analysis at service completion times of buffers with output interruptions

Zeitschrift far Operations Research, Volume 29, page 301 - 313

Analysis at Service Completion Times of Buffers with Output Interruptions

By H. Bruneel, Ghent 1

Received October 1984

Abstract: In an earlier paper [Bruneel, 1983a]we investigated a discrete-time buffer system with infinite waiting room and one single output channel, subjected to stochastic interruptions.For this system, the probability generating function of the number of messages in the buffer at service completion times was derived, under rather general assumptions concerning the arrival process and the server interruption process. However, at a certain point in the analysis, we tacitly introduced an approximation, by neglecting the (statistical) information available on the first customer of a busy period.

In the present paper we show that this simplifying assumption is not really necessary in order to keep the problem analytically tractable, and the exact (but rather tedious) derivations are given. As a result, an expression is found for the probability generating function of the buffer contents, which is even less complicated than the one obtained in the previous paper.

The results of the two studies are compared and axe found to be substantially different from each other for realistic values of the parameters of the system.

Zusammenfassung: In einer friiheren Arbeit [Bruneel, 1983a] wurde ein Puffersystem in diskreter Zeit mit einem unbeschr/inkten Speicherraum und einem zuf~illigen Unterbrechungen ausgesetz- ten Ausgabekanal untersucht. Fiir dieses System wurde die erzeugende Funktion der Anzahl der Auftr~ige im Puffer zu den Zeitpunkten der Abfertigung eines Auftrags berechnet, wobei recht allgemeine Annahmen fiber den Ankunftstrom und den Prozet~ der Bedienungs-Unterbrechungen zugrundelagen. Dabei wurde auch implizit eine Approximation benutzt, indem (statistische) In- formation fiber den ersten Kunden einer Arbeitsperiode nicht beriicksichtigt wurde.

In der vorliegenden Arbeit wird nun gezeigt, dai~ mit betr/ichtlichem Mehraufwand die Analyse auch ohne diese Approximation durchftihrbar ist.

Der resultierende Ausdruck ftir die erzeugende Funktion des Pufferinhalts ist sogar einfacher. Ein Vergleich zeigt, da~ die neuen Ergebnisse bei realistischer Parameterwahl yon den friiher er- zielten betr~ichtlich abweichen.

1 Introduction

A discrete-time buffer system with one single output channel and an infinite

waiting room is considered. In such a system, messages arrive at the buffer via (one

of) the input channels, then wait in the buffer until all the earlier arrived messages

have been transmitted, and are finally taken out of the buffer themselves for trans-

mission via the output channel. It is assumed that all the messages have the same

constant length and that the output channel transmits data with constant speed, so

that the service time of the messages is constant. This constant time period is taken as the (discrete) uni t of time, and is called a clock time period. It is further assumed that the data transmission is synchronous, i.e., the time axis is divided in time intervals, called(time)slots or clock time intervals, the length of which is one

clock time period, so that the transmission of a message can only start at the begin-

1 Herwig Bruneel, Department of Computer Science, Ghent State University, Sint-Pietersnieuw- straat 41, B-9000 Ghent, Belgium.

0340-9422/85/070301-31352.50 �9 1985 Physica-Verlag, Heidelberg

302 H. Bruneel

ning of a slot and end at the end of a slot. This implies that a message cannot leave the buffer system at the end of the time slot during which it entered the system. Therefore, when the buffer is empty at the beginning of a time slot, no message can leave the system at the end of this slot, even if some arrivals have occurred during this slot.

Discrete-time buffer systems have been studied quite extensively in the scientific literature, both for the case of a finite waiting room [Chu, 1970a+b; Chu/Liang;Dor; Maritsas/Hartley; Hill] and the case of an infinite waiting room [Kobayashi/Konheim; Chu/Konheim; Fredrikson; Barath-Kumar; Bruneel, 1983b; Towsley/l~olf; Konheim ]; both for one single output channel and for multiple output channels [e.g. Chu, 1970a].

orrivals ~ INFINITE BUFFER I ~176 ~ departures output chonnel

input channels Fig. 1. Buffer system with one single randomly interrupted output channel.

