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OR Spectrum

, Volume 34, Issue 1, pp 293–310 | Cite as

An analytical method for the calculation of the number of units at the arrival instant in a discrete time G/G/1-queueing system with batch arrivals

  • Marc Schleyer
Regular Article

Abstract

In this paper, we analyze a G/G/1-queueing system with batch arrivals in the discrete time domain. On the basis of the waiting time distribution, we present an analytical approach for the calculation of the number of units in the queue at the arrival instant. This approach is exact within an ε-environment. In addition, the computing times are low such that a huge set of computations can be performed in a short time. We close our paper with a set of numerical experiments. Our research is motivated through various applications in the field of logistics, where it is of important interest to know the distribution of the number of units waiting to be proceeded at the moment of a new arrival.

Keywords

Queueing systems Performance evaluation Batch arrivals Buffer dimensioning 

List of symbols

A

Random variable describing the inter-arrival time

ai

Probability that a realization of A is i time units (i = 1, . . . , a max)

Y

Random variable describing the batch size

yi

Probability that a realization of Y is i time units i (i = 1, . . . , y max)

B

Random variable describing the service time

bi

Probability that a realization of B is i time units i (i = 1, . . . , b max)

λ

Arrival rate

μ

Service rate

ρ

Utilization

W

Random variable describing the waiting time of an individual unit if units arrive singly

Wbatch

Random variable describing the waiting time of an individual unit if units arrive in batches

wbatch,i

Probability that a realization of W batch is i time units (i = 0, 1, . . .)

\({W_{{\rm batch}}^{{\rm I}}}\)

Random variable describing the waiting time of the whole batch

\({w_{{\rm batch},i}^{{\rm I}}}\)

Probability that a realization of \({W_{{\rm batch}}^{{\rm I}}}\) is i time units (i = 0, 1, . . .)

\({W_{{\rm batch}}^{{\rm II}}}\)

Random variable describing the waiting time of an individual unit during the service of the batch itself

\({w_{{\rm batch},i}^{{\rm II}}}\)

Probability that a realization of \({W_{{\rm batch}}^{{\rm II}}}\) is i time units (i = 0, 1, . . .)

N(t)

Random variable describing the number of units in the queue at time instant t

τn

Arrival instant of the nth unit

δn

Departure instant of the nth unit

N(τ)

Random variable describing the number of units in the queue at the arrival time instant

Wn

Random variable describing the waiting time of the nth unit if units arrive singly

Wbatch,n(l)

Random variable describing the waiting time of the nth unit depending on position l within its batch (l = 1, . . . , y max)

V(l)

Random variable describing the sojourn time of an individual unit depending on position l within its batch (l = 1, . . . , y max)

vi(l)

Probability that a realization of V(l) is i time units (i = 1, 2, . . .)

qm

Probability that an arbitrary chosen unit is element of a batch of size m (m = 1, . . . , y max)

ol

Probability that an arbitrary chosen unit is located at position l within its batch (l = 1, . . . , y max)

P(Q = m|O = l)

Probability that an arbitrary chosen unit is element of a batch of size m under the condition that this unit is at position l (m = 1, . . . , y max; l = 1, . . . , y max)

P(O = l|Q = m)

Probability that an arbitrary chosen unit is located at position l under the condition that this unit is element of a batch of size m (m = 1, . . . , y max; l = 1, . . . , y max)

\({\alpha^{n}_{k,i}(l)}\)

Probability that the time interval between τ n+k+1 and τ n is i time units depending on the position l of unit n (i = 1, 2, . . .; l = 1, . . . , y max)

\({\Phi^{n}_{k,r}(l)}\)

Probability that the time interval [τ n+k+1τ n ](l) is composed of r batch inter-arrival time intervals (r = 1, 2 . . .; l = 1, . . . , y max)

ηi

Probability that the number of units at the arrival instant is i (i = 0, 1, . . .)

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Institut für Fördertechnik und Logistiksysteme, Karlsruher Institut für TechnologieKarlsruheGermany

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