An alternative to Wiener-Hopf methods for the study of bounded processes

1964 ◽  
Vol 1 (1) ◽  
pp. 85-120 ◽  
Author(s):  
J. Keilson
Keyword(s):  
Waiting Time ◽  
Finite Interval ◽  
Time Process ◽  
First Passage ◽  
Discrete Parameter ◽  
Waiting Time Process ◽  
Image Position ◽  
Wald's Identity ◽  
Independent Identically Distributed ◽  
The Continuum

Homogeneous additive processes on a finite or semi-infinite interval have been studied in many forms. Wald's identity for the first passage process on the finite interval (see for example Miller, 1961), the waiting time process of Lindley (1952), and a variety of problems in the theory of queues, dams, and inventories come to mind. These processes have been treated by and large by methods in the complex plane. Lindley's discrete parameter process on the continuum, for example, described by where the ξn are independent identically distributed random variables, has been discussed by Wiener-Hopf methods in recent years by Lindley (1952), Smith (1953), Kemperman (1961), Keilson (1961), and many others. A review of earlier studies is given by Kemperman.

An alternative to Wiener-Hopf methods for the study of bounded processes

1964 ◽  
Vol 1 (01) ◽  
pp. 85-120 ◽  
Author(s):  
J. Keilson
Keyword(s):  
Waiting Time ◽  
Finite Interval ◽  
Time Process ◽  
First Passage ◽  
Discrete Parameter ◽  
Waiting Time Process ◽  
Image Position ◽  
Wald's Identity ◽  
Independent Identically Distributed ◽  
The Continuum

Homogeneous additive processes on a finite or semi-infinite interval have been studied in many forms. Wald's identity for the first passage process on the finite interval (see for example Miller, 1961), the waiting time process of Lindley (1952), and a variety of problems in the theory of queues, dams, and inventories come to mind. These processes have been treated by and large by methods in the complex plane. Lindley's discrete parameter process on the continuum, for example, described by where the ξ n are independent identically distributed random variables, has been discussed by Wiener-Hopf methods in recent years by Lindley (1952), Smith (1953), Kemperman (1961), Keilson (1961), and many others. A review of earlier studies is given by Kemperman.

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 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.

A Correlated Queue

1969 ◽  
Vol 6 (01) ◽  
pp. 122-136 ◽  
Author(s):  
B.W. Conolly ◽  
N. Hadidi
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Busy Period ◽  
The State ◽  
Queueing Model ◽  
Time Process ◽  
Waiting Time Process ◽  
The Subject ◽  
Service Pattern ◽  
Arrival Pattern

A “correlated queue” is defined to be a queueing model in which the arrival pattern influences the service pattern or vice versa. A particular model of this nature is considered in this paper. It is such that the service time of a customer is directly proportional to the interval between his own arrival and that of his predecessor. The initial busy period, state and output processes are analyzed in detail. For completeness, a sketch is also given of the analysis of the waiting time process which forms the subject of another paper. The results are used in the analysis of the state and output processes.

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, ∞).

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.

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.

A Correlated Queue

1969 ◽  
Vol 6 (1) ◽  
pp. 122-136 ◽  
Author(s):  
B.W. Conolly ◽  
N. Hadidi
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Busy Period ◽  
The State ◽  
Queueing Model ◽  
Time Process ◽  
Waiting Time Process ◽  
The Subject ◽  
Service Pattern ◽  
Arrival Pattern

A “correlated queue” is defined to be a queueing model in which the arrival pattern influences the service pattern or vice versa. A particular model of this nature is considered in this paper. It is such that the service time of a customer is directly proportional to the interval between his own arrival and that of his predecessor. The initial busy period, state and output processes are analyzed in detail. For completeness, a sketch is also given of the analysis of the waiting time process which forms the subject of another paper. The results are used in the analysis of the state and output processes.

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