During the last decade several researchers have concerned themselves with buffer systems, subjected to random server interruptions (Fig. 1). Kekre, Saxena and Khalid have described and analyzed several finite waiting room models [Kekre/Saxena; Kekre/ Saxena/Khalid; Kekre/Khalid, 1980, 1981 ], resulting in a numerical and graphical characterisation of the buffer behavior. Several infinite waiting room models have been presented as well [Bruneel, 1983a;Hsu; Georganas;BruneeI, 1983c, 1984 a + b; Heines], not only for the single output channel case but also for the multiple output channel case [e.g Georganas; Bruneel, 1984a]. These studies differ in the choice of the arrival process of the messages, the type of model for the server interruption process and the choice of the time instants at which the buffer contents is observed (arbitrary clock times, service opportunity times, times of departure). However, they have one feature in common: the probability generating function of the number of messages in the buffer is derived analytically as an explicit function of the parameters which characterize the system.

The present paper belongs to the latter category and deals with a discrete-time buffer system with one single, randomly interrupted, output channel, which is analyzed at service completion times. The arrival process of the messages is general and the model for the server interruption process is the one presented in our earlier paper [Bruneel, 1983a]. However, the analysis leading to the distribution of the number of messages in the buffer, is now carried through in an exact manner, without the introduction of any approximations, as opposed to our previous study.

The details of the model, as well as some definitions and terminology are given in the next section. Section 3 presents the improved, more accurate derivation of the probability generating function of the number of messages in the buffer. Finally, the results are discussed and compared to the ones found in [Bruneel, 1983a] in section 4.

Analysis at Service Completion Times of Buffers 303

2 Mathematical Description of the Buffer System

The arrival stream of messages into the buffer is described by means of a sequence of i.i.d, stochastic variables, which denote the total numbers of arriving messages (via all the input channels) during the consecutive time slots. The common probability density function of these stochastic variables is denoted by c (n) and the corresponding probability generating function by C (z). That is,

c (n) = Prob [n messages arrive during a time slot],

C(z)=n~=Oc(n)zn.

The interruption process of the output line is described as follows. The output line is assumed to be in one of two possible states: A or B, i.e., available or blocked respectively. Time slots during with the server state is A or B are called A-slots and B-slots respectively. Time intervals during which the server state remains A or B are called A- times and B-times respectively. As time goes by, A-times and B-times occur alterna- tely, depending on whether the output channel is available or not. In the model the lengths of A-times and B-times (expressed in integer numbers of slots) are considered as two mutually independent sequences of i.i.d, random variables. The common density function of the B-times is denoted by b (n) and the corresponding probability generating function by B (z). Analogously the symbol a (n) is used for the density of the A-times. Thus,

a (n) = Prob [length of an A-time equals n slots],

b (n) = Prob [length of a B-time equals n slots],

B (z) = ~ b (n) z n. n=l

The analysis is carried out for arbitrarily distributed B-times and for geometrically distributed A-times, i.e.,

a(n)=(1--a)a n'l, n>~ l.

This particular distribution is chosen mainly to reduce the complexity of the mathematical analysis, as explained in [Bruneel, 1983a].

It is clear that the evolution of the server state in time can be described by means of strings of characters, taken from the alphabet (A, B), where each A or B in the string corresponds to the occurrence of an A-slot or a B-slot respectively. For example, the sequence of two A-slots, followed by six B-slots, followed by five A-slots would be "represented" by the string A 2 B 6 AS. If x and y are strings built up from the alphabet {.4, B}, i.e., x, y E (A, B}*, then we define p (x) as the probability of having a sequence of slots represented by the string x and p (x [y) as the probability of having a sequence of slots represented by the string x if we just had a sequence of slots re-

304 H. Bruneel

presented by the stringy, immediately before this. In [Bruneel, 1983a] it was shown that, for the system described here, the following relationships apply:

