The steady state of a multi-server mixed queue

1973 ◽  
Vol 5 (03) ◽  
pp. 614-631 ◽  
Author(s):  
N. B. Slater ◽  
T. C. T. Kotiah
Keyword(s):  
Steady State ◽  
Statistical Mechanics ◽  
Generating Functions ◽  
Queueing System ◽  
Linear Equations ◽  
Weighted Sum ◽  
System Of Linear Equations ◽  
Different Types ◽  
Multi Server ◽  
Steady State Probabilities

In a multi-server queueing system in which the customers are of several different types, it is useful to define states which specify the types of customers being served as well as the total number present. Analogies with some problems in statistical mechanics are found fruitful. Certain generating functions are defined in such a way that they satisfy a system of linear equations. Solution of the associated eigenvector problem shows that the steady-state probabilities for states in which all the servers are busy can be represented by a weighted sum of geometric probabilities.

The steady state of a multi-server mixed queue

1973 ◽  
Vol 5 (3) ◽  
pp. 614-631 ◽  
Author(s):  
N. B. Slater ◽  
T. C. T. Kotiah
Keyword(s):  
Steady State ◽  
Statistical Mechanics ◽  
Generating Functions ◽  
Queueing System ◽  
Linear Equations ◽  
Weighted Sum ◽  
System Of Linear Equations ◽  
Different Types ◽  
Multi Server ◽  
Steady State Probabilities

In a multi-server queueing system in which the customers are of several different types, it is useful to define states which specify the types of customers being served as well as the total number present. Analogies with some problems in statistical mechanics are found fruitful. Certain generating functions are defined in such a way that they satisfy a system of linear equations. Solution of the associated eigenvector problem shows that the steady-state probabilities for states in which all the servers are busy can be represented by a weighted sum of geometric probabilities.

scholarly journals M/M/1 Queueing system with Customer Balking and Reneging

2019 ◽  
Vol 8 (9) ◽  
pp. 2636-2646
Keyword(s):  
Steady State ◽  
Performance Measures ◽  
Generating Functions ◽  
Queueing System ◽  
Probability Generating Functions ◽  
Multiple Vacation ◽  
Steady State Probabilities ◽  
Explicit Expressions

This paper deals with an M/M/1 queueing system with customer balking and reneging. Balking and reneging of the customers are assumed to occur due to non-availability of the server during vacation and breakdown periods. Steady state probabilities for both the single and multiple vacation scenarios are obtained by employing probability generating functions. We evaluate the explicit expressions for various performance measures of the queueing system.

scholarly journals An analysis of steady-state distribution in M/M/1 queueing system with balking based on concept of statistical mechanics

2019 ◽  
Author(s):  
IKUO ARIZONO ◽  
Yasuhiko Takemoto
Keyword(s):  
Steady State ◽  
Statistical Mechanics ◽  
Queue Length ◽  
Queueing System ◽  
Analysis Model ◽  
Steady State Distribution ◽  
Steady State Analysis ◽  
Extended Analysis ◽  
The Moment ◽  
State Analysis

The phenomenon of balking has been considered frequently in the steady-state analysis of the M/M/1 queueing system. Balking means the phenomenon that a customer who arrives at a queueing system leaves without joining a queue, since he/she is disgusted with the waiting queue length at the moment of his/her arrival. In the traditional studies for the steady-state analysis of the M/M/1 queueing system with balking, it has been typically assumed that the arrival rates obey an inverse proportional function for the waiting queue length. In this study, based on the concept of the statistical mechanics, we have a challenge to extend the traditional steady-state analysis model for the M/M/1 queueing system with balking. As the result, we have defined an extended analysis model for the M/M/1 queueing system under the consideration of the change in the directivity strength of balking. In addition, the procedure for estimating the strength of balking in this analysis model using the observed data in the M/M/1 queueing system has been also constructed.

Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts

1987 ◽  
Vol 24 (2) ◽  
pp. 486-494 ◽  
Author(s):  
Thomas Hanschke
Keyword(s):  
Steady State ◽  
Generating Functions ◽  
Explicit Formulas ◽  
Steady State Probabilities

In this paper we study the M/M/2/2 queue with repeated attempts. It is shown that the part generating functions of the steady state probabilities can be expressed in of generalized hypergeometric unctions.

scholarly journals On a bulk queueing system with impatient customers

1998 ◽  
Vol 3 (6) ◽  
pp. 539-554 ◽  
Author(s):  
Lotfi Tadj ◽  
Lakdere Benkherouf ◽  
Lakhdar Aggoun
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Queueing System ◽  
Impatient Customers ◽  
Ergodicity Condition ◽  
Bulk Service ◽  
Bulk Arrival ◽  
Steady State Probabilities ◽  
System Characteristics

We consider a bulk arrival, bulk service queueing system. Customers are served in batches ofrunits if the queue length is not less thanr. Otherwise, the server delays the service until the number of units in the queue reaches or exceeds levelr. We assume that unserved customers may get impatient and leave the system. An ergodicity condition and steady-state probabilities are derived. Various system characteristics are also computed.

Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts

1987 ◽  
Vol 24 (02) ◽  
pp. 486-494 ◽  
Author(s):  
Thomas Hanschke
Keyword(s):  
Steady State ◽  
Generating Functions ◽  
Explicit Formulas ◽  
Steady State Probabilities

In this paper we study the M/M/2/2 queue with repeated attempts. It is shown that the part generating functions of the steady state probabilities can be expressed in of generalized hypergeometric unctions.

scholarly journals Markov chains with quasitoeplitz transition matrix: applications

1990 ◽  
Vol 3 (2) ◽  
pp. 141-152
Author(s):  
A. M. Dukhovny
Keyword(s):  
Steady State ◽  
Markov Chains ◽  
Generating Functions ◽  
Transition Matrix ◽  
Queueing Systems ◽  
Warm Up ◽  
Hitting Probabilities ◽  
Steady State Probabilities ◽  
Transient And Steady State

Application problems are investigated for the Markov chains with quasitoeplitz transition matrix. Generating functions of transient and steady state probabilities, first zero hitting probabilities and mean times are found for various particular cases, corresponding to some known patterns of feedback ( “warm-up,” “switch at threshold” etc.), Level depending dams and queue-depending queueing systems of both M/G/1 and MI/G/1 types with arbitrary random sizes of arriving and departing groups are studied.

scholarly journals Rhombic alternative tableaux, assemblees of permutations, and the ASEP

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Olya Mandelshtam ◽  
Xavier Viennot
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Generating Functions ◽  
Finite Lattice ◽  
One Dimensional ◽  
Open Boundaries ◽  
Bijective Proof ◽  
Insertion Algorithm ◽  
International Audience ◽  
Steady State Probabilities

International audience In this paper, we introduce therhombic alternative tableaux, whose weight generating functions providecombinatorial formulae to compute the steady state probabilities of the two-species ASEP. In the ASEP, there aretwo species of particles, oneheavyand onelight, on a one-dimensional finite lattice with open boundaries, and theparametersα,β, andqdescribe the hopping probabilities. The rhombic alternative tableaux are enumerated by theLah numbers, which also enumerate certainassembl ́ees of permutations. We describe a bijection between the rhombicalternative tableaux and these assembl ́ees. We also provide an insertion algorithm that gives a weight generatingfunction for the assemb ́ees. Combined, these results give a bijective proof for the weight generating function for therhombic alternative tableaux.

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