The virtual waiting-time and related processes

1986 ◽  
Vol 18 (02) ◽  
pp. 558-573 ◽  
Author(s):  
D. R. Cox ◽  
Valerie Isham
Keyword(s):  
Waiting Time ◽  
Arrival Rate ◽  
Transient Behaviour ◽  
Decay Properties ◽  
State Dependent ◽  
Continuous State Space ◽  
Virtual Waiting Time ◽  
Continuous State ◽  
Waiting Time Process ◽  
Properties Of Soil

The virtual waiting-time process of Takács is one of the simplest examples of a stochastic process with a continuous state space in continuous time in which jump transitions interrupt periods of deterministic decay. Properties of the process are reviewed, and the transient behaviour examined in detail. Several generalizations of the process are studied. These include two-sided jumps, periodically varying ‘arrival’ rate and the presence of a state-dependent decay rate; the last case is motivated by the properties of soil moisture in hydrology. Throughout, the emphasis is on the derivation of simple interpretable results.

The virtual waiting-time and related processes

1986 ◽  
Vol 18 (2) ◽  
pp. 558-573 ◽  
Author(s):  
D. R. Cox ◽  
Valerie Isham
Keyword(s):  
Waiting Time ◽  
Arrival Rate ◽  
Transient Behaviour ◽  
Decay Properties ◽  
State Dependent ◽  
Continuous State Space ◽  
Virtual Waiting Time ◽  
Continuous State ◽  
Waiting Time Process ◽  
Properties Of Soil

The virtual waiting-time process of Takács is one of the simplest examples of a stochastic process with a continuous state space in continuous time in which jump transitions interrupt periods of deterministic decay. Properties of the process are reviewed, and the transient behaviour examined in detail. Several generalizations of the process are studied. These include two-sided jumps, periodically varying ‘arrival’ rate and the presence of a state-dependent decay rate; the last case is motivated by the properties of soil moisture in hydrology. Throughout, the emphasis is on the derivation of simple interpretable results.

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.

The stable M/G/1 queue in heavy traffic and its covariance function

1977 ◽  
Vol 9 (01) ◽  
pp. 169-186 ◽  
Author(s):  
Teunis J. Ott
Keyword(s):  
Brownian Motion ◽  
Waiting Time ◽  
Covariance Function ◽  
Busy Period ◽  
Stationary Process ◽  
Heavy Traffic ◽  
Time Process ◽  
Period Distribution ◽  
Virtual Waiting Time ◽  
Waiting Time Process

Let X(t) be the virtual waiting-time process of a stable M/G/1 queue. Let R(t) be the covariance function of the stationary process X(t), B(t) the busy-period distribution of X(t); and let E(t) = P{X(t) = 0|X(0) = 0}. For X(t) some heavy-traffic results are given, among which are limiting expressions for R(t) and its derivatives and for B(t) and E(t). These results are used to find the covariance function of stationary Brownian motion on [0, ∞).

Complementary generating functions for the MX/GI/1/k and GI/My/1/K queues and their application to the comparison of loss probabilities

1990 ◽  
Vol 27 (3) ◽  
pp. 684-692 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Length Distribution ◽  
Direct Proof ◽  
Interarrival Time ◽  
Convex Order ◽  
Queue Length Distribution ◽  
Virtual Waiting Time ◽  
Loss Probabilities ◽  
Waiting Time Process

A direct proof is presented for the fact that the stationary system queue length distribution just after the service completion epochs in the Mx/GI/1/k queue is given by the truncation of a measure on Z+ = {0, 1, ·· ·}. The related truncation formulas are well known for the case of the traffic intensity ρ < 1 and for the virtual waiting time process in M/GI/1 with a limited waiting time (Cohen (1982) and Takács (1974)). By the duality of GI/MY/1/k to Mx/GI/1/k + 1, we get a similar result for the system queue length distribution just before the arrival of a customer in GI/MY/1/k. We apply those results to prove that the loss probabilities of Mx/GI/1/k and GI/MY/1/k are increasing for the convex order of the service time and interarrival time distributions, respectively, if their means are fixed.

Limiting diffusions for the conditioned M/G/1 queue

1974 ◽  
Vol 11 (02) ◽  
pp. 355-362 ◽  
Author(s):  
Douglas P. Kennedy
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Traffic Intensity ◽  
Time Process ◽  
Brownian Excursion ◽  
Virtual Waiting Time ◽  
Waiting Time Process

The virtual waiting time process, W(t), in the M/G/1 queue is investigated under the condition that the initial busy period terminates but has not done so by time n ≥ t. It is demonstrated that, as n → ∞, W(t), suitably scaled and normed, converges to the unsigned Brownian excursion process or a modification of that process depending whether ρ ≠ 1 or ρ = 1, where ρ is the traffic intensity.

An algebraic approach to the waiting time process in GI/M/S

1973 ◽  
Vol 10 (01) ◽  
pp. 181-191 ◽  
Author(s):  
Jacqueline Loris-Teghem
Keyword(s):  
Waiting Time ◽  
Algebraic Approach ◽  
Markovian Process ◽  
Time Process ◽  
Two Dimensional ◽  
Transient Behaviour ◽  
Waiting Time Process ◽  
The Times

The transient behaviour of the waiting time process in GI/M/S is studied algebraically by means of a two-dimensional Markovian process {(vn, ln } , where the variables vn denote the times of full occupation immediately after the arrival instants Tn and where ln = max {0,– 1 + number of idle servers at (Tn + 0)}.

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.

scholarly journals On the virtual waiting time for an M/G/1 retrial queue with two types of calls

1993 ◽  
Vol 6 (1) ◽  
pp. 11-23 ◽  
Author(s):  
Bong Dae Choi ◽  
Dong Hwan Han ◽  
Guennadi Falin
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Retrial Queue ◽  
Arrival Rate ◽  
Switching System ◽  
Stieltjes Transform ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
Virtual Waiting Time ◽  
Incoming Call

We consider an M/G/1 retrial queueing system with two types of calls which models a telephone switching system. In the case that arriving calls are blocked due to the channel being busy, the outgoing calls are queued in priority group whereas the incoming calls enter the retrial group in order to try service again after a random amount of time. In this paper we find the Laplace-Stieltjes transform of the distribution of the virtual waiting time for an incoming call. When the arrival rate of outgoing calls is zero, it is shown that our result is consistent with the known result for a retrial queueing system with one type of call.

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