Prob [B-time = n] for n t> 1 p (A I x AB n) = Prob [B-time/> n]

f o r n = 0 and (1)

p (B [ x AB n) =

Prob [B-time/> n + 1] Prob [B-time/> n]

forn~> 1

forn = 0

independently of the string x E (.4, B)*. That is, the probability of having an A-slot or a B-slot (at a given point in time) is dependent only on the number of B-slots just before this time since the last A-slot. Other interesting results concern the probabilities that an arbitrary slot is an A-slot or a B-slot, i.e., p (A) or p (B) respectively. They are given by

E [A-time] p (,4) = E---[A-time] + E [B-time]

and (2) E [B-time]

p (B) = E [A-time] + E [B-time] "

3 The Number of Messages in the Buffer at Service Completion Times

Let q denote the nonnegative integer random variable which indicates the number of messages in the buffer at service completion times, when a stochastic equilibrium has been reached, i.e., the number of messages left behind in the buffer by a departure in steady state. It was shown in [Bruneel, 1983a] that the probability generating function Q (z) o fq is given by the expression

Q (z) = Q ( o ) . z w (z) - v (z) z - v (z ) ( 3 )

Here V (z) and IV (z) are the probability generating functions of the random variables v and w respectively, which are defined as follows. The random variable v denotes the number of arrivals to the buffer between the departures of two consecutive messages, provided the second message was already waiting in the buffer at the departure time of the first. The stochastic variable w denotes the number of arrivals to the buffer during the total time the first customer of a busy period remains in the buffer.

InBruneel [1983a] it was shown that V(z) takes the form

V (z) = C (z) (a + (1 --a) B (C (z))) (4)


and that W (z) can be expressed as

c (z) - c ( o ) . R (C (z)) w(z) = z [1 - c ( o ) ]

(5)

where the function R ( ' ) denotes the generating function of the "discrete system time" r of the first customer of a busy period, i.e., the number of full slots a message, which finds the buffer empty upon arrival, remains in the buffer. In Bruneel [1983a] an expression for R (z) was derived under the simplifying assumption that the first customer of a busy period arrives during a time slot which is totally arbitrary. That is, it was assumed that the evolution in time of the server state before and during this time slot can be described by a string x E {.4, B)* with the probability p (x), defined in the previous section. This assumption may be a good first approximation, but it is not fully correct, as we shall see below. The point is that the time slot during which the first customer of a busy period arrives, - let us call this slotI1 - is not completely arbitrary, because there is some a priori information about the server state duringI1. Therefore, the probability of having a sequence of A-slots and B-slots represented by the string x (up to time slot 11 ) is not necessarily given by p (x).

Let C1 denote the first customer of a busy period, i.e., the first customer arriving during time slot 11, and Co the last customer of the previous busy period. Let I0 indi- cate the time slot at the end of which Co has left the buffer; then we are assured that lo is an A-slot. The (discrete) distance between the slots lo and 11 is a positive integer stochastic variable K (Fig. 2), with density function

Prob [K= k] = [ 1 - - C ( o ) ] " [C(o)] k'l, k>~l. (6)

Here C (o) is the value of the generating function C (z) at z = 0, and denotes the probability of having no arrivals during a time slot. The number of B-slots since the last A-slot, up to time slot 11, is also a stochastic variable, which will be indicated by the symbol J. The random variable J can take all nonnegative integer values, the value

departure C O

2- 0 2" 1

I I I I I - - i hme

I I I ~ ~.1 I K I

arrival C I

Fig. 2. Definition of the customers Co and C1, the time slots I o and I 1 and the discrete distance K.

J = 0 corresponding of course to the case where 11 is an A-slot itself. It is clear that the probability distribution of J is crucial with respect to the derivation ofR (z), because the number of B-slots since the last A-slot, up to time slot 11, determines the probabilities of having A-slots or B-slots after 11.

Let us define the quantities f ( ] ) and f ( / I k) as

f ( ] )= Vrob [J = j l, j /> 0,

306 H. Bruneel

i.e., the density function of J, and

f ( ] [k) = P r o b [ J = ] I K = k ] , k ~ 1, O<~]<~k,

i.e., the conditional density of J given K.

Then f(]) can be expressed as

~ Prob[K=k]f(olk) f o r ] = 0 k=l

f q ) =

Prob [K = k] f q [k) for ] i> 1. k=]

It thus suffices to determine f ( ] I k) for all allowed values of] and k in order to obtain the probability distribution of J.

For all] ~> 1 and k--->], we have

f(] l k) = f(o I k - ] ) p (B ] [xA)

provided we define

;(o I o)= 1.

Here p (B / I xA) can be expressed as

p(s / IxA)= ~I 1 p(s IxAB m) m=0

(1 - -a) Prob [B-time ~>]]

using the equations (1).

We thus obtain

f ( ] I k) = (1 - -a) Prob [B-time >~]]f(o I k --])

for]~> 1 andk>~].

From (6) - ( 8 ) , f ( ] ) for/t> 1 is found as

f(]) = ~, [1 - -C(o)] [C(o)] k'l (1 - -a) Prob [B-time >~]]f(o I k--]) k=]

= (1 --a) [C(o)] ]'1 Prob [B-time ~>]] i~0 [1 - -C(o)] [C(o)]if(o 10

= (1 -- a) [C (o)] 1"1 Prob [B-time ~>]] [1 -- C (o) + C ( o ) f (o)].

(7)

(8)

(9)


The only remaining unknown parameter in this expression is f ( o ) ; it can be computed 0 o

from the normalization equation E f ( i ) = 1. As a result we obtain after a few mathe- i=0

matical derivations

f(o) = [1 -- C(o)] [a + (1 --a)B (C (o))1 1 --C(o) [,~ + (1 --~)B (C (o))]

(10)

Using (10) in (9), we further obtain

[1 -- C(o)] (1 - -a ) f ( l ) = 1 -- C (o) [a + (1 -- a) B (C (o))] [C (o)] y'l Prob [B-time ~>j] (11)

f o r / ~ 1.

The probability density function of the discrete system time r of the first customer of a busy period, can now be derived as follows:

Prob [r = n ] = Prob [evolution of server state after time slot I1 can be represented by string B n'l A ]

= ~ Prob [B n'l A afterI1 IJ =]] " Prob [ J = ] ] j=0

= ~ p ( B n - I A IxABi) f ( i ) , (12) i=0

forn ~> 1.

Here p (B n-1 A I x AB j) can be expressed as a function of known quantities, using the equations (1). The following cases have to be considered:

(i) n = l a n d ] = 0

p (B n-1 A [ xAB ]) = p (A [ xA) = o~;

(ii) n = l a n d 1 > 0

b (y) p (B n'l A [xAB j) =p (A [xAB i) = Prob [B-time i>/'] '

n > l and j - - O

p (B n-x A I xAB ]) = p (B n'l A [xA)

n-2 = p ( B IxA)"m~l p (B IxAB m) " p (A IxaB n-x)

= (1 - -a ) b (n -- 1);

(iii)

308 H. Bruneel

(iv) n > l a n d ] > 0

p (1t n'l A I xAB ]) =/+fI "2 p (B I xABm) �9 p (,4 I xAB ]+n-l) m=]

b q + n - - 1 ) Prob [B-time ~>]]"

By use of these expressions, along with equations (10) - (11), in (12), the density of r is found as follows:

(i) Prob [r = 1] = otf(o) + b (]) Prob [B-time ~>/]f(J) ]=1

= a [o~ + (1 --a)B (C(o))]M + (1 --a)M ~ [C(o)] ]'1 b (]) (13) /=1

where we have introduced the notation M for the constant given by

1 -- C (o) M =

1 --C(o) [a + (1 - - a ) B (C (o))] (14)

(ii) For values of n > 1

~=1 b ( ]+n- -1 ) Prob [r = n] = (1 --a) b (n -- 1)f(o) + / Prob [B-time/>/'] f (])

= (1 --a) b (n-- 1) [a + (1 --a)B (C(o))]M

+ ( 1 - - a ) M ~ [C(o)] ]-1 b ( j + n - - 1 ) . ]=1

(15)

From (13) and (15) we can now derive an expression for the generating function R (z).

