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

1975 ◽  
Vol 7 (3) ◽  
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.

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

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.

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

1975 ◽  
Vol 7 (3) ◽  
pp. 647-655 ◽  
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.

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

1975 ◽  
Vol 7 (03) ◽  
pp. 647-655
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.

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.

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.

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.

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