R
EVUE FRANÇAISE D’
AUTOMATIQUE,
INFORMATIQUE,
RECHERCHE OPÉRATIONNELLE
. R
ECHERCHE OPÉRATIONNELLEE. J. V ANDERPERRE
On the busy period for the M|G|1 queue with-finite waiting room
Revue française d’automatique, informatique, recherche opé- rationnelle. Recherche opérationnelle, tome 8, noV1 (1974), p. 41-44
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
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ON THE BUSY PERIOD FOR THE M|G|1 QUEUE WITH FINITE WAITING ROOM
by E. J, VÂNDERPERRE (*)
Summary. — The Laplace-Stieltjes transform (L.5.T.) of the probability distribution function of the busy period for the M\G 1 queueing System with finite waiting room can be
obtained by using a supplementary variable method (a Markov characterization).
In the present paper > it is shown that applying this method, it is rather easy to analyse queueing Systems with a finite waiting room and with Poisson input.
1. INTRODUCTION
For the well known model of the M|G|l queue with finite waiting room, Cohen [1] obtained the L.S.T. of the distribution function of the busy period in a rather simple way using a Markov renewal branehing argument.
In this paper, we use a simple method based on the inclusion of a supple- mentary variable (Markov characterization). Moreover, the method is not restricted to queueing Systems with Poisson input.
2. MATHEMATTCAL ANALYSES OF THE SYSTEM
Dénote by X"*1 the mean interarrivai time of customers and by F(.) the distribution function of the service times. Lét QK dénote the duration of a busy period of the M|G|l queue with K waiting places. Acustomer who finds ail waiting places occupied, cannot enter the system. He départs and never returns (overflow).
Moreover, let Xt be the number of customers waiting in the system at any instant of time t and xt the past service time of the customer being served at time t. It is assumed that a customer arrives at time / = 0 and that he meets an empty system.
(1) Département de sciences économiques, Université de Hasselt, Belgique.
Revue Française d^Automatique, Informatique et Recherché Opérationnelle n° jan. 1974, V-l.
42 E. J. VANDERPERRE
The busy period 0* is the time between t = 0 and the first instant at which the system becomes empty (the absorbing state).
For i = 0, 1,..., K and /, x > 0 we define
x)dx M Pr[Xt = i,x^xt<x + dx\xo = 0]
By simple probabilistic arguments, we have for i = 0, ls . . . K— 1
- x, 0 - F(x)}
P& x) - £" Pj(t - x, 0 +) { 1 - E e-*" ^ } [1 - F(*)]
The Laplace transform with respect to t of pt (f, x) will be denoted by pï(s9 x) ; Re s > 0.
If we consider the events in which xt = 0 +5 then we obtain the following set of récurrence relations for the boundary values of the functions pf (s, x) ;
(i)
£
o+)= 2- Pîfco
where Sof is the Kronecker symbol.
Let
/•oc
m * Jo e-«e-x*
i+l-J
Res ^ 0
i> 0
The L.S.T. of the probability distribution function of the busy period 6K will be denoted by E { c~s^ }.
Clearly
(2) E {*-**}= Ms)pS(s9O+)
Revue Française d'Automatique, Informatique et Recherche Opérationnelle
The foUowing column K-vectors will be used.
f = [1,0, ...,0, ...,0]T
and the K x K matrix function M>) Ms) 0 0 Ms) Ms) Ms) 0
Ms) M') À® Ms
P(s) =
o
fi+1(s) ƒ,(*) ƒ,_!(*) . . . M*) Ms)
where
By (1) we obtain
0 0 0
0 0 0
0 0 0
oo
ao(s)
where / is the unit matrix.
It can be shown that the'matrix [ 7 ~ P ( r ) ]90 < Re 5 < 00, is nonsingular.
For AT > 0 we define det [7—P(s)] = AK(^) and A0(*y) = 1. By (2) and Cramer5s rule, we have
(3)
But (4)
Let (5)
E { e" & } = I.
;~ '
n» janvier 1974, V-l.
4 4 E. J. VANDERPERRE
By (4) and (5) we obtain after some simplifications
(6) YK(S) =J E * fj(s)YK+i-j(s) + aK.t(s) K>-\
Let
CO Y ( * Z ) « * S YK(S)ZK \Z\ < \9(s)\
where ç>(s) is the smallest root of the f unctional équation Z —f(s + X —XZ) = 0 Re s ^ 0.
By (6), (7) and algebra, we find
If O dénote a circle with center at the origin of the complex Z-plane and with radius \z\ < |pO)|, Re s > 0, then by (3), (5) and (8) we obtain for K= 1,2, . . . R e j ^ 0,
ï - {ï -M } ö^f ^ ï ( i - zy'iz-fis + E {e-4* } = füJ?
i — {î —fis) }^^=n — zf
1[z—fis + x — xz)]"
1It is easy to show that the result agrées with Cohen's Theorem [1] p. 825.
REFERENCES
[1] COHEN J. W., On the busy periods for the M\G\1 queue withfinite and with infinité waiting room, J. Appl. Prob., 8 (1971), 821-827.
[2] VANDERPERRE E. J., The[busy period ofa repairman attending a n + 1 unit parallel System, R.AJ.R.O, V-2 (1973), 124-126.
Revue Française d'Automatique, Informatique et Recherche Opérationnelle n° jan. 1974, V-l.