R ( z ) = ~ P r o b [ r = n ] z n rt=l

= [a + (1 --a)B (C(o))]M [otz + ~ (1 --a) b (n -- 1) z n ] n=2

+(1 - - a ) ,M ~ ~ b(]+n- -1)[C(o) lJ - l z n n=l ]=1

= [ ~ + ( 1 -cOB (C(o))]Mz [a + (1 --e,)B (z)] k

+(1 --a)Mz ~ b (k)z k Y, [C(o)] ]'1 z "] k=l ]=1

where k = n + ] -- 1. A few additional mathematical manipulations finally lead to

R (z) = Mz { [ee + (1-- a) B ( C (o ) ) ] [a + (1-- a) B (z) ] + ( l -- a) B (z) - B (C (~ ) } �9 z - c ( o ) "

(16)


The equation (16) can now be combined with the equations (3) - (5) in order to find the correct expression for Q (z), the probability generating function of the number of messages in the buffer at service completion times. This expression appears to take the following rather simple form:

Q (z) = D" C (z)- 1 - C (z) [a + (1 - a) B (C (z))] z --C(z) [a + (1 - - a )B (C (z))]

where the unkown constant D is defined by

O ~ [a + (1 --oOB (C (o))1 Q (o) 1 - C ( o ) [a + (1 - -a)B (C (o))] "

(17)

The parameter D can be determined from the normalisation equation Q (1) = 1, so that we eventually obtain

[ 1 - p(A) ~ C(z) 1 --C(z) [a + (1 - -a)B (C (z)) l Q(z)= E[C] ] z- -C(z)[ot+(1 --a)B(C(z))]

(18)

where E [C] denotes the mean arrival rate (per slot), i.e., the average number of arrivals per time slot, and p (A) is the long-run probability of having the output channel of the buffer available (see equation (2)).

Notice that, since we are dealing with a single output buffer, the generating function Q (z) also describes the number of messages in the buffer as observed by the messages upon arrival in the system.

4 Discussion of the Results

The only difference between the present analysis and the previous analysis lies in the expression for R (z). In Bruneel [ 1983a] we derived the approximating expression

R* (z) = p (A) z {1 + (1 -- a) ~ [B (z) -- 1]}, (19)

under the assumption that the time slot Ii is an arbitrary slot, for which no a priori information concerning the server state is available. As opposed to the expression (16), this approximating expression does not depend on the parameter C (o), which denotes the probability of having no arrivals during a time slot. Moreover, it can be shown that as the value of C (o) approaches unity, the expression (16) becomes equi- valent to the expression (19). This should not be too surprising, since if C (o) ~ 1, the mean value of the discrete distance K between the time slots lo and 11 (defined in the previous section) goes to infinity, and the correlation between the server states during lo and I1 disappears, so that I1 becomes an arbitrary time slot. The above makes clear that the largest deviations of the approximate results from the exact results may be expected for low values of C (o), i.e., for high values of the mean arrival rate of the messages.

310 H. Bruneel

It is our experience that in many practical cases the influence of the approximation on the results of the analysis is quite substantial. Therefore, when it comes to actually design a buffer with a finite waiting room, it may be important to use the exact results instead of the approximate ones. Let us consider a particular case which illus- trates the possible large deviations of the approximate results from the exact ones. Suppose the B-times have a geometric distribution with parameter/3, i.e.,

b(n)=(1-~)[3 n'l, n>/1

or, equivalently,

(z) = ( I - t~) z 1 --/3z

For this special case, R (z) is given by

R (z) = z" {(1 -- ~) (1 --/3) + Is + (1 -- a --/3) C (o)] [a + (1 -- a --/3) z]} [1 + (1 --a-- /3) C(o)] (1 --/3z)

and the expected value E [r] of the discrete system time r of the first customer of a busy period, by

