A.Wolisz,J.Izydorczyk
1. Introduction
Recently an increasing interest can be observed in system performance evaluation,equally for computer systems,production systems,transportation systems etc.Among tools utilized in sy
stem performance evaluation^the queueing theory approach can be pointed out as one of the moBt powerful and therefore this area of science is being extensively developed.
A
lot of new queueing models reflecting special features of investigated real - life systems have been studied.For example investigations in the area of computer systems performance evaluation gave reason for a rapid developement of priority queueing models and time-sharing models.This application, as well as industrial engineering and computer networks^ stimulated also a big affort in the area of multi - stage queueing systems and networks of queues.Among the relatively new,nonclassic service systems one can distinguish an interesting group of queueing models, with service time of every customer being dependent on waiting time spent in the queue by this very customer.Such an phenomenon occurs in numerous real-life situations.
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Ал example which animated the majority of investigations in this area is the material flow between steel-plant and
blooming mill in metalurgical industry.
Obtained in the LD- conversion process steel is poured into special moulds and left for some predetermined time period justified by metalurgical reasons in order to assure proper crystallization.After this period moulds are removed in a strip
ping bay,and ingots are transported to the soaking pits department in order to be heated.The heating should assure an / theoretically/
constant temperature in every point of the in g o t ,slightly lower than the melting temperature.This is necessary for assuring proper rolling conditions.
The stream of ingots as well as the heating duration has a stochastic nature,thus a queue of ingots waiting for charging into the pits forms in front of the soaking pits division.
Waiting ingots are subject to extensive heat losses and the heating duration is evidently dependent on waiting time.
As an example of such a dependence in the area of computer system performance evaluation one can point out ( c f j L I B U 7 4 7 ) flight control systems.In the case when a request for trajectory correction is w a i t i n g in the queue of jobs /perhaps identical in nature but issued by other planes/ it^s service time may be increased, as a need for either more sophisticated position calcu
lations or additional radar measurements will probably occurs.
In this p a p e r we shall present state of the art in thp area of queueing systems with waiting time dependent service times.
Additionally we shall give some examples illustrating special features of such systems.The error which would occurs had the dependence of service time upon waiting time been neglected will also be investigated.
2 .Basic definitions
In further considerations we shall deal with service systems having following featuresî
a/ Demands arrive one by one in time epochs t^ , t^ ,...,t^ , •••
and time periods a = t .j - t are indep e n d e n t ,identically distributed random variables with distribution function A O O , thus the input process is a renewal process,
b/ Demands are served individually,service time b^ of every de
mand is dependent on it's waiting time w and given by a condi
tional distribution function B(x/w).
We shall introduce for such systems a modified Kendall notation
* / B w / n / N / К / DISC, (2,1) where o(,n,N,K,DISC have the classical meaningj [KLEI 75] and stand respectively for the shape of distribution A(x) ,number of parallel service stations,dimension of the source,the maximal number of demands accepted by the system and the service discipline,while Bw states,that the system has service time dependent upon waiting time,this dependence being given by an additionally defined
conditional distribution function B(x/w).
Further we shall be interested in the throughput of the considered systems.Let us define the maximal throughput Т^13С of a given system under the scheduling discipline DISC as
a supremum of the demand's stream intensity Л ,for which the considered system is ergodic.
In this paper the following notation will be used:
Let r be a random variable.The probability density function(p.d.f) of this variable will be denoted r(x),distribution function R(x) and the Laplace^- Stiiltjes transform of this distribution function r(s) , r(s) = / e“ st d RCt) .
Mean value of the variable r will be denoted E(r).
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It is worth mentioning that for classical systems with B(x/w) « B(*), TDISC is independent of queueing state conditions for the FIFO queueing discipline,while all so far obtained results for other queueing disciplines will be presented in section 6 .
The first results in this area have been reported in [SUGA 65] where a system GI/B / n with
W .
BCx/w)= HCv-fíwl) , y">0
was investigated.In this case b= describes the deterministic dependence of service time upon waiting time.
The function has following features:
*(0) > 0
Imi 'f(w) « 'f(oo) <
00 W-» оOIt has also been assumed,that the mean interarrival time E(a) is finite,and demands are served on n identical,independently operating servers.
A more general dependence of service time upon waiting time was considered in .[CALL 73].
A class of discrete systems was investigeted with variables a, w, b^, standing for interarrival time,waiting time and service time being discrete,with integer,nonnegative values.
Additionally it was assumed that
Pr{ bw - a = ]} > O for W - O t<L...
It has been proved,that the considered system is ergodic if a/ E(bw )< oo for w = 0,1,2.., (3.4)
b/ lim sup C ( b w ) < E(a) . (3.5)
w-> oo
The author conjectured that a similar condition holds also for the continous case,which has been later investigated in [TWEE 75] . Let us define a random variable z^ = b^ - a , having a p.d.f.
zw(x). We shall assume that г^(х)>0 for all x e C-6,é) where 8 is a positive constant independent of w . Under these assumptions it has been proved that the considered system is ergodic,if (i) b (x) is continous in w for every fixed x.
(ii) there exist positive constants N,B,e such that ECbw )^ В < , w ^
E-(bw )^ E(aO-£ , w > N .
Condition (ii) can be rewritten in the form
(ii)’ the mean service time E ( b w)is a continous function in w and is bounded for every w ,
(ii)> lim sup E ( b ) < E ( a ) w -> oo
which together with (i) give an analogy to conditions (3.4)and(3 .5).
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4. Steady - atate analysis for FIFO discipline»
1и this section we shall present a methodology for analysis of open,single server queueing systems with waiting time dependent service times in the case of PIPO discipline.This description will be followed by a list of special cases which have so far been solved in detailes.
Let us investigate two consecutively carved demands, "n"
and "n+1 " having waiting times w*1 ,wn+^ and service times the distribution function of Cn+1 ) -th demand waiting time W л+1Сх)сап be expressed in terms of W л(х) as
exists and satisfies an integral equation
cO oO