Analysis of a Loop Transmission System with Round-Robin Scheduling of Services

A finite population. multi-queue model is developed for a loop transmission system. Approximate expressions for the state transition matrix and other system variables are derived in recursive forms. I t is also shown that a number of useful system parameters, such as average message response time. average cycle time, and average response time conditioned on message length. can be obtained. The analytical results have been validated by simulation. Introduction In the literature, most system models involving many queues with a single server assume that customers are drawn from an infinite population, such that arrival and service processes are mutually independent. Published works on this type of model include Liebowitz [ I ] ~ Kruskal [ 21. Cooper and Murray [ 31, Eisenberg [ 41. and Konheim and Meister [ 51. These papers differ mainly in the service disciplines and the degree of generality of the models assumed. In some physical systems, however, the arrival and service processes are not independent. For instance, in an interactive environment. arrivals at a queue occur in batches and no new arrival occurs until the previous entry has been completely served. To our knowledge, only one author has considered a system model of this type: Runnenberg [6] assumes that the server patrols the queues in cyclic order, completely exhausting each before advancing to the next. In contrast. we assume a round-robin scheduling of services. That is, the server provides only one unit of service for each visit to a queue. The motivation that brought this problem to our attention is its application to loop transmission systems. In this case the server is a processor that controls the loop; the batches are the messages, each of which consists of a number of characters to be transmitted; the queues are the terminals attached to the loop. We believe that the model described in this paper can be useful in a number of other applications also. The model consists of N queues distributed around a loop (Fig. 1 ) . The queues are served in cyclic order by a traveling server. The arrivals to the queues occur as messages, where the number of characters in a message varies according to a geometric distribution with mean R . The server “walks” from one queue to the next, 486 servicing exactly one character for each visit to a queue. The service time per character, t,, and the walking time between adjacent queues. t,, are both assumed to be constants. At each queue the time interval from the completion of service of one message to the arrival of the next is exponentially distributed with mean l / h . For this model, several recursive expressions have been developed to obtain the state transition matrix and other important system variables. Here, the state of the system is defined as the total number of non-empty queues in the system at the instants of termination of the service quanta. The choice of this definition is made in the interest of limiting the number of possible states to the total number of queues, N : because a more complete specification of the system would require 2N possible states. Such a definition necessitates an approximation in the derivation of the recursive expressions. However, comparison with simulation results (Tables 1 and 2) shows that this approximation is quite acceptable. From these recursive expressions, it is shown that a number of useful system parameters, such as average message response time, average cycle time, and average response time conditioned on message length. can be readily derived. Theory The solution for the system parameters is based on three stationary matrices: State transition: P = [ p i j ] , Walking time: W = [ w i j], and Cycle time: c’ = [Cij]. The terms “walking time” and “cycle time” as used above are to be interpreted, respectively, as the elapsed times between successive service quanta and between R. M. WU AND YEN-BIN CHