scholarly journals Discussion on the transient behavior of single server Markovian multiple variant vacation queues

2021 ◽  
Vol 31 (1) ◽  
Author(s):  
Manickam Vadivukarasi ◽  
Kaliappan Kalidass
Keyword(s):  
Waiting Time ◽  
Transient Behavior ◽  
System Size ◽  
Time Dependent ◽  
Stochastic Decomposition ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Decomposition Structure ◽  
Important Time ◽  
Time Dependent Solution

In this paper, we consider an M/M/1 queue where beneficiary visits occur singly. Once the beneficiary level in the system becomes zero, the server takes a vacation immediately. If the server finds no beneficiaries in the system, then the server is allowed to take another vacation after the return from the vacation. This process continues until the server has exhaustively taken all the J vacations. The closed form transient solution of the considered model and some important time dependent performance measures are obtained. Further, the steady state system size distribution is obtained from the time-dependent solution. A stochastic decomposition structure of waiting time distribution and expression for the additional waiting time due to the presence of server vacations are studied. Numerical assessments are presented.

Convex ordering of the attained waiting times in single-server queues and related problems

1991 ◽  
Vol 28 (02) ◽  
pp. 433-445 ◽  
Author(s):  
Masakiyo Miyazawa ◽  
Genji Yamazaki
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Waiting Times ◽  
Service Discipline ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Convex Order ◽  
Convex Ordering ◽  
Single Server Queues ◽  
Service Disciplines

The attained waiting time of customers in service of the G/G/1 queue is compared for various work-conserving service disciplines. It is proved that the attained waiting time distribution is minimized (maximized) in convex order when the discipline is FCFS (PR-LCFS). We apply the result to characterize finiteness of moments of the attained waiting time in the GI/GI/1 queue with an arbitrary work-conserving service discipline. In this discussion, some interesting relationships are obtained for a PR-LCFS queue.

scholarly journals Time-Dependent Solution of a Full Hydrodynamic Model Including Convective Terms and Viscous Effect

VLSI Design ◽  
1998 ◽  
Vol 6 (1-4) ◽  
pp. 173-176 ◽  
Author(s):  
Deyin Xu ◽  
Ting-Wei Tang ◽  
Sergei S. Kucherenko
Keyword(s):  
Steady State ◽  
Hydrodynamic Model ◽  
Plasma Oscillation ◽  
Transient Behavior ◽  
Time Dependent ◽  
Viscous Effect ◽  
Turn On ◽  
Time Dependent Solution

Sub-picosecond turn-on transient behavior of ballistic diodes (N+ - N - N+ structures) is studied by solving a system of time-dependent hydrodynamic (HD) equations. Convective terms as well as viscous effect are included in the study. The simulation result indicates that the diode undergoes approximately one quarter of a plasma oscillation before it relaxes to the steady-state value through collisions.

scholarly journals Note sur un modele de file GI/G/1 a service autonome (avec vacances du serveur)

1987 ◽  
Vol 19 (1) ◽  
pp. 289-291 ◽  
Author(s):  
Christine Fricker
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Conditional Distribution ◽  
Stochastic Decomposition ◽  
Analytic Method ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Vacation Model ◽  
The Law

Keilson and Servi introduced in [5] a variation of a GI/G/1 queue with vacation, in which at the end of a service time, either the server is not idle, and he starts serving the first customer in the queue with probability p, or goes on vacation with probability 1 – p, or he is idle, and he takes a vacation. At the end of a vacation, either customers are present, and the server starts serving the first customer, or he is idle, and he takes a vacation. The case p = 1, called the GI/G/1/V queue, was studied analytically by Gelenbe and Iasnogorodski [3] (see also [4]) and then by Doshi [1] and Fricker [2] who obtained stochastic decomposition results on the waiting-time of the nth customer extending the law decomposition result of [3]. Keilson and Servi [5] give a more complete analytic method of treating both the GI/G/1/V model and the Bernoulli vacation model: instead of the waiting time, they use a bivariate process at the service and vacation initiation epochs and the waiting-time distribution is computed as a conditional distribution of the above. In this note the law decomposition result is obtained from a reduction to the GI/G/1/V model with a modified service-time distribution just using the waiting time, with simple path arguments so that by [1] and [2] stochastic decomposition results are valid, which extend the result of [5].

