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Buffers with correlated input and output interruptions
Volume3, Nuraber 3
OPERAgHONSRESEARCHLErTERS
August 1984
BUFFERS WITH CORRELATED INPUT AND OUTPUT INTERRUPTIONS
Herwig BRUNEEL Dt'paro~u.nJr of ( "omputer Scwnt ,'. Glwm Start Unirer.~il)'. Belgium
Recciv~dApril 1984 RevisedJune 1984
the letter considers a discrete buffered system ~ith one randomly interrupted output channel and stochastic interruptionsof the arrival stream which are correlated to the oulput interruptions. The hehavlourof thi~ buffer system is Mudicd, and in p~rilcularthe influenceof the correlation is derived. queues * modelling*correlation
I. Introduction Discrete-time queuing models are used to describe a large variety of communication or computer systems (or subsystems) that involve the synchronous transmission of data via some kind of communication channel. Examples are computer terminals, statistical multiplexers, nodes of a communication network, integrated voice-data systems (see e.g. [1]), and many others. In all these applications a buffer is used for the temporary storage of data-units ('messages') awaiting transmission via the communication channel. In many cases the communication channel is not permanently available for the transmission of data from the buffer. This may be caused by external elements (environmental noise, atmospheric circumstances) resulting in failures of the channel. The service interruptions may, however, also be built in, e.g., in an integrated voice-data system, the channel is blocked for the data, whenever voice packets are to be transmitted. On the other hand interruptions are also possible in the arrival stream of messages into the buffer, e.g. when the arrival stream consists of the overflow traffic from some other communication system, or when it is the output of a demultiplexing facility. Of course, the arrival interruptions can also be due to failures, just as the service interruptions. This work was supported by the Belgian National Fund for ScientificResearch (N.F,W.O.).
It is clear that, in some cases, the arrival interruptions and the service interruptions do not occur independently from each other, especially when both types of interruptions are determined by a common mechanism, e.g. environmental circumstances. However, to the best of the authors knowledge, ao studies exist of buffer systems with correlated input and output interruptions. The present paper presents a simple mathematical model which makes it possible to deal with this kind of correlation. The analysis leads to an explicit expression for the probability generating function of the number of messages in the buffer. From this result all the important performance measures (such as the mean queue length in the buffer) and the influence of the correlation on buffer behaviour can be derived.
2. The investigated system (Fig. !) A buffered communication system with synchronous transmission mode (i.e. transmissions can start at discrete clock times only), and fixed length messages, is considered. The messages enter the system via the input line(s), according to a probability density function e ( n ) H Prob[ n messages arrive during one clock
0167-6377/84/$3.00 © 1984. ElsevierScience Publishers B.V.(North-Holland)
time period]
149
V) ume 3, Number
OPERATIONSRESEARCHLE'ITERS
orrtvats
departures np
Fig. l. lions.
ine
Buffersystem with random inpul and output interrup-
August 1984
3. Analysis of bnffer behaviour Let us define the random variable u as the buffer occupancy (expressed in messages) in stochastic equilibrium, at the beginning of a clock time period, Let p ( n ) denote the probability density function of u, i.e.
and a probability generating function
p ( n ) ~ Proh[u = n],
e(z)~ ~ e(n)z",
and pA(n) and p . ( n ) the joint probabilities, defined as
whenever the input switch S~ is closed. They are transmitted from the system via the output channel, at the rate of one message per clock time period, whenever the output switch So is closed. Messages awaiting transmission are stored in a bnffer, which is assumed to be of infinite size. The output line of the system is subjected to random interruptions (switch So), which are characterised by two (independent) parameters a and ~O, defined as
pa(n) ~ Problu = n and output line available during previons clock time period]; pR(n) & Proh[u = n and output line blocked during previous clock time periodl.
a ~ Prob[So closed during k-th clock time period[
The goal of this section is to determine P(z), which is the generating function of the equilibrium buffer occupancy u at random clock times. Using the definitions and notations, introduced above, one can show that the following relationships exist:
S0 closed during ( k - 1)-th clock-time period] and ~ Prob[ So open during k-th clock time period I So open during (k - 1)-th clock time period]. In other words: both the 'available periods' and the 'blocked periods' of the output line (see also Brnneel [1,2]) are geometrically distributed random variables with mean values 1 / ( 1 - ~ ) and 1 / ( 1 - , 8 ) respectively. The long-run probability of an available output line is ,,
o
1-#
In these circumstam:es it is clear that p(n) = pA(n) +pa(n). If P(:), PA(r.) and Pi~(z) denote the Z-transforms of p(n), p^(n) and p , ( n ) respeclively, then we also have
P(:) = PA{z)-~ Fa(z).
(3)
p~(O) = [(1 - A ) +1:(0)1
× {~[ p~(o)+ p^(1)l - fl)[ pn(0) +pB(1)] } ;
+(1
pA(n)= (l -fA)[ap:,(n + l)
(4)
+(1 - - ] ] ) p . ( n + l)]
+/^{[~p~(o) +11 - / ~ ) p ~ ( o ) l e ( . )
(])
1 -a-#'
n+l
The arrival steam is subjected to random interruptions as well (switch S,), which depend on the state of the output line: A (available) or B (blocked). More precisely, it is assumed that
+ E [~p~(-,) m- I
+(1
-fl)pe(m)]e(n+l-m)}
(5)
for n >~ 1;
Prob[ $, closed IS0 closed] = &
pB(,t) = (1 --fB)[(1 -
and
a)pA(n) +flpB(n)]
+/a E [(1 -.)p,(m)
Prob[S, closedlS0 open] = r e , where f,,, and fB are two independent parameters. The long-run probability of an available arrival stream is easily seen to be given by
for n >~0.
f a of, + ( ! - o)f B.
The equations (4)-(6) can be used to derive
150
(2)
+flpB(m)]e(n-m) (6)
Vohllne3, Number 3
relationships between the Z-transforms P,,(.-) and P . ( : L From (4) and (5) one obtains
e,(:) = [I -.& +/~E(:)] × ap.~(.')+(I-fl)P,(:)+C(.--
(7) [I - f n + f.E(:)]
x[O
The unknown constant (" can be calculated from tile aormalisation condition P ( 1 ) = 1 (Kleinrock [51), which yields
c= e -Ix.
I)
whereas (6) leads to P.(:)=
1 -_a-p
(8)
-,~)e,d:)+Pv,(:)].
Combining the equations (7l and (8). expressions can be derived for P~.(z) and P , ( z ) . which, upon substitution in (3). yield the following expression for the generating function P(z):
I)[1 -£, +/~E(:)]
-
#+(l---fl)X/A
+Af" (2- a - ill(o-/h)
(" ~ ,,pA(O) + (l -- B )p,(O).
= c(:
(to)
The bchaviour of the described buffer system is completely specified by the equations (9) and (10). For instance, the mean buffer occupancy N at random clock times, can be found from the formula N = d P ( : t / d : ] : _ ~. yielding
The parameter C, introduced in (7). is an unknown constant, defined as
e(:)
August 1984
OPERATIONS RESEARCHLE'II"ERS
+
rE"(1) 2(o-fh~"
(ll) Here X and E"(1) denote the first and second derivatives of E(z ) respectively, evaluated at z = I.
The physical meaning of A is the mean number of arrivals per clock time interval, when the arrival stream is available.
× (l+(l-a-fl)[1-fs+fuE(z)]}
x {.-- ~[1 -f,, +f,,E(:)]
4. Discussion of results
-/~=[1 -f. +/,,E(~)]
Equation (1) makes clear that the same value of o can be found for different choices of a and ,8. In earlier work [1,3] we showed that the bchaviour of the buffer system is highly dependent on the actual values of o and ,8. even for a constant value of
x
[1 - f .
+f.E(-,)]
(9)
} -'
200-
100
0.
I
O.
,
,
,
I
0.5
'
I
I.
1.5
.
Fig, 2. Mean buffer occupancy N against mean arrival rate A for the case of Peisson arrivals; a = 0.995. /8 = 0,98, f = 0.4 and K = 0.5/0.6/0.7/0.8/0.9/1.0forcurvesa/b/c/d/e/f.