E [r] = dR (z) z= = 1 + ~ (1 - -~ --/3) + (1 - - a --/3) ( 2 - - a --/3) C(o) az 1 (1 -t3)[1 +(1 -~- /3 ) C(o)]

In most practical cases the mean values of A-times and B-times are on the order of magnitude 102 up to l0 s [see e.g.Bruneel, 1983c, 1984b]. This means that realistic values of a and/3 lie between 0.99 and unity, so that the denominator factor 1 + (1 - a --/3) C (o) takes very small values in the vicinity of C (o) = 1. It follows that considerable changes in the value o f E [r] are to be expected even for small deviations of C (o) from unity. The range of possible values for the parameter C (o) depends on the particular shape of the arrival process and can usually be derived from the condition for a stochastic equilibrium which, for the system studied here, demands that the mean number of arrivals per time slot be strictly less than the long-run probability of having the output channel available, i.e., the quantity p (A) introduced in section 2. Notice that in the case of geometric B-times with parameter/3, p (A) can be expressed as

1 - /3 P ( A ) - 2 _ o t _ / 3 �9

Example: Poisson arrival process.

In this case c (n) = e "x ~kn n-T-' n > 0 ;

C (z) = e "x(1-z)"

C (o) = e "x.


The mean arrival rate per slot is X and thus the equilibrium condition reads

1 - - f l X <

2 - - a --I~ "

The range of possible values for C (o) in these circumstances, is determined by

1-#

e 2-~-~ < C ( o ) ~ < 1,

and clearly includes values which deviate considerably from unity. Hence, substantial differences are to be expected between actual and approximate value of E Jr]. This conjecture is confirmed by Fig. 3 which gives a plot o f E [r] versus X for realistic values of the parameters a and ft. Notice that the approximate value o f E [r] is the one corresponding to X = 0.

~pproximc#e 100-

50-

o I I f = o o o t 0.2 0.3 )~

Fig. 3. Exact and approximate value of the mean system time for the first customer of a busy period, E [r], versus the mean arnval rate X, for geometri c A-times and B-times with parameters ~ = 0.99 and fl = 0.995, and a Poisson arrival process.

The mean buffer occupancy at service completion times, for this special case, is found as

- - d Q ( z ) z=1 q P o i s s o n - d z

1--e--fl X+[I+fl+2(1--a--f l)XlX (20) = X + 2 - - o e - - f l 2 [1 - - f l - - ( 2 - - ~ - - f l ) X]

In Fig. 4 a plot is given of (/Poisson versus X, for the same set o f values for a and fl

as in Fig. 3. In the same figure we have plotted the mean buffer contents qPoisson as

obtained when the approximate expression (19) for R (z) is used instead of the exact expression (16). The plots make clear that the approximate analysis leads to an over- estimation of the expected buffer contents, which is in accordance with the over- estimation o f E [r], illustrated by Fig. 3. However, the deviation ofg/Poisson from

312 H. Bruneel

Mean buffer occupancy

200- q poisson ~~~Poisso n

100 -

I I I =-

ol 02 03 X Exact (qPoisson) and approximate (q*Poisson) value of the mean buffer occupancy versus the mean arrival rate X, for geometric A-times and B-times with parameters a = 0.99 and # = 0.996, and a Poisson arrival process.

0 0.0

Fig. 4

qPoisson-* is not excessive, in spite of the very large difference between the exact and

approximate values o f E [r], for "large" values o f X. The reason is that for large values o f ;% the first arrivals of a busy period form only a small fraction of the total amount of arrivals, so that the influence o f a deviation in E [r] on the mean buffer occupancy is minimized.

Similar results are obtained for other choices of the arrival process.

Acknowledgement This work was supported by the Belgian National Fund for Scientific Research

(N.F.W.O.).


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analysis at service completion times of buffers with output interruptions

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