obtained.

Analysis of the busy period for the M/M/c queue: an algorithmic approach

2001 ◽  
Vol 38 (1) ◽  
pp. 209-222 ◽  
Author(s):  
J. R. Artalejo ◽  
M. J. Lopez-Herrero
Keyword(s):  
Busy Period ◽  
Queueing System ◽  
Linear Equations ◽  
Time Interval ◽  
System Of Linear Equations ◽  
Recursive Procedure ◽  
Algorithmic Approach ◽  
Period Distribution ◽  
Algorithmic Analysis ◽  
Number Of Customers

This paper presents an algorithmic analysis of the busy period for the M/M/c queueing system. By setting the busy period equal to the time interval during which at least one server is busy, we develop a first step analysis which gives the Laplace-Stieltjes transform of the busy period as the solution of a finite system of linear equations. This approach is useful in obtaining a suitable recursive procedure for computing the moments of the length of a busy period and the number of customers served during it. The maximum entropy formalism is then used to analyse what is the influence of a given set of moments on the distribution of the busy period and to estimate the true busy period distribution. Our study supplements a recent work of Daley and Servi (1998) and other studies where the busy period of a multiserver queue follows a different definition, i.e., a busy period is the time interval during which all servers are engaged.

scholarly journals M/M/1 Multiple Vacation Queueing Systems with Differentiated Vacations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Oliver C. Ibe ◽  
Olubukola A. Isijola
Keyword(s):  
Steady State ◽  
Busy Period ◽  
Queueing System ◽  
Queueing Systems ◽  
Steady State Solution ◽  
State Solution ◽  
Multiple Vacation ◽  
Service Times

We consider a multiple vacation queueing system in which a vacation following a busy period has a different distribution from a vacation that is taken without serving at least one customer. For ease of analysis it is assumed that the service times are exponentially distributed and the two vacation types are also exponentially distributed but with different means. The steady-state solution is obtained.

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