151
Volume 3, Number 3
OPERATIONS RESEARCH LETTERS
higher values of a and fl, i,e., higher values of the mean lengths of "available periods' and "blocked periods', give rise to higher mean buffer occupancies (,~). Equation (2) learns that, for a given value of a+ different choices for fA and fB may lead to the same value o f f . Let us investigate the influence on buffer behaviour of the actual values of fA and f a , for a constant value o f f ( a n d constant values of tx and fl, and hence o). To that end. we introduce a new parameter K, defined as K A o f ^ I f . As a function of this new parameter K, and the constant value f , fA and f a can be expressed as fA = K'J/o andfB =(1 -K)f/(1-o). It is clear that for a constant value of the long-run probability f of having the arrivial stream available, the correlation between input and output interruptions is completely determined by the value of K, "Large' or "small' values of K correspond to a positive or negative correlation respectively; the case without correlation, i.e. f,~ = f B = f , is found for K = o. In Fig. 2 the mean buffer occupancy ?~ (as calculated from O l D is plotted against the mean arrival rate ~., for the case of a Polsson arrival process (see e.g. Hsu [4|), constant values ~ = 0.995, o:
152
August 1984
fl = 0.98 (hence a = 0.8), and f = 0.4, and variable values for the parameter K. The curves a, b and c correspond to a negative correlation ( K < o ), curve d to the case without correlation ( K = o ). curves e and f to a positive correlation ( K > o). The relative positions of the curves show that the mean buffer occupancy is higher (lower) in case of negative (positive) correlation between input and output interruptions, which is in total agreement with the intuition. The quantitative differences between the curves follow from equation (11).
References
It] H. Bruneel. "Analysis of butter behaviour tar an integrated voice.data system", Electron. Lea, 19, 72-74 (1983). 121 H. BruneeL "Buffers with stochastic output interruptions", Electron Left. 19, 735-737 (1983). ]31 H. Brun~¢l, "On the behavior of buffers with random server interruptions", Perfarmam'e Evaluation S, 165-175 (1983)+ 141 J, Hsu. "Buffer behavior with Poisson arrival and geometric output processes", IEEE Trans. Commun. 22, 1940-1041 (1974).
[5] L. Kleinrock+ Queueing Systems, Volume I: Theory, Wiley, New York, 1975.
OPERAgHONSRESEARCHLErTERS
August 1984
BUFFERS WITH CORRELATED INPUT AND OUTPUT INTERRUPTIONS
Herwig BRUNEEL Dt'paro~u.nJr of ( "omputer Scwnt ,'. Glwm Start Unirer.~il)'. Belgium
Recciv~dApril 1984 RevisedJune 1984
the letter considers a discrete buffered system ~ith one randomly interrupted output channel and stochastic interruptionsof the arrival stream which are correlated to the oulput interruptions. The hehavlourof thi~ buffer system is Mudicd, and in p~rilcularthe influenceof the correlation is derived. queues * modelling*correlation
I. Introduction Discrete-time queuing models are used to describe a large variety of communication or computer systems (or subsystems) that involve the synchronous transmission of data via some kind of communication channel. Examples are computer terminals, statistical multiplexers, nodes of a communication network, integrated voice-data systems (see e.g. [1]), and many others. In all these applications a buffer is used for the temporary storage of data-units ('messages') awaiting transmission via the communication channel. In many cases the communication channel is not permanently available for the transmission of data from the buffer. This may be caused by external elements (environmental noise, atmospheric circumstances) resulting in failures of the channel. The service interruptions may, however, also be built in, e.g., in an integrated voice-data system, the channel is blocked for the data, whenever voice packets are to be transmitted. On the other hand interruptions are also possible in the arrival stream of messages into the buffer, e.g. when the arrival stream consists of the overflow traffic from some other communication system, or when it is the output of a demultiplexing facility. Of course, the arrival interruptions can also be due to failures, just as the service interruptions. This work was supported by the Belgian National Fund for ScientificResearch (N.F,W.O.).
It is clear that, in some cases, the arrival interruptions and the service interruptions do not occur independently from each other, especially when both types of interruptions are determined by a common mechanism, e.g. environmental circumstances. However, to the best of the authors knowledge, ao studies exist of buffer systems with correlated input and output interruptions. The present paper presents a simple mathematical model which makes it possible to deal with this kind of correlation. The analysis leads to an explicit expression for the probability generating function of the number of messages in the buffer. From this result all the important performance measures (such as the mean queue length in the buffer) and the influence of the correlation on buffer behaviour can be derived.
