Conditioned limit theorems for waiting-time processes of the M/G/1 queue

1983 ◽  
Vol 20 (03) ◽  
pp. 675-688 ◽  
Author(s):  
G. Hooghiemstra
Keyword(s):  
Waiting Time ◽  
Limit Theorems ◽  
Random Variable ◽  
Weak Limit ◽  
Random Time ◽  
Time Process ◽  
Brownian Excursion ◽  
Time Index ◽  
Waiting Time Process ◽  
Number Of Customers

This paper is on conditioned weak limit theorems for imbedded waiting-time processes of an M/G/1 queue. More specifically we study functional limit theorems for the actual waiting-time process conditioned by the event that the number of customers in a busy period exceeds n or equals n. Attention is also paid to the actual waiting-time process with random time index. Combined with the existing literature on the subject this paper gives a complete account of the conditioned limit theorems for the actual waiting-time process of an M/G/1 queue for arbitrary traffic intensity and for a rather general class of service-time distributions. The limit processes that occur are Brownian excursion and meander, while in the case of random time index also the following limit occurs: Brownian excursion divided by an independent and uniform (0, 1) distributed random variable.

Conditioned limit theorems for waiting-time processes of the M/G/1 queue

1983 ◽  
Vol 20 (3) ◽  
pp. 675-688 ◽  
Author(s):  
G. Hooghiemstra
Keyword(s):  
Waiting Time ◽  
Limit Theorems ◽  
Random Variable ◽  
Weak Limit ◽  
Random Time ◽  
Time Process ◽  
Brownian Excursion ◽  
Time Index ◽  
Waiting Time Process ◽  
Number Of Customers

This paper is on conditioned weak limit theorems for imbedded waiting-time processes of an M/G/1 queue. More specifically we study functional limit theorems for the actual waiting-time process conditioned by the event that the number of customers in a busy period exceeds n or equals n. Attention is also paid to the actual waiting-time process with random time index.Combined with the existing literature on the subject this paper gives a complete account of the conditioned limit theorems for the actual waiting-time process of an M/G/1 queue for arbitrary traffic intensity and for a rather general class of service-time distributions.The limit processes that occur are Brownian excursion and meander, while in the case of random time index also the following limit occurs: Brownian excursion divided by an independent and uniform (0, 1) distributed random variable.

Limit theorems for periodic queues

1977 ◽  
Vol 14 (3) ◽  
pp. 566-576 ◽  
Author(s):  
J. Michael Harrison ◽  
Austin J. Lemoine
Keyword(s):  
Waiting Time ◽  
Limit Theorems ◽  
Arrival Rate ◽  
Random Variable ◽  
Rate Function ◽  
Asymptotic Distributions ◽  
Single Server ◽  
Time Process ◽  
Poisson Input ◽  
Waiting Time Process

Consider a single-server queue with service times distributed as a general random variable S and with non-stationary Poisson input. It is assumed that the arrival rate function λ (·) is periodic with average value λ and that ρ = λE(S) < 1. Both weak and strong limit theorems are proved for the waiting-time process W = {W1, W2, · ··} and the server load (or virtual waiting-time process) Z = {Z(t), t ≧ 0}. The asymptotic distributions associated with Z and W are shown to be related in various ways. In particular, we extend to the case of periodic Poisson input a well-known formula (due to Takács) relating the limiting virtual and actual waiting-time distributions of a GI/G/1 queue.

Limit theorems for periodic queues

1977 ◽  
Vol 14 (03) ◽  
pp. 566-576 ◽  
Author(s):  
J. Michael Harrison ◽  
Austin J. Lemoine
Keyword(s):  
Waiting Time ◽  
Limit Theorems ◽  
Arrival Rate ◽  
Random Variable ◽  
Rate Function ◽  
Asymptotic Distributions ◽  
Single Server ◽  
Time Process ◽  
Poisson Input ◽  
Waiting Time Process

Consider a single-server queue with service times distributed as a general random variable S and with non-stationary Poisson input. It is assumed that the arrival rate function λ (·) is periodic with average value λ and that ρ = λE(S) &lt; 1. Both weak and strong limit theorems are proved for the waiting-time process W = {W 1, W 2, · ··} and the server load (or virtual waiting-time process) Z = {Z(t), t ≧ 0}. The asymptotic distributions associated with Z and W are shown to be related in various ways. In particular, we extend to the case of periodic Poisson input a well-known formula (due to Takács) relating the limiting virtual and actual waiting-time distributions of a GI/G/1 queue.

