Queues with random service output: the case of Poisson arrivals

1974 ◽  
Vol 11 (04) ◽  
pp. 771-784 ◽  
Author(s):  
Jacob Grinstein ◽  
Michael Rubinovitch
Keyword(s):  
Queueing Models ◽  
Time Interval ◽  
Single Server ◽  
Independent Increments ◽  
Virtual Waiting Time ◽  
Two Factors ◽  
Waiting Time Process ◽  
Poisson Arrivals ◽  
Busy Server ◽  
Independent Identically Distributed

A general class of single server queueing models is formulated. They distinguish between two factors that may influence the duration of service times: variability in the service requirements of customers, and variability (over time) in the service output of the server. Accordingly, we assume that the demands for service of successive customers form a sequence of independent, identically distributed random variables and that the amount of service produced by a busy server in a time interval is determined by the increment of a process with stationary independent increments over that interval. The results include the distribution of the busy period and the limiting distribution of the queue length. We also investigate the potential waiting process which is an extension of virtual waiting time process in existing queueing models.

Queues with random service output: the case of Poisson arrivals

1974 ◽  
Vol 11 (4) ◽  
pp. 771-784 ◽  
Author(s):  
Jacob Grinstein ◽  
Michael Rubinovitch
Keyword(s):  
Queueing Models ◽  
Time Interval ◽  
Single Server ◽  
Independent Increments ◽  
Virtual Waiting Time ◽  
Two Factors ◽  
Waiting Time Process ◽  
Poisson Arrivals ◽  
Busy Server ◽  
Independent Identically Distributed

A general class of single server queueing models is formulated. They distinguish between two factors that may influence the duration of service times: variability in the service requirements of customers, and variability (over time) in the service output of the server. Accordingly, we assume that the demands for service of successive customers form a sequence of independent, identically distributed random variables and that the amount of service produced by a busy server in a time interval is determined by the increment of a process with stationary independent increments over that interval. The results include the distribution of the busy period and the limiting distribution of the queue length. We also investigate the potential waiting process which is an extension of virtual waiting time process in existing queueing models.

On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals

1971 ◽  
Vol 8 (3) ◽  
pp. 494-507 ◽  
Author(s):  
E. K. Kyprianou
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Queueing System ◽  
Random Variable ◽  
Single Server ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Poisson Arrivals ◽  
Image Position ◽  
Quasi Stationary Distribution

We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt, and the service time has distribution dB(t), 0 < t < ∞. Let W(t) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable denote the first busy period initiated by a waiting time u at time t = 0.

On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals

1971 ◽  
Vol 8 (03) ◽  
pp. 494-507 ◽  
Author(s):  
E. K. Kyprianou
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Queueing System ◽  
Random Variable ◽  
Single Server ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Poisson Arrivals ◽  
Image Position ◽  
Quasi Stationary Distribution

We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt, and the service time has distribution dB(t), 0 &lt; t &lt; ∞. Let W(t) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable denote the first busy period initiated by a waiting time u at time t = 0.

scholarly journals Periodic queues in heavy traffic

1989 ◽  
Vol 21 (02) ◽  
pp. 485-487 ◽  
Author(s):  
G. I. Falin
Keyword(s):  
Wiener Process ◽  
Waiting Time ◽  
Diffusion Approximation ◽  
Queueing System ◽  
Heavy Traffic ◽  
Time Process ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Periodic Queues ◽  
Poisson Arrivals

An analytic approach to the diffusion approximation in queueing due to Burman (1979) is applied to the M(t)/G/1/∞ queueing system with periodic Poisson arrivals. We show that under heavy traffic the virtual waiting time process can be approximated by a certain Wiener process with reflecting barrier at 0.

A limit theorem for priority queues in heavy traffic

1973 ◽  
Vol 10 (04) ◽  
pp. 907-912 ◽  
Author(s):  
J. Michael Harrison
Keyword(s):  
Limit Theorem ◽  
Queueing System ◽  
Heavy Traffic ◽  
Critical Value ◽  
Single Server ◽  
Traffic Condition ◽  
Transient Distribution ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Priority Queueing

A single server, two priority queueing system is studied under the heavy traffic condition where the system traffic intensity is either at or near its critical value. An approximation is developed for the transient distribution of the low priority customers' virtual waiting time process. This result is stated formally as a limit theorem involving a sequence of systems whose traffic intensities approach the critical value.

A limit theorem for priority queues in heavy traffic

1973 ◽  
Vol 10 (4) ◽  
pp. 907-912 ◽  
Author(s):  
J. Michael Harrison
Keyword(s):  
Limit Theorem ◽  
Queueing System ◽  
Heavy Traffic ◽  
Critical Value ◽  
Single Server ◽  
Traffic Condition ◽  
Transient Distribution ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Priority Queueing

A single server, two priority queueing system is studied under the heavy traffic condition where the system traffic intensity is either at or near its critical value. An approximation is developed for the transient distribution of the low priority customers' virtual waiting time process. This result is stated formally as a limit theorem involving a sequence of systems whose traffic intensities approach the critical value.

scholarly journals Periodic queues in heavy traffic

1989 ◽  
Vol 21 (2) ◽  
pp. 485-487 ◽  
Author(s):  
G. I. Falin
Keyword(s):  
Wiener Process ◽  
Waiting Time ◽  
Diffusion Approximation ◽  
Queueing System ◽  
Heavy Traffic ◽  
Time Process ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Periodic Queues ◽  
Poisson Arrivals

An analytic approach to the diffusion approximation in queueing due to Burman (1979) is applied to the M(t)/G/1/∞ queueing system with periodic Poisson arrivals. We show that under heavy traffic the virtual waiting time process can be approximated by a certain Wiener process with reflecting barrier at 0.

Diffusion Approximation for Efficiency-Driven Queues When Customers Are Patient

2020 ◽  
Vol 68 (4) ◽  
pp. 1265-1284 ◽  
Author(s):  
Shuangchi He
Keyword(s):  
Diffusion Approximation ◽  
Single Server ◽  
Service Time Distribution ◽  
Service Levels ◽  
Infinite Dimensional ◽  
Virtual Waiting Time ◽  
Multiple Servers ◽  
Server Queue ◽  
Waiting Time Process ◽  
Functional Central Limit

The analysis of queues with multiple servers is typically challenging when the service time distribution is general. Such analysis usually involves an infinite-dimensional process for tracking service ages or residual service times. In “Diffusion Approximation for Efficiency-Driven Queues When Customers Are Patient,” He demonstrates from a macroscopic perspective that, if customers are relatively patient and the system is overloaded, the dynamics of a many-server queue could be as simple as the dynamics of a single-server queue. In particular, the virtual waiting time process can be captured by a one-dimensional diffusion process, which enables us to obtain simple formulas for performance measures, such as service levels and effective abandonment fractions. To justify this diffusion model, a functional central limit theorem is established for the superposition of stationary renewal processes.

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 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.

Sign in / Sign up

Export Citation Format

Share Document