scholarly journals A second look at a queueing system with moving average input process

1965 ◽  
Vol 5 (1) ◽  
pp. 100-106 ◽  
Author(s):  
P. D. Finch ◽  
C. Pearce
Keyword(s):  
Queueing System ◽  
Moving Average ◽  
Time Interval ◽  
Single Server ◽  
Input Process ◽  
Second Look ◽  
Queue Discipline ◽  
Image Position

We consider a single-server queueing system with first-come first-served queue discipline in which (i) customers arrive at the instants 0 = A0 < A1 < A2 < …, with time interval between the mth and (m+1)th arrivals

The departure process from a queueing system

1976 ◽  
Vol 80 (2) ◽  
pp. 283-285 ◽  
Author(s):  
F. P. Kelly
Keyword(s):  
Queueing System ◽  
Arrival Process ◽  
Single Server ◽  
Departure Process ◽  
Common Distribution ◽  
Poisson Arrival ◽  
Queue Discipline ◽  
Common Distribution Function ◽  
Image Position ◽  
Poisson Arrival Process

Consider a single-server queueing system with a Poisson arrival process at rate λ and positive service requirements independently distributed with common distribution function B(z) and finite expectationwhere βλ < 1, i.e. an M/G/1 system. When the queue discipline is first come first served, or last come first served without pre-emption, the stationary departure process is Poisson if and only if G = M (i.e. B(z) = 1 − exp (−z/β)); see (8), (4) and (2). In this paper it is shown that when the queue discipline is last come first served with pre-emption the stationary departure process is Poisson whatever the form of B(z). The method used is adapted from the approach of Takács (10) and Shanbhag and Tambouratzis (9).

scholarly journals The single server queueing system with Non-recurrent input-process and Erlang service time

1963 ◽  
Vol 3 (2) ◽  
pp. 220-236 ◽  
Author(s):  
P. D. Finch
Keyword(s):  
Service Time ◽  
Queueing System ◽  
Single Server ◽  
Input Process ◽  
Service Times ◽  
Image Position

We consider a single server queueing system in which customers arrive at the instants t0, t1, …, tm, …. We write τm = tm+1 − tm, m ≧ 0. There is a single server with distribution of service times B(x) given by where k is an integer not less than unity.

The single-server queue with service depending on queue size and with the preemptive-resume last-come–first-served queue discipline

1987 ◽  
Vol 24 (03) ◽  
pp. 758-767
Author(s):  
D. Fakinos
Keyword(s):  
Queueing System ◽  
Limiting Distribution ◽  
Single Server ◽  
Queue Size ◽  
Single Server Queue ◽  
Preemptive Resume ◽  
Server Queue ◽  
Queue Discipline ◽  
Service Times

This paper studies theGI/G/1 queueing system assuming that customers have service times depending on the queue size and also that they are served in accordance with the preemptive-resume last-come–first-served queue discipline. Expressions are given for the limiting distribution of the queue size and the remaining durations of the corresponding services, when the system is considered at arrival epochs, at departure epochs and continuously in time. Also these results are applied to some particular cases of the above queueing system.

scholarly journals Concavity of the throughput of tandem queueing systems with finite buffer storage space

1990 ◽  
Vol 22 (03) ◽  
pp. 764-767 ◽  
Author(s):  
Ludolf E. Meester ◽  
J. George Shanthikumar
Keyword(s):  
Queueing System ◽  
Queueing Systems ◽  
Sample Path ◽  
Concave Function ◽  
Time Interval ◽  
Single Server ◽  
Finite Buffer ◽  
Buffer Storage ◽  
Unlimited Supply ◽  
Number Of Customers

We consider a tandem queueing system with m stages and finite intermediate buffer storage spaces. Each stage has a single server and the service times are independent and exponentially distributed. There is an unlimited supply of customers in front of the first stage. For this system we show that the number of customers departing from each of the m stages during the time interval [0, t] for any t ≧ 0 is strongly stochastically increasing and concave in the buffer storage capacities. Consequently the throughput of this tandem queueing system is an increasing and concave function of the buffer storage capacities. We establish this result using a sample path recursion for the departure processes from the m stages of the tandem queueing system, that may be of independent interest. The concavity of the throughput is used along with the reversibility property of tandem queues to obtain the optimal buffer space allocation that maximizes the throughput for a three-stage tandem queue.

scholarly journals A queueing system with moving average input process and batch arrivals

1965 ◽  
Vol 5 (4) ◽  
pp. 434-442 ◽  
Author(s):  
C. Pearce
Keyword(s):  
Queueing System ◽  
Moving Average ◽  
Limiting Distribution ◽  
Input Process ◽  
Batch Arrivals

In a recent paper by P. D. Finch and myself [1], the solution for the limiting distribution of a moving average queueing system was obtained. In this paper the system is generalised to the case of batch arrivals in batches of size ρ > 1.

