The continuity of the single server queue

1972 ◽  
Vol 9 (2) ◽  
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.

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.

The single server queue with Poisson input and semi-Markov service times

1966 ◽  
Vol 3 (1) ◽  
pp. 202-230 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Markov Process ◽  
Time Dependence ◽  
Stationary Solutions ◽  
Single Server ◽  
Single Server Queue ◽  
Poisson Input ◽  
Server Queue ◽  
Semi Markov Process ◽  
Poisson Arrivals ◽  
Service Times

We assume that the successive service times in a single server queue with Poisson arrivals form an m-state semi-Markov process.The results for the M/G/1 queue are extended to this case. Both the time-dependence and the stationary solutions are discussed.

On ross's conjectures about queues with non-stationary poisson arrivals

1982 ◽  
Vol 19 (01) ◽  
pp. 245-249 ◽  
Author(s):  
D. P. Heyman
Keyword(s):  
Poisson Process ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Average Delay ◽  
Server Queue ◽  
Poisson Arrivals

Ross (1978) conjectured that the average delay in a single-server queue is larger when the arrival process is a non-stationary Poisson process than when it is a stationary Poisson process with the same rate. We present an example where equality obtains. When the number of waiting-positions is finite, Ross conjectured that the proportion of lost customers is greater in the nonstationary case. We present a counterexample to this conjecture.

On ross's conjectures about queues with non-stationary poisson arrivals

1982 ◽  
Vol 19 (1) ◽  
pp. 245-249 ◽  
Author(s):  
D. P. Heyman
Keyword(s):  
Poisson Process ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Average Delay ◽  
Server Queue ◽  
Poisson Arrivals

Ross (1978) conjectured that the average delay in a single-server queue is larger when the arrival process is a non-stationary Poisson process than when it is a stationary Poisson process with the same rate. We present an example where equality obtains. When the number of waiting-positions is finite, Ross conjectured that the proportion of lost customers is greater in the nonstationary case. We present a counterexample to this conjecture.

A single-server queue with limited virtual waiting time

1974 ◽  
Vol 11 (03) ◽  
pp. 612-617 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Waiting Time ◽  
Single Server ◽  
Single Server Queue ◽  
Limiting Distributions ◽  
Poisson Input ◽  
Virtual Waiting Time ◽  
General Service Times ◽  
Server Queue ◽  
Service Times ◽  
General Service

The limiting distributions of the actual waiting time and the virtual waiting time are determined for a single-server queue with Poisson input and general service times in the case where there are two types of services and no customer can stay in the system longer than an interval of length m.

A single-server queue with limited virtual waiting time

1974 ◽  
Vol 11 (3) ◽  
pp. 612-617 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Waiting Time ◽  
Single Server ◽  
Single Server Queue ◽  
Limiting Distributions ◽  
Poisson Input ◽  
Virtual Waiting Time ◽  
General Service Times ◽  
Server Queue ◽  
Service Times ◽  
General Service

The limiting distributions of the actual waiting time and the virtual waiting time are determined for a single-server queue with Poisson input and general service times in the case where there are two types of services and no customer can stay in the system longer than an interval of length m.

The single server queue with Poisson input and semi-Markov service times

1966 ◽  
Vol 3 (01) ◽  
pp. 202-230 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Markov Process ◽  
Time Dependence ◽  
Stationary Solutions ◽  
Single Server ◽  
Single Server Queue ◽  
Poisson Input ◽  
Server Queue ◽  
Semi Markov Process ◽  
Poisson Arrivals ◽  
Service Times

We assume that the successive service times in a single server queue with Poisson arrivals form an m-state semi-Markov process. The results for the M/G/1 queue are extended to this case. Both the time-dependence and the stationary solutions are discussed.

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