The c — Server Queue with Constant Service Times and a Versatile Markovian Arrival Process

1982 ◽  
pp. 31-70 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Markovian Arrival Process ◽  
Arrival Process ◽  
Server Queue ◽  
Service Times ◽  
Constant Service Times

The infinite server queue with semi-Markovian arrivals and negative exponential services

1972 ◽  
Vol 9 (1) ◽  
pp. 178-184 ◽  
Author(s):  
Marcel F. Neuts ◽  
Shun-Zer Chen
Keyword(s):  
Discrete Time ◽  
Infinite Number ◽  
Queue Length ◽  
Markovian Arrival Process ◽  
Arrival Process ◽  
Infinite Server ◽  
Server Queue ◽  
Service Times ◽  
Negative Exponential ◽  
Stationary Probabilities

The queue with an infinite number of servers with a semi-Markovian arrival process and with negative exponential service times is studied. The queue length process and the type of the last customer to join the queue before time t are studied jointly, both in continuous and in discrete time. Limiting stationary probabilities are also obtained.

The infinite server queue with semi-Markovian arrivals and negative exponential services

1972 ◽  
Vol 9 (01) ◽  
pp. 178-184 ◽  
Author(s):  
Marcel F. Neuts ◽  
Shun-Zer Chen
Keyword(s):  
Discrete Time ◽  
Infinite Number ◽  
Queue Length ◽  
Markovian Arrival Process ◽  
Arrival Process ◽  
Infinite Server ◽  
Server Queue ◽  
Service Times ◽  
Negative Exponential ◽  
Stationary Probabilities

The queue with an infinite number of servers with a semi-Markovian arrival process and with negative exponential service times is studied. The queue length process and the type of the last customer to join the queue before time t are studied jointly, both in continuous and in discrete time. Limiting stationary probabilities are also obtained.

The continuity of the single server queue

1972 ◽  
Vol 9 (02) ◽  
pp. 370-381 ◽  
Author(s):  
Douglas P. Kennedy
Keyword(s):  
Metric Spaces ◽  
Arrival Process ◽  
Single Server ◽  
Implicit Assumption ◽  
Single Server Queue ◽  
Virtual Waiting Time ◽  
Convergence Of Probability Measures ◽  
Server Queue ◽  
Poisson Arrivals ◽  
Service Times

In many applications of queueing theory assumptions of either Poisson arrivals or exponential service times are made. The implicit assumption is that if the actual arrival process approximates a Poisson process and the service times are close to exponential, then the quantities of interest in the real queueing system (viz. the virtual waiting time, queue length, idle times, etc.), will approximate those of the idealized model. The continuity of the single server queue acting as functionals of the arrival and service processes is established. The proof involves an application of the theory of weak convergence of probability measures on metric spaces.

Bounds for the expected delays in some tandem queues

1980 ◽  
Vol 17 (03) ◽  
pp. 831-838 ◽  
Author(s):  
Shun-Chen Niu
Keyword(s):  
Upper Bound ◽  
Renewal Process ◽  
Arrival Process ◽  
Tandem Queues ◽  
Service Distribution ◽  
Deterministic Service ◽  
Service Times ◽  
Constant Service Times ◽  
General Service

Tandem queues are analyzed. An upper bound for the stationary expected delay in front of the second server is found for a sequence of two queues in tandem where the first server has deterministic service times, the second server has general service distribution, and the arrival process is an arbitrary renewal process. The result is extended to the case of n queues in tandem where all the servers except the last one have constant service times.

Limiting diffusion approximations for the many server queue and the repairman problem

1965 ◽  
Vol 2 (02) ◽  
pp. 429-441 ◽  
Author(s):  
Donald L. Iglehart
Keyword(s):  
Transition Probabilities ◽  
Point Of View ◽  
Death Process ◽  
Arrival Process ◽  
Service Discipline ◽  
Birth And Death Processes ◽  
Queueing Process ◽  
Server Queue ◽  
Service Times ◽  
Number Of Customers

We shall consider a many server (multiple channels in parallel) queueing process in which customers arrive at the queue according to a Poisson process. The service times are assumed to be independent and exponentially distributed. As usual, the service times are independent of the arrival process. We assume that no server is idle if there is a customer waiting, but that otherwise the service discipline is arbitrary. If there are n servers and we let Xn (t) denote the number of customers waiting or being served at time t, then it is well known that Xn (t) is a birth and death process with stationary transition probabilities. A very comprehensive analysis of this many server queue from the point of view of birth and death processes has been carried out by Karlin and McGregor [5].

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