2. The investigated system (Fig. !) A buffered communication system with synchronous transmission mode (i.e. transmissions can start at discrete clock times only), and fixed length messages, is considered. The messages enter the system via the input line(s), according to a probability density function e ( n ) H Prob[ n messages arrive during one clock
0167-6377/84/$3.00 © 1984. ElsevierScience Publishers B.V.(North-Holland)
time period]
149
V) ume 3, Number
OPERATIONSRESEARCHLE'ITERS
orrtvats
departures np
Fig. l. lions.
ine
Buffersystem with random inpul and output interrup-
August 1984
3. Analysis of bnffer behaviour Let us define the random variable u as the buffer occupancy (expressed in messages) in stochastic equilibrium, at the beginning of a clock time period, Let p ( n ) denote the probability density function of u, i.e.
and a probability generating function
p ( n ) ~ Proh[u = n],
e(z)~ ~ e(n)z",
and pA(n) and p . ( n ) the joint probabilities, defined as
whenever the input switch S~ is closed. They are transmitted from the system via the output channel, at the rate of one message per clock time period, whenever the output switch So is closed. Messages awaiting transmission are stored in a bnffer, which is assumed to be of infinite size. The output line of the system is subjected to random interruptions (switch So), which are characterised by two (independent) parameters a and ~O, defined as
pa(n) ~ Problu = n and output line available during previons clock time period]; pR(n) & Proh[u = n and output line blocked during previous clock time periodl.
a ~ Prob[So closed during k-th clock time period[
The goal of this section is to determine P(z), which is the generating function of the equilibrium buffer occupancy u at random clock times. Using the definitions and notations, introduced above, one can show that the following relationships exist:
S0 closed during ( k - 1)-th clock-time period] and ~ Prob[ So open during k-th clock time period I So open during (k - 1)-th clock time period]. In other words: both the 'available periods' and the 'blocked periods' of the output line (see also Brnneel [1,2]) are geometrically distributed random variables with mean values 1 / ( 1 - ~ ) and 1 / ( 1 - , 8 ) respectively. The long-run probability of an available output line is ,,
o
1-#
In these circumstam:es it is clear that p(n) = pA(n) +pa(n). If P(:), PA(r.) and Pi~(z) denote the Z-transforms of p(n), p^(n) and p , ( n ) respeclively, then we also have
P(:) = PA{z)-~ Fa(z).
(3)
p~(O) = [(1 - A ) +1:(0)1
× {~[ p~(o)+ p^(1)l - fl)[ pn(0) +pB(1)] } ;
+(1
pA(n)= (l -fA)[ap:,(n + l)
(4)
+(1 - - ] ] ) p . ( n + l)]
+/^{[~p~(o) +11 - / ~ ) p ~ ( o ) l e ( . )
(])
1 -a-#'
n+l
The arrival steam is subjected to random interruptions as well (switch S,), which depend on the state of the output line: A (available) or B (blocked). More precisely, it is assumed that
+ E [~p~(-,) m- I
+(1
-fl)pe(m)]e(n+l-m)}
(5)
for n >~ 1;
Prob[ $, closed IS0 closed] = &
pB(,t) = (1 --fB)[(1 -
and
a)pA(n) +flpB(n)]
+/a E [(1 -.)p,(m)
Prob[S, closedlS0 open] = r e , where f,,, and fB are two independent parameters. The long-run probability of an available arrival stream is easily seen to be given by
for n >~0.
f a of, + ( ! - o)f B.
The equations (4)-(6) can be used to derive
150
(2)
+flpB(m)]e(n-m) (6)
Vohllne3, Number 3
relationships between the Z-transforms P,,(.-) and P . ( : L From (4) and (5) one obtains
e,(:) = [I -.& +/~E(:)] × ap.~(.')+(I-fl)P,(:)+C(.--
(7) [I - f n + f.E(:)]
x[O
The unknown constant (" can be calculated from tile aormalisation condition P ( 1 ) = 1 (Kleinrock [51), which yields
c= e -Ix.