On the waiting-time process in a single-line queue with repeated calls

1986 ◽  
Vol 23 (1) ◽  
pp. 185-192 ◽  
Author(s):  
G. I. Falin
Keyword(s):  
Waiting Time ◽  
Limit Theorems ◽  
Queueing System ◽  
Queueing Systems ◽  
Retrial Queue ◽  
Time Process ◽  
Light Traffic ◽  
Repeated Calls ◽  
Waiting Time Process ◽  
Mean Variance

Waiting time in a queueing system is usually measured by a period from the epoch when a subscriber enters the system until the service starting epoch. For repeated orders queueing systems it is natural to measure the waiting time by the number of repeated attempts, R, which have to be made by a blocked primary call customer before the call enters service. We study this problem for the M/M/1/1 retrial queue and derive expressions for mean, variance and generating function of R. Limit theorems are stated for heavy- and light-traffic cases.

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.

On the waiting-time process in a single-line queue with repeated calls

1986 ◽  
Vol 23 (01) ◽  
pp. 185-192 ◽  
Author(s):  
G. I. Falin
Keyword(s):  
Waiting Time ◽  
Limit Theorems ◽  
Queueing System ◽  
Queueing Systems ◽  
Retrial Queue ◽  
Time Process ◽  
Light Traffic ◽  
Repeated Calls ◽  
Waiting Time Process ◽  
Mean Variance

Waiting time in a queueing system is usually measured by a period from the epoch when a subscriber enters the system until the service starting epoch. For repeated orders queueing systems it is natural to measure the waiting time by the number of repeated attempts,R, which have to be made by a blocked primary call customer before the call enters service. We study this problem for theM/M/1/1 retrial queue and derive expressions for mean, variance and generating function ofR.Limit theorems are stated for heavy- and light-traffic cases.

Assembly-like queues

1973 ◽  
Vol 10 (2) ◽  
pp. 354-367 ◽  
Author(s):  
J. Michael Harrison
Keyword(s):  
Waiting Time ◽  
Limit Theorems ◽  
Theoretic Model ◽  
Single Server ◽  
Time Process ◽  
Single Server Queue ◽  
Multiple Input ◽  
Server Queue ◽  
Waiting Time Process ◽  
Load Conditions

A queueing theoretic model of an assembly operation is introduced. The model, consisting of K ≧ 2 renewal input processes and a single server, is a multiple input generalization of the GI/G/1 queue. The server requires one input item of each type k = 1,…, K for each of his services. It is shown that the model is inherently unstable in the following sense. The associated vector waiting time process Wn cannot converge in distribution to a non-defective limit, regardless of how well balanced the input and service processes may be. Limit theorems are developed for appropriately normalized versions of Wn under the various possible load conditions. Another waiting time process, equivalent to that in a single-server queue whose input is the minimum of K renewal processes, is also identified. It is shown to converge in distribution to a particular limit under certain load conditions.

Assembly-like queues

1973 ◽  
Vol 10 (02) ◽  
pp. 354-367 ◽  
Author(s):  
J. Michael Harrison
Keyword(s):  
Waiting Time ◽  
Limit Theorems ◽  
Theoretic Model ◽  
Single Server ◽  
Time Process ◽  
Single Server Queue ◽  
Multiple Input ◽  
Server Queue ◽  
Waiting Time Process ◽  
Load Conditions

A queueing theoretic model of an assembly operation is introduced. The model, consisting of K ≧ 2 renewal input processes and a single server, is a multiple input generalization of the GI/G/1 queue. The server requires one input item of each type k = 1,…, K for each of his services. It is shown that the model is inherently unstable in the following sense. The associated vector waiting time process Wn cannot converge in distribution to a non-defective limit, regardless of how well balanced the input and service processes may be. Limit theorems are developed for appropriately normalized versions of Wn under the various possible load conditions. Another waiting time process, equivalent to that in a single-server queue whose input is the minimum of K renewal processes, is also identified. It is shown to converge in distribution to a particular limit under certain load conditions.

Limiting diffusions for the conditioned M/G/1 queue

1974 ◽  
Vol 11 (2) ◽  
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.

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