On the transient state probabilities for a queueing model where potential customers are discouraged by queue length

1974 ◽  
Vol 11 (02) ◽  
pp. 345-354 ◽  
Author(s):  
Bent Natvig
Keyword(s):  
Queue Length ◽  
Transient State ◽  
Exact Expression ◽  
Single Server ◽  
Exact Formulation ◽  
Queueing Process ◽  
State Dependent ◽  
Contemporary Work ◽  
Potential Customers ◽  
Low Traffic

Earlier work by Hadidi and Conolly and contemporary work by the author point to the great operational advantages of state-dependent queueing models. Let pin (t) be the state probabilities and p∗ in the corresponding L.T.'s relative to the single server birth-and-death queueing process with parameters λn = λ/(n + 1), n ≥ 0, μn = μ, n ≥ 1. We have obtained an exact formulation of p ∗ i0 , p ∗ in (n ≥ 1) being determined recursively. An exact expression for p 10(t) is given in the case of low traffic intensities, and this has been approximated efficiently. Numerical evaluations show that the steady-state is reached very rapidly.

On the transient state probabilities for a queueing model where potential customers are discouraged by queue length

1974 ◽  
Vol 11 (2) ◽  
pp. 345-354 ◽  
Author(s):  
Bent Natvig
Keyword(s):  
Queue Length ◽  
Transient State ◽  
Exact Expression ◽  
Single Server ◽  
Exact Formulation ◽  
Queueing Process ◽  
State Dependent ◽  
Contemporary Work ◽  
Potential Customers ◽  
Low Traffic

Earlier work by Hadidi and Conolly and contemporary work by the author point to the great operational advantages of state-dependent queueing models. Let pin (t) be the state probabilities and p∗in the corresponding L.T.'s relative to the single server birth-and-death queueing process with parameters λn = λ/(n + 1), n ≥ 0, μn = μ, n ≥ 1. We have obtained an exact formulation of p∗i0, p∗in (n ≥ 1) being determined recursively. An exact expression for p10(t) is given in the case of low traffic intensities, and this has been approximated efficiently. Numerical evaluations show that the steady-state is reached very rapidly.

scholarly journals A bulk queueing system under N-policy with bilevel service delay discipline and start-up time

1993 ◽  
Vol 6 (4) ◽  
pp. 359-384 ◽  
Author(s):  
David C. R. Muh
Keyword(s):  
Queue Length ◽  
Queueing System ◽  
Probability Generating Function ◽  
Single Server ◽  
Numerical Examples ◽  
Queueing Process ◽  
Poisson Input ◽  
Service Delay ◽  
Start Up ◽  
Bulk Arrival

The author studies the queueing process in a single-server, bulk arrival and batch service queueing system with a compound Poisson input, bilevel service delay discipline, start-up time, and a fixed accumulation level with control operating policy. It is assumed that when the queue length falls below a predefined level r(≥1), the system, with server capacity R, immediately stops service until the queue length reaches or exceeds the second predefined accumulation level N(≥r). Two cases, with N≤R and N≥R, are studied.The author finds explicitly the probability generating function of the stationary distribution of the queueing process and gives numerical examples.

On the multiserver queue with finite waiting room and controlled input

1985 ◽  
Vol 17 (2) ◽  
pp. 408-423 ◽  
Author(s):  
Jewgeni Dshalalow
Keyword(s):  
Markov Chain ◽  
Queue Length ◽  
Limiting Distribution ◽  
Embedded Markov Chain ◽  
Single Server ◽  
Simple Relationship ◽  
Waiting Room ◽  
Original Process ◽  
State Dependent ◽  
Server System

In this paper we study a multi-channel queueing model of type with N waiting places and a non-recurrent input flow dependent on queue length at the time of each arrival. The queue length is treated as a basic process. We first determine explicitly the limit distribution of the embedded Markov chain. Then, by introducing an auxiliary Markov process, we find a simple relationship between the limiting distribution of the Markov chain and the limiting distribution of the original process with continuous time parameter. Here we simultaneously combine two methods: solving the corresponding Kolmogorov system of the differential equations, and using an approach based on the theory of semi-regenerative processes. Among various applications of multi-channel queues with state-dependent input stream, we consider a closed single-server system with reserve replacement and state-dependent service, which turns out to be dual (in a certain sense) in relation to our model; an optimization problem is also solved, and an interpretation by means of tandem systems is discussed.

scholarly journals On a single-server queue with fixed accumulation level, state dependent service, and semi-Markov modulated input flow

1992 ◽  
Vol 15 (3) ◽  
pp. 593-600 ◽  
Author(s):  
Jewgeni H. Dshalalow ◽  
Gary Russell
Keyword(s):  
Single Server ◽  
Input Stream ◽  
Idle Period ◽  
Queueing Process ◽  
State Dependent ◽  
Special Cases ◽  
Server Queue ◽  
Semi Markov Process ◽  
The Mean ◽  
Markov Modulated

The authors study the queueing process in a single-server queueing system with state dependent service and with the input modulated by a semi-Markov process embedded in the queueing process. It is also assumed that the server capacity isr≥1and that any service act will not begin until the queue accumulates at leastrunits. In this model, therefore, idle periods also depend upon the queue length.The authors establish an ergodicity criterion for the queueing process and evaluate explicitly its stationary distribution and other characteristics of the system, such as the mean service cycle, intensity of the system, intensity of the input stream, distribution of the idle period, and the mean busy period. Various special cases are treated.