I)
whereas (6) leads to P.(:)=
1 -_a-p
(8)
-,~)e,d:)+Pv,(:)].
Combining the equations (7l and (8). expressions can be derived for P~.(z) and P , ( z ) . which, upon substitution in (3). yield the following expression for the generating function P(z):
I)[1 -£, +/~E(:)]
-
#+(l---fl)X/A
+Af" (2- a - ill(o-/h)
(" ~ ,,pA(O) + (l -- B )p,(O).
= c(:
(to)
The bchaviour of the described buffer system is completely specified by the equations (9) and (10). For instance, the mean buffer occupancy N at random clock times, can be found from the formula N = d P ( : t / d : ] : _ ~. yielding
The parameter C, introduced in (7). is an unknown constant, defined as
e(:)
August 1984
OPERATIONS RESEARCHLE'II"ERS
+
rE"(1) 2(o-fh~"
(ll) Here X and E"(1) denote the first and second derivatives of E(z ) respectively, evaluated at z = I.
The physical meaning of A is the mean number of arrivals per clock time interval, when the arrival stream is available.
× (l+(l-a-fl)[1-fs+fuE(z)]}
x {.-- ~[1 -f,, +f,,E(:)]
4. Discussion of results
-/~=[1 -f. +/,,E(~)]
Equation (1) makes clear that the same value of o can be found for different choices of a and ,8. In earlier work [1,3] we showed that the bchaviour of the buffer system is highly dependent on the actual values of o and ,8. even for a constant value of
x
[1 - f .
+f.E(-,)]
(9)
} -'
200-
100
0.
I
O.
,
,
,
I
0.5
'
I
I.
1.5
.
Fig, 2. Mean buffer occupancy N against mean arrival rate A for the case of Peisson arrivals; a = 0.995. /8 = 0,98, f = 0.4 and K = 0.5/0.6/0.7/0.8/0.9/1.0forcurvesa/b/c/d/e/f.
151
Volume 3, Number 3
OPERATIONS RESEARCH LETTERS
higher values of a and fl, i,e., higher values of the mean lengths of "available periods' and "blocked periods', give rise to higher mean buffer occupancies (,~). Equation (2) learns that, for a given value of a+ different choices for fA and fB may lead to the same value o f f . Let us investigate the influence on buffer behaviour of the actual values of fA and f a , for a constant value o f f ( a n d constant values of tx and fl, and hence o). To that end. we introduce a new parameter K, defined as K A o f ^ I f . As a function of this new parameter K, and the constant value f , fA and f a can be expressed as fA = K'J/o andfB =(1 -K)f/(1-o). It is clear that for a constant value of the long-run probability f of having the arrivial stream available, the correlation between input and output interruptions is completely determined by the value of K, "Large' or "small' values of K correspond to a positive or negative correlation respectively; the case without correlation, i.e. f,~ = f B = f , is found for K = o. In Fig. 2 the mean buffer occupancy ?~ (as calculated from O l D is plotted against the mean arrival rate ~., for the case of a Polsson arrival process (see e.g. Hsu [4|), constant values ~ = 0.995, o:
152
August 1984
fl = 0.98 (hence a = 0.8), and f = 0.4, and variable values for the parameter K. The curves a, b and c correspond to a negative correlation ( K < o ), curve d to the case without correlation ( K = o ). curves e and f to a positive correlation ( K > o). The relative positions of the curves show that the mean buffer occupancy is higher (lower) in case of negative (positive) correlation between input and output interruptions, which is in total agreement with the intuition. The quantitative differences between the curves follow from equation (11).
References
It] H. Bruneel. "Analysis of butter behaviour tar an integrated voice.data system", Electron. Lea, 19, 72-74 (1983). 121 H. BruneeL "Buffers with stochastic output interruptions", Electron Left. 19, 735-737 (1983). ]31 H. Brun~¢l, "On the behavior of buffers with random server interruptions", Perfarmam'e Evaluation S, 165-175 (1983)+ 141 J, Hsu. "Buffer behavior with Poisson arrival and geometric output processes", IEEE Trans. Commun. 22, 1940-1041 (1974).
[5] L. Kleinrock+ Queueing Systems, Volume I: Theory, Wiley, New York, 1975.