Some applications of the theory of infinite capacity service systems to a single server system with linearly state dependent service

1971 ◽  
Vol 8 (1) ◽  
pp. 202-207 ◽  
Author(s):  
B. W. Conolly
Keyword(s):  
Queueing System ◽  
Service Systems ◽  
Single Server ◽  
Simple Manner ◽  
State Dependent ◽  
Poisson Arrivals ◽  
Negative Exponential ◽  
Server System

SummaryA certain single server queueing system with negative exponential service with mean rate nμ, when the system contains n customers, and Poisson arrivals, is formally equivalent to the infinite capacity system M/M/∞. This equivalence is exploited to yield in a very simple manner results for the single server system which were previously obtained by difficult analysis (see Hadidi (1969)).

Some applications of the theory of infinite capacity service systems to a single server system with linearly state dependent service

1971 ◽  
Vol 8 (01) ◽  
pp. 202-207
Author(s):  
B. W. Conolly
Keyword(s):  
Queueing System ◽  
Service Systems ◽  
Single Server ◽  
Simple Manner ◽  
State Dependent ◽  
Poisson Arrivals ◽  
Negative Exponential ◽  
Server System

Summary A certain single server queueing system with negative exponential service with mean rate nμ, when the system contains n customers, and Poisson arrivals, is formally equivalent to the infinite capacity system M/M/∞. This equivalence is exploited to yield in a very simple manner results for the single server system which were previously obtained by difficult analysis (see Hadidi (1969)).

scholarly journals Insensitive Bounds for the Moments of the Sojourn Times in M/GI Systems Under State-Dependent Processor Sharing

2010 ◽  
Vol 42 (01) ◽  
pp. 246-267 ◽  
Author(s):  
Andreas Brandt ◽  
Manfred Brandt
Keyword(s):  
Waiting Times ◽  
The State ◽  
Single Server ◽  
Processor Sharing ◽  
Actual Number ◽  
Service Capacity ◽  
State Dependent ◽  
Poisson Arrivals ◽  
Service Times ◽  
The Given

We consider a system with Poisson arrivals and independent and identically distributed service times, where requests in the system are served according to the state-dependent (Cohen's generalized) processor-sharing discipline, where each request receives a service capacity that depends on the actual number of requests in the system. For this system, we derive expressions as well as tight insensitive upper bounds for the moments of the conditional sojourn time of a request with given required service time. The bounds generalize and extend corresponding results, recently given for the single-server processor-sharing system in Cheung et al. (2006) and for the state-dependent processor-sharing system with exponential service times by the authors (2008). Analogous results hold for the waiting times. Numerical examples for the M/M/m-PS and M/D/m-PS systems illustrate the given bounds.

scholarly journals Stochastic analysis of the departure and quasi-input processes in a versatile single-server queue

1996 ◽  
Vol 9 (2) ◽  
pp. 171-183 ◽  
Author(s):  
J. R. Artalejo ◽  
A. Gomez-Corral
Keyword(s):  
Markov Chain ◽  
Stochastic Analysis ◽  
Queueing System ◽  
Single Server ◽  
Structural Form ◽  
Single Server Queue ◽  
Server Queue ◽  
Negative Exponential ◽  
Orbit Size ◽  
Negative Arrivals

This paper is concerned with the stochastic analysis of the departure and quasi-input processes of a Markovian single-server queue with negative exponential arrivals and repeated attempts. Our queueing system is characterized by the phenomenon that a customer who finds the server busy upon arrival joins an orbit of unsatisfied customers. The orbiting customers form a queue such that only a customer selected according to a certain rule can reapply for service. The intervals separating two successive repeated attempts are exponentially distributed with rate α+jμ, when the orbit size is j≥1. Negative arrivals have the effect of killing some customer in the orbit, if one is present, and they have no effect otherwise. Since customers can leave the system without service, the structural form of type M/G/1 is not preserved. We study the Markov chain with transitions occurring at epochs of service completions or negative arrivals. Then we investigate the departure and quasi-input processes.

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.