scholarly journals Transient analysis of a queue where potential customers are discouraged by queue length

2001 ◽  
Vol 7 (5) ◽  
pp. 433-454 ◽  
Author(s):  
P. R. Parthasarathy ◽  
N. Selvaraju
Keyword(s):  
Queue Length ◽  
Transient Analysis ◽  
Queueing System ◽  
Steady State Solution ◽  
Transient Solution ◽  
State Dependent ◽  
Transient Solutions ◽  
Natural Measure ◽  
The Mean ◽  
Potential Customers

The transient solution is obtained analytically using continued fractions for a state-dependent birth-death queue in which potential customers are discouraged by the queue length. This queueing system is then compared with the well-known infinite server queueing system which has the same steady state solution as the model under consideration, whereas their transient solutions are different. A natural measure of speed of convergence of the mean number in the system to its stationarity is also computed.

scholarly journals On a first passage problem in general queueing systems with multiple vacations

1992 ◽  
Vol 5 (2) ◽  
pp. 177-192 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Preliminary Analysis ◽  
Queueing System ◽  
Queueing Systems ◽  
Compound Poisson Process ◽  
Single Server ◽  
New Techniques ◽  
Queueing Process ◽  
Multiple Vacation ◽  
State Dependent ◽  
First Passage Problem

The author studies a generalized single-server queueing system with bulk arrivals and batch service, where the server takes vacations each time the queue level falls below r(≥1) in accordance with the multiple vacation discipline. The input to the system is assumed to be a compound Poisson process modulated by the system and the service is assumed to be state dependent. One of the essential part in the analysis of the system is the employment of new techniques related to the first excess level processes. A preliminary analysis of such processes and recent results of the author on modulated processes enabled the author to obtain all major characteristics for the queueing process explicitly. Various examples and applications are discussed.

The transient state probabilities for a queueing model where potential customers are discouraged by queue length

1981 ◽  
Vol 18 (2) ◽  
pp. 499-506 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
Queue Length ◽  
Transition Probabilities ◽  
Transient State ◽  
Death Process ◽  
Queueing Model ◽  
Potential Customers ◽  
Birth Death

Exact expressions are derived for the transition probabilities of the birth-death process with parameters and which serves as a queueing model where potential customers are discouraged by queue length.

The transient state probabilities for a queueing model where potential customers are discouraged by queue length

1981 ◽  
Vol 18 (02) ◽  
pp. 499-506 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
Queue Length ◽  
Transition Probabilities ◽  
Transient State ◽  
Death Process ◽  
Queueing Model ◽  
Potential Customers ◽  
Birth Death

Exact expressions are derived for the transition probabilities of the birth-death process with parametersandwhich serves as a queueing model where potential customers are discouraged by queue length.

On the multiserver queue with finite waiting room and controlled input

1985 ◽  
Vol 17 (02) ◽  
pp. 408-423 ◽  
Author(s):  
Jewgeni Dshalalow
Keyword(s):  
Markov Chain ◽  
Queue Length ◽  
Limiting Distribution ◽  
Embedded Markov Chain ◽  
Single Server ◽  
Simple Relationship ◽  
Waiting Room ◽  
Original Process ◽  
State Dependent ◽  
Server System

In this paper we study a multi-channel queueing model of type with N waiting places and a non-recurrent input flow dependent on queue length at the time of each arrival. The queue length is treated as a basic process. We first determine explicitly the limit distribution of the embedded Markov chain. Then, by introducing an auxiliary Markov process, we find a simple relationship between the limiting distribution of the Markov chain and the limiting distribution of the original process with continuous time parameter. Here we simultaneously combine two methods: solving the corresponding Kolmogorov system of the differential equations, and using an approach based on the theory of semi-regenerative processes. Among various applications of multi-channel queues with state-dependent input stream, we consider a closed single-server system with reserve replacement and state-dependent service, which turns out to be dual (in a certain sense) in relation to our model; an optimization problem is also solved, and an interpretation by means of tandem systems is discussed.

State-dependent signalling in queueing networks

1994 ◽  
Vol 26 (02) ◽  
pp. 436-455 ◽  
Author(s):  
W. Henderson ◽  
B. S. Northcote ◽  
P. G. Taylor
Keyword(s):  
Queueing Networks ◽  
State Dependence ◽  
Product Form ◽  
Batch Size ◽  
Single Server ◽  
Negative Customers ◽  
Affect Control ◽  
State Dependent ◽  
Special Cases ◽  
Equilibrium Distributions

It has recently been shown that networks of queues with state-dependent movement of negative customers, and with state-independent triggering of customer movement have product-form equilibrium distributions. Triggers and negative customers are entities which, when arriving to a queue, force a single customer to be routed through the network or leave the network respectively. They are ‘signals' which affect/control network behaviour. The provision of state-dependent intensities introduces queues other than single-server queues into the network. This paper considers networks with state-dependent intensities in which signals can be either a trigger or a batch of negative customers (the batch size being determined by an arbitrary probability distribution). It is shown that such networks still have a product-form equilibrium distribution. Natural methods for state space truncation and for the inclusion of multiple customer types in the network can be viewed as special cases of this state dependence. A further generalisation allows for the possibility of signals building up at nodes.

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