On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1

1980 ◽  
Vol 17 (3) ◽  
pp. 802-813 ◽  
Author(s):  
A. De Meyer ◽  
J. L. Teugels
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Service Time ◽  
Busy Period ◽  
Queueing System ◽  
Traffic Intensity

For the distribution function of the busy period in the M/G/l queueing system with traffic intensity less than one it is shown that the tail varies regularly at infinity iff the tail of the service time varies regularly at infinity.

On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1

1980 ◽  
Vol 17 (03) ◽  
pp. 802-813 ◽  
Author(s):  
A. De Meyer ◽  
J. L. Teugels
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Service Time ◽  
Busy Period ◽  
Queueing System ◽  
Traffic Intensity

For the distribution function of the busy period in the M/G/l queueing system with traffic intensity less than one it is shown that the tail varies regularly at infinity iff the tail of the service time varies regularly at infinity.

On a property of a refusals stream

1997 ◽  
Vol 34 (03) ◽  
pp. 800-805 ◽  
Author(s):  
Vyacheslav M. Abramov
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Busy Period ◽  
Elementary Proof ◽  
Queueing System ◽  
Queueing Systems ◽  
Single Server ◽  
Wide Family

This paper consists of two parts. The first part provides a more elementary proof of the asymptotic theorem of the refusals stream for an M/GI/1/n queueing system discussed in Abramov (1991a). The central property of the refusals stream discussed in the second part of this paper is that, if the expectations of interarrival and service time of an M/GI/1/n queueing system are equal to each other, then the expectation of the number of refusals during a busy period is equal to 1. This property is extended for a wide family of single-server queueing systems with refusals including, for example, queueing systems with bounded waiting time.

Characterization for the queueing system M/G/∞

1973 ◽  
Vol 74 (1) ◽  
pp. 141-143 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Distribution Function ◽  
Time Distribution ◽  
Queueing System ◽  
Arrival Process ◽  
Service Time Distribution ◽  
Traffic Intensity ◽  
Infinitely Divisible ◽  
Waiting Room ◽  
Room Size ◽  
Arrival Intensity

Consider a queueing system M/G/s with the arrival intensity λ, the service time distribution function B(t) (B(0) < 1) having a finite mean and the waiting room size N ≤ ∞. If s < ∞ and N = ∞, we shall also assume that its relative traffic intensity is less than 1. Since the arrival process of this system is Poisson, it is immediate that in this case the distribution of the number of arrivals during an interval is infinitely divisible.

Queues with semi-Markovian arrivals

1967 ◽  
Vol 4 (02) ◽  
pp. 365-379 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
Finite Number ◽  
Waiting Time ◽  
Busy Period ◽  
Queueing System ◽  
Interarrival Time ◽  
Single Server ◽  
Queue Size

A queueing system with a single server is considered. There are a finite number of types of customers, and the types of successive arrivals form a Markov chain. Further, the nth interarrival time has a distribution function which may depend on the types of the nth and the n–1th arrivals. The queue size, waiting time, and busy period processes are investigated. Both transient and limiting results are given.

Functional limit theorems for the queue GI/G/1 in light traffic

1971 ◽  
Vol 3 (02) ◽  
pp. 269-281 ◽  
Author(s):  
Donald L. Iglehart
Keyword(s):  
Service Time ◽  
Limit Theorems ◽  
Queueing System ◽  
Deterministic System ◽  
Traffic Intensity ◽  
Random Vectors ◽  
Functional Limit ◽  
Light Traffic ◽  
Functional Limit Theorems ◽  
Natural Measure

We consider a single GI/G/1 queueing system in which customer number 0 arrives at time t 0 = 0, finds a free server, and experiences a service time v 0. The nth customer arrives at time t n and experiences a service time v n . Let the interarrival times t n - t n-1 = u n , n ≧ 1, and define the random vectors X n = (v n-1, u n ), n ≧ 1. We assume the sequence of random vectors {X n : n ≧ 1} is independent and identically distributed (i.i.d.). Let E{u n } = λ-1 and E{v n } = μ-1, where 0 &lt; λ, μ &lt; ∞. In addition, we shall always assume that E{v 0 2} &lt; ∞ and that the deterministic system in which both v n and u n are degenerate is excluded. The natural measure of congestion for this system is the traffic intensity ρ = λ/μ. In this paper we shall restrict our attention to systems in which ρ &lt; 1. Under this condition, which we shall refer to as light traffic, our system is of course stable.