A STATE-DEPENDENT POLLING MODEL WITH k-LIMITED SERVICE

2009 ◽  
Vol 23 (2) ◽  
pp. 385-408 ◽  
Author(s):  
E. M. M. Winands ◽  
I. J. B. F. Adan ◽  
G. J. van Houtum ◽  
D. G. Down
Keyword(s):  
Priority Queue ◽  
Service Discipline ◽  
Single Server ◽  
Polling Model ◽  
State Dependent ◽  
Production Environments ◽  
Poisson Arrivals ◽  
Queue Model ◽  
Sojourn Time Distributions ◽  
Limited Service

We consider a two-queue model with state-dependent setups, in which a single server alternately serves the two queues. The high-priority queue is served exhaustively, whereas the low-priority queue is served according to the k-limited strategy. A setup at a queue is incurred only if there are customers waiting at the polled queue. We obtain the transforms of the queue length and sojourn time distributions under the assumption of Poisson arrivals, generally distributed service times, and generally distributed setup times. The interest for this model is fueled by an application in the field of logistics. It is shown how the results of this analysis can be applied in the evaluation of a stochastic two-item single-capacity production system. From these results we can conclude that significant cost reductions are possible by bounding the production runs of the low-priority item, which indicates the potential of the k-limited service discipline as priority rule in production environments.

The generalised state-dependent queue: the busy period Erlangian

1974 ◽  
Vol 11 (03) ◽  
pp. 618-623
Author(s):  
B. W. Conolly
Keyword(s):  
Laplace Transform ◽  
Generating Function ◽  
Busy Period ◽  
Continued Fraction ◽  
Queueing System ◽  
Joint Probability ◽  
Single Server ◽  
System State ◽  
State Dependent ◽  
The Laplace Transform

A continued fraction representation is presented of the Laplace transform of the generating function of the fundamental joint probability and density of busy period length measured in customers served and duration in time. The setting is the single server Erlang queueing system where the parameters of negative exponentially distributed arrival and service times have a general dependence on instantaneous system state.

scholarly journals Insensitive Bounds for the Moments of the Sojourn Times in M/GI Systems Under State-Dependent Processor Sharing

2010 ◽  
Vol 42 (1) ◽  
pp. 246-267 ◽  
Author(s):  
Andreas Brandt ◽  
Manfred Brandt
Keyword(s):  
Waiting Times ◽  
The State ◽  
Single Server ◽  
Processor Sharing ◽  
Actual Number ◽  
Service Capacity ◽  
State Dependent ◽  
Poisson Arrivals ◽  
Service Times ◽  
The Given

We consider a system with Poisson arrivals and independent and identically distributed service times, where requests in the system are served according to the state-dependent (Cohen's generalized) processor-sharing discipline, where each request receives a service capacity that depends on the actual number of requests in the system. For this system, we derive expressions as well as tight insensitive upper bounds for the moments of the conditional sojourn time of a request with given required service time. The bounds generalize and extend corresponding results, recently given for the single-server processor-sharing system in Cheung et al. (2006) and for the state-dependent processor-sharing system with exponential service times by the authors (2008). Analogous results hold for the waiting times. Numerical examples for the M/M/m-PS and M/D/m-PS systems illustrate the given bounds.

scholarly journals MX/G/1 Vacation Queueing System with Two Types of Repair facilities and Server Timeout

2019 ◽  
Vol 8 (9) ◽  
pp. 2040-2043
Keyword(s):  
Size Distribution ◽  
Exponential Distribution ◽  
Service Time ◽  
Queueing System ◽  
Batch Size ◽  
Single Server ◽  
Poisson Arrivals ◽  
Server Failure ◽  
System Length ◽  
General Service

We consider a single server vacation queue with two types of repair facilities and server timeout. Here customers are in compound Poisson arrivals with general service time and the lifetime of the server follows an exponential distribution. The server find if the system is empty, then he will wait until the time ‘c’. At this time if no one customer arrives into the system, then the server takes vacation otherwise the server commence the service to the arrived customers exhaustively. If the system had broken down immediately, it is sent for repair. Here server failure can be rectified in two case types of repair facilities, case1, as failure happens during customer being served willstays in service facility with a probability of 1-q to complete the remaining service and in case2 it opts for new service also who joins in the head of the queue with probability q. Obtained an expression for the expected system length for different batch size distribution and also numerical results are shown

scholarly journals A transient solution to an M/M/1 queue: a simple approach

1987 ◽  
Vol 19 (04) ◽  
pp. 997-998 ◽  
Author(s):  
P. R. Parthasarathy
Keyword(s):  
Queueing System ◽  
Simple Approach ◽  
Time Dependent ◽  
Single Server ◽  
Transient Solution ◽  
Poisson Arrivals ◽  
Service Times ◽  
Time Dependent Solution

A time-dependent solution for the number in a single-server queueing system with Poisson arrivals and exponential service times is derived in a direct way.

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