32 zyxwvutsrqpon IEEE TRANSACTIONS ON COMMUNICATIONS,.VOL. COM-35, NO. I, JANUARY 1987 On Discrete Buffers in a Two-State Environment HERWIG BRUNEEL Abstract-A zyxwvutsrqp discrete buffered system with infinite buffer size, zyxwvutsr one single outputchannel,andperiodicopportunities for service (synchro- nous transmission) is considered in a two-state environment. The output channel is subjected toa random interruption process, which is character- ized by a Bernoulli sequence of independent random variables, with probabilities dependent on the environment state.The environment states have random sojourn times with “mixture of geometries"-type distribu- tions. The arrival process is dependent on the environment state, but arbitrary. For this system, expressions are derived for the probability generating functions of the number of messages in the buffer at various time instants. A number of special cases and possible applicatiQns of the model are discussed, and an extended example is given’as an illustration of the study. I. INTRODUCTION D ISCRETE-time buffer systems with synchronous transmission have been investigated quite extensively in the past peveral years. Much work has been done on the case where the output lines of the buffer are available for all times, e.g., [I], [2]; however,lately,manyresearchershavecon- cerned themselves with buffers where the output lines are subjected to random interruptions in time or where messages have to be retransmitted a stochastic number of times, due to errors, e.g., [3]-[15]. Both finite [I], [2], [6]-[9] and infinite [3]-[5], [lo]-[15] waiting rooms have been treated. A Poisson arrival process has been considered in [3]-[6], a hybrid input process (Poisson and compound Poisson) has been treated in [7]-[9], whereas the arrival process is arbitrary in [lo]-[15]. However, most zyxwvutsr of: these analyses have -one feature in common: the numbers of arrivals durjng consecutive ‘clock time intervals are i.i.d. random variables, i.e., the arrival process is uncorrelated. Only a few researchers have used a different kind of arrival process. Bruneel [14] has ,treated a buffer system in which the arrival stream is subjected to random interruptions in time, i.e., where “available periods” (duringwhichi.i.d.groupsarrive in theconsecutiveclock time intervals) and “blocked periods” (during which no arrivals occur) of stochastic length occur alternately. The analysis remains, however, restricted to gesmetrically distrib- uted available periods and arbitrarily distributed but finite blocked periods. Towsley [13] has considered a statistical multiplexer in a two-state environment, with different arrival processes for the two environment states. Unfortunately, his analysis is only valid for Markovian environments, i.e., for geometrically distributed sojourn times for the two environ- ment states, except for a few pqrticular choices ,for the parameters of the system. The present analysis is the result of an attempt to generalize Paper approved by the Editor for Computer Communication of the IEEE Communications Society. Manuscript received December 2, 1982; revised January 15, 1986. This paper was presented at the Second National Congress on Quantitative Methods for Decision Making, Brussels. Belgium, December 1983. This work was supported by the Belgian National Fund for Scientific Research (NFWO). The .author is with the Department of Computer Science, Ghent State University, B-9000 Ghent, Belgium. IEEE Log Number 861 1699. zyxwvutsrqpo . . Towsley’s results to more general distributions for the sojourn times of the environment states. We consider environment state sojourn times with probability density functions that can be written as mixtures of finite numbers of geometric densities. Although such mixtures do not yield the full generality that we would want, they can be used as approxima- tions for a large class of discrete probability distributions whose coefficient of variation ‘is larger than for a simple geometric distribution. The extension from a geometric density to a mixture of geometrics can be considered as the discrete-time equivalent of the generalization from an expo- nential to a hyperexponential distribution in continuous time, a technique which has been applied successfully to the interar- rivaltimeandservicetimedistributions in .continuous-time queueing systems (see, e.g., [16]). The details of the model are explained in Section 11. Some preliminary terminology is introduced in Section 111, whereas the actual analysis is carried through in Section IV. Sections V and VI contain a discussionofthemodelanditsanalysis, along with a number of special cases and practical applications of the study. Finally, an example is worked out in detail in Section VII. 11. THE SYSTEM UNDER CONSIDERATION A diagram of the investigated system is depicted in Fig. 1. Messages arrive in a stochastic manner via one of the input channels, wait in the buffer for some time, and are then taken out via the output channel. We assume that the messages have a fixed length ‘and that the output channel transmits data with constant speed. Hence, the transmission time of the messages is constant. The clock time period is defined as the time required to transmit one message. The data transmission is synchronous, i.e., the data are taken out synchronously from the buffer for transmission at each discrete clock time. However, this can only happen if the switch S is closed, i.e.,if the output channel is available. Whenever the switch S is open, no transmission can take place. The “environment” of the buffer is described as follows: there are two possible’ environment states, which will be indicated by A and B. As time goes by, the environment state is A for a number of clock time periods, then B for a number of clocktimeperiods,then A again,and so on. Thetime intervals (expressed in clock time periods) during which the environment state is zyxw A or B will be called A-times and B- times, respectively. The A-times and B-times occur alter- nately and take positive integer values only. We assume that the A-times are i.i.d. random variables with probability density function pA(n), that the B-times are i.i.d. random variables with probability density function pa(n), and that the A-times and B-times are mutually independent. Throughout this paper it will be assumed that both PA (n) and p~(n) can be written as mixtures of a finite number of geometric density functions: rA pA(n)=c zyxw a;(l -cY;)(cY;)-’ (1) z i= I ‘B Pdn) = b;(l - Pimi)”- (2) z i= 1 0090-6778/87/0100-0032$01 .OO zyxwvut 0 1987 IEEE