Single-server queues with impatient customers

1984 ◽  
Vol 16 (4) ◽  
pp. 887-905 ◽  
Author(s):  
F. Baccelli ◽  
P. Boyer ◽  
G. Hebuterne
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Distribution Functions ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Input Process ◽  
Poisson Input ◽  
Virtual Waiting Time ◽  
Random Threshold ◽  
The Stability

We consider a single-server queueing system in which a customer gives up whenever his waiting time is larger than a random threshold, his patience time. In the case of a GI/GI/1 queue with i.i.d. patience times, we establish the extensions of the classical GI/GI/1 formulae concerning the stability condition and the relation between actual and virtual waiting-time distribution functions. We also prove that these last two distribution functions coincide in the case of a Poisson input process and determine their common law.

A note on the queueing system GI/Ek/1

1969 ◽  
Vol 6 (3) ◽  
pp. 708-710 ◽  
Author(s):  
P. D. Finch
Keyword(s):  
Time Distribution ◽  
Queueing System ◽  
Single Server ◽  
Service Time Distribution ◽  
Alternative Procedure ◽  
Single Server Queue ◽  
Server Queue ◽  
Image Position ◽  
The Right ◽  
Similar Evaluation

In this note we adopt the notation and terminology of Kingman (1966) without further comment. For the general single server queue one has For the queueing system GI/Ek/1 it is possible to make use of the particular nature of the service time distribution to evaluate the right-hand side of Equation (1) in terms of the k roots of a certain equation. This evaluation is carried out in detail in Prabhu (1965) to which reference should be made for the technicalities involved. A similar evaluation applies to the limiting distribution when it exists. However, the resulting expression again involves the k roots of a certain equation. In this note we draw attention to an alternative procedure which does not involve the calculation of roots. We remark that a similar, but slightly different, procedure can be used in the study of the queueing system Ek/GI/1. Details of this will be presented in a separate note.

scholarly journals Concavity of the throughput of tandem queueing systems with finite buffer storage space

1990 ◽  
Vol 22 (3) ◽  
pp. 764-767 ◽  
Author(s):  
Ludolf E. Meester ◽  
J. George Shanthikumar
Keyword(s):  
Queueing System ◽  
Queueing Systems ◽  
Sample Path ◽  
Concave Function ◽  
Time Interval ◽  
Single Server ◽  
Finite Buffer ◽  
Buffer Storage ◽  
Unlimited Supply ◽  
Number Of Customers

We consider a tandem queueing system with m stages and finite intermediate buffer storage spaces. Each stage has a single server and the service times are independent and exponentially distributed. There is an unlimited supply of customers in front of the first stage. For this system we show that the number of customers departing from each of the m stages during the time interval [0, t] for any t ≧ 0 is strongly stochastically increasing and concave in the buffer storage capacities. Consequently the throughput of this tandem queueing system is an increasing and concave function of the buffer storage capacities. We establish this result using a sample path recursion for the departure processes from the m stages of the tandem queueing system, that may be of independent interest. The concavity of the throughput is used along with the reversibility property of tandem queues to obtain the optimal buffer space allocation that maximizes the throughput for a three-stage tandem queue.

The single-server queue with service depending on queue size and with the preemptive-resume last-come–first-served queue discipline

1987 ◽  
Vol 24 (3) ◽  
pp. 758-767 ◽  
Author(s):  
D. Fakinos
Keyword(s):  
Queueing System ◽  
Limiting Distribution ◽  
Single Server ◽  
Queue Size ◽  
Single Server Queue ◽  
Preemptive Resume ◽  
Server Queue ◽  
Queue Discipline ◽  
Service Times

This paper studies the GI/G/1 queueing system assuming that customers have service times depending on the queue size and also that they are served in accordance with the preemptive-resume last-come–first-served queue discipline. Expressions are given for the limiting distribution of the queue size and the remaining durations of the corresponding services, when the system is considered at arrival epochs, at departure epochs and continuously in time. Also these results are applied to some particular cases of the above queueing system.

Limit theorems for the single server queue with traffic intensity one

1970 ◽  
Vol 7 (01) ◽  
pp. 227-233 ◽  
Author(s):  
N. U. Prabhu
Keyword(s):  
Limit Theorems ◽  
Random Variables ◽  
Single Server ◽  
Traffic Intensity ◽  
Arrival Times ◽  
Scale Parameters ◽  
Server Queue ◽  
Queue Discipline ◽  
Image Position ◽  
Queueing Processes

We consider a single server queueing system with inter-arrival times {un , n ≧ 1}, and service times {υn , n ≧ 1} and the queue-discipline, ‘first-come, first-served’. It is assumed that {un } and {υn } are two independent renewal processes, and 0 &gt; E(un ) =a &lt; ∞, 0 &lt; E(υn ) = b &lt; ∞. The traffic intensity is P ρ b/a(0 &gt; ρ &gt; ∞). This paper is concerned with the case ρ = 1, where it is known that the various queueing processes such as the queue-length Q(t) and waiting time W(t) diverge to + ∞ in distribution as t → ∞. Borovkov [1], [2] and Brody [3] have obtained limit distributions for Q(t) and W(t) with appropriate location and scale parameters in the cases P ≧ 1. Here we investigate random variables related to the busy and idle periods in the system. To explain our approach, we consider the random variables Xn = υn – un (n ≧ 1). Let S 0 ≡ 0, Sn = X 1 + X 2 + ··· + Xn (n ≧ 1), and define the sequence {Nk , k ≧ 0} as

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