Limit theorems for periodic queues

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.

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

Journal of Applied Probability ◽

10.1017/s0021900200023913 ◽

1983 ◽

Vol 20 (03) ◽

pp. 675-688 ◽

Cited By ~ 2

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.

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Assembly-like queues

Journal of Applied Probability ◽

10.2307/3212352 ◽

1973 ◽

Vol 10 (2) ◽

pp. 354-367 ◽

Cited By ~ 90

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.

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Assembly-like queues

Journal of Applied Probability ◽

10.1017/s0021900200095358 ◽

1973 ◽

Vol 10 (02) ◽

pp. 354-367 ◽

Cited By ~ 15

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.

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

Journal of Applied Probability ◽

10.2307/3213902 ◽

1983 ◽

Vol 20 (3) ◽

pp. 675-688 ◽

Cited By ~ 14

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.

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The general bulk queue as a matrix factorisation problem of the Wiener-Hopf type. Part II.

Advances in Applied Probability ◽

10.2307/1426133 ◽

1975 ◽

Vol 7 (3) ◽

pp. 647-655 ◽

Cited By ~ 4

Author(s):

John Dagsvik

Keyword(s):

Waiting Time ◽

Service Time ◽

Time Distribution ◽

Single Server ◽

Service Time Distribution ◽

Time Process ◽

Waiting Time Process ◽

Bulk Queue ◽

Erlang Distributions ◽

Matrix Factorisation

In a previous paper (Dagsvik (1975)) the waiting time process of the single server bulk queue is considered and a corresponding waiting time equation is established. In this paper the waiting time equation is solved when the inter-arrival or service time distribution is a linear combination of Erlang distributions. The analysis is essentially based on algebraic arguments.

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On queues with periodic Poisson input

Journal of Applied Probability ◽

10.1017/s0021900200034239 ◽

1981 ◽

Vol 18 (04) ◽

pp. 889-900 ◽

Cited By ~ 9

Author(s):

Austin J. Lemoine

Keyword(s):

Waiting Time ◽

Direct Proof ◽

Poisson Model ◽

Fixed Time ◽

Time Of Day ◽

Asymptotic Distributions ◽

Single Server ◽

Service Time Distribution ◽

Asymptotic Results ◽

Poisson Input

This paper is concerned with asymptotic results for a single-server queue having periodic Poisson input and general service-time distribution, and carries forward the analysis of this model initiated in Harrison and Lemoine. First, it is shown that a theorem of Hooke relating the stationary virtual and actual waiting-time distributions for the GI/G/1 queue extends to the periodic Poisson model; it is then pointed out that Hooke's theorem leads to the extension (developed in [3]) of a related theorem of Takács. Second, it is demonstrated that the asymptotic distribution for the server-load process at a fixed ‘time of day' coincides with the distribution for the supremum, over the time horizon [0,∞), of the sum of a stationary compound Poisson process with negative drift and a continuous periodic function. Some implications of this characterization result for the computation and approximation of the asymptotic distributions are then discussed, including a direct proof, for the periodic Poisson case, of a recent result of Rolski comparing mean asymptotic customer waiting time with that of a corresponding M/G/1 system.

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

Journal of Applied Probability ◽

10.2307/3214127 ◽

1986 ◽

Vol 23 (1) ◽

pp. 185-192 ◽

Cited By ~ 18

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.

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Waiting time and workload in queues with periodic Poisson input

Journal of Applied Probability ◽

10.1017/s0021900200027376 ◽

1989 ◽

Vol 26 (02) ◽

pp. 390-397 ◽

Cited By ~ 2

Author(s):

Austin J. Lemoine

Keyword(s):

Differential Equation ◽

Waiting Time ◽

Arrival Rate ◽

Time Of Day ◽

Single Server ◽

Single Server Queue ◽

Poisson Input ◽

Service Distribution ◽

Server Queue ◽

General Service

This paper develops moment formulas for asymptotic workload and waiting time in a single-server queue with periodic Poisson input and general service distribution. These formulas involve the corresponding moments of waiting-time (workload) for the M/G/1 system with the same average arrival rate and service distribution. In certain cases, all the terms in the formulas can be computed exactly, including moments of workload at each ‘time of day.' The approach makes use of an asymptotic version of the Takács [12] integro-differential equation, together with representation results of Harrison and Lemoine [3] and Lemoine [6].

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The general bulk queue as a matrix factorisation problem of the Wiener-Hopf type. Part I.

Advances in Applied Probability ◽

10.1017/s0001867800040830 ◽

1975 ◽

Vol 7 (03) ◽

pp. 636-646

Author(s):

John Dagsvik

Keyword(s):

Waiting Time ◽

Single Server ◽

System Of Equations ◽

Time Process ◽

Operator Equations ◽

Algebraic Setting ◽

Waiting Time Process ◽

Bulk Queue ◽

The Relationship ◽

Matrix Factorisation

The relationship between the Wiener-Hopf factorisation of matrices and the solution of systems of certain operator equations is discussed in an algebraic setting. It is shown that the study of the waiting time process of the nth arriving group of the general single server bulk queue leads to equations of that type. This system of equations may be considered as an extension of Lindley's waiting-time equation.

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