Functional limit theorems for the queue GI/G/1 in light traffic

1971 ◽  
Vol 3 (2) ◽  
pp. 269-281 ◽  
Author(s):  
Donald L. Iglehart
Keyword(s):  
Service Time ◽  
Limit Theorems ◽  
Queueing System ◽  
Deterministic System ◽  
Traffic Intensity ◽  
Random Vectors ◽  
Functional Limit ◽  
Light Traffic ◽  
Functional Limit Theorems ◽  
Natural Measure

We consider a single GI/G/1 queueing system in which customer number 0 arrives at time t0 = 0, finds a free server, and experiences a service time v0. The nth customer arrives at time tn and experiences a service time vn. Let the interarrival times tn - tn-1 = un, n ≧ 1, and define the random vectors Xn = (vn-1, un), n ≧ 1. We assume the sequence of random vectors {Xn : n ≧ 1} is independent and identically distributed (i.i.d.). Let E{un} = λ-1 and E{vn} = μ-1, where 0 < λ, μ < ∞. In addition, we shall always assume that E{v02} < ∞ and that the deterministic system in which both vn and un are degenerate is excluded. The natural measure of congestion for this system is the traffic intensity ρ = λ/μ. In this paper we shall restrict our attention to systems in which ρ < 1. Under this condition, which we shall refer to as light traffic, our system is of course stable.

A note on the queueing system GI/G/∞

1968 ◽  
Vol 64 (2) ◽  
pp. 477-479 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Distribution Function ◽  
Service Time ◽  
Queueing System ◽  
Expected Value ◽  
Arrival Times ◽  
Service Times

Consider a queueing system GI/G/∞ in which (i) the inter-arrival times are distributed with distribution function A(t) (A(O +) = 0) (ii) the service times have distribution function B(t) such that the expected value of the service time is β(>∞).

A note on losses in M/GI/1/n queues

1999 ◽  
Vol 36 (4) ◽  
pp. 1240-1243 ◽  
Author(s):  
Rhonda Righter
Keyword(s):  
Simple Proof ◽  
Service Time ◽  
Busy Period ◽  
Queueing System ◽  
Interarrival Time ◽  
The Mean

Let Ln be the number of losses during a busy period of an M/GI/1/n queueing system. We develop a coupling between Ln and Ln+1 and use the resulting relationship to provide a simple proof that when the mean service time equals the mean interarrival time, ELn = 1 for all n. We also show that Ln is increasing in the convex sense when the mean service time equals the mean interarrival time, and it is increasing in the increasing convex sense when the mean service time is less than the mean interarrival time.

scholarly journals On Distributions of One Class of Random Sums and their Applications

2020 ◽  
Vol 8 (1) ◽  
pp. 153-164
Author(s):  
Ivan Matsak ◽  
Mikhail Moklyachuk
Keyword(s):  
Distribution Function ◽  
Busy Period ◽  
Queueing System ◽  
Single Server ◽  
Redundant System ◽  
Random Sums ◽  
Sums Of Random Variables ◽  
Geiger Muller Counter ◽  
First Moment ◽  
Event Occurrence

We propose results of the investigation of properties of the random sums of random variables. We consider the case, where the number of summands is the first moment of an event occurrence. An integral equation is presented that determines distributions of random sums. With the help of the obtained results we analyse the distribution function of the time during which the Geiger-Muller counter will not lose any particles, the distribution function of the busy period of a redundant system with renewal, and the distribution function of the sojourn times of a single-server queueing system.

scholarly journals Bounds for the variance of the busy period of the M/G/∞ queue

1984 ◽  
Vol 16 (4) ◽  
pp. 929-932 ◽  
Author(s):  
M. F. Ramalhoto
Keyword(s):  
Distribution Function ◽  
Lower Bounds ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Random Variable ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Service Time Distribution ◽  
Lower And Upper Bounds

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.

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