Single-server queues with impatient customers

1984 ◽  
Vol 16 (4) ◽  
pp. 887-905 ◽  
Author(s):  
F. Baccelli ◽  
P. Boyer ◽  
G. Hebuterne
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Distribution Functions ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Input Process ◽  
Poisson Input ◽  
Virtual Waiting Time ◽  
Random Threshold ◽  
The Stability

We consider a single-server queueing system in which a customer gives up whenever his waiting time is larger than a random threshold, his patience time. In the case of a GI/GI/1 queue with i.i.d. patience times, we establish the extensions of the classical GI/GI/1 formulae concerning the stability condition and the relation between actual and virtual waiting-time distribution functions. We also prove that these last two distribution functions coincide in the case of a Poisson input process and determine their common law.

Exponential spectra as a tool for the study of server-systems with several classes of customers

1978 ◽  
Vol 15 (01) ◽  
pp. 162-170 ◽  
Author(s):  
J. Keilson
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Heavy Traffic ◽  
Transient Behavior ◽  
Single Server ◽  
Idle State ◽  
Priority Systems ◽  
Completely Monotone ◽  
Server Systems ◽  
Server System

For a single-server system having several Poisson streams of customers with exponentially distributed service times, busy period densities, waiting time densities, and idle state probabilities are completely monotone. The exponential spectra for such densities are of importance for understanding the transient behavior of such systems. Algorithms are given for the computation of such spectra. Applications to heavy traffic situations and priority systems are also discussed.

scholarly journals An M/G/1 queue with multiple types of feedback and gated vacations

1997 ◽  
Vol 34 (03) ◽  
pp. 773-784 ◽  
Author(s):  
Onno J. Boxma ◽  
Uri Yechiali
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Queue Length ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Server Queue ◽  
Poisson Arrivals

This paper considers a single-server queue with Poisson arrivals and multiple customer feedbacks. If the first service attempt of a newly arriving customer is not successful, he returns to the end of the queue for another service attempt, with a different service time distribution. He keeps trying in this manner (as an ‘old' customer) until his service is successful. The server operates according to the ‘gated vacation' strategy; when it returns from a vacation to find K (new and old) customers, it renders a single service attempt to each of them and takes another vacation, etc. We study the joint queue length process of new and old customers, as well as the waiting time distribution of customers. Some extensions are also discussed.

Note sur un modele de file GI/G/1 a service autonome (avec vacances du serveur)

1987 ◽  
Vol 19 (01) ◽  
pp. 289-291
Author(s):  
Christine Fricker
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Conditional Distribution ◽  
Stochastic Decomposition ◽  
Analytic Method ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Vacation Model ◽  
The Law

Keilson and Servi introduced in [5] a variation of a GI/G/1 queue with vacation, in which at the end of a service time, either the server is not idle, and he starts serving the first customer in the queue with probability p, or goes on vacation with probability 1 – p, or he is idle, and he takes a vacation. At the end of a vacation, either customers are present, and the server starts serving the first customer, or he is idle, and he takes a vacation. The case p = 1, called the GI/G/1/V queue, was studied analytically by Gelenbe and Iasnogorodski [3] (see also [4]) and then by Doshi [1] and Fricker [2] who obtained stochastic decomposition results on the waiting-time of the nth customer extending the law decomposition result of [3]. Keilson and Servi [5] give a more complete analytic method of treating both the GI/G/1/V model and the Bernoulli vacation model: instead of the waiting time, they use a bivariate process at the service and vacation initiation epochs and the waiting-time distribution is computed as a conditional distribution of the above. In this note the law decomposition result is obtained from a reduction to the GI/G/1/V model with a modified service-time distribution just using the waiting time, with simple path arguments so that by [1] and [2] stochastic decomposition results are valid, which extend the result of [5].

Exponential expansion for the tail of the waiting-time probability in the single-server queue with batch arrivals

1988 ◽  
Vol 20 (4) ◽  
pp. 880-895 ◽  
Author(s):  
J. C. W. Van Ommeren
Keyword(s):  
Waiting Time ◽  
Interarrival Time ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Batch Arrivals ◽  
Exponential Expansion ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Server Queue

This paper deals with the single-server queue with batch arrivals. We show that under suitable conditions the waiting-time distribution of an individual customer has an asymptotically exponential expansion. Computationally useful characterizations of the amplitude factor and the decay parameter are given for the practically important case in which the interarrival time and the service time have phase-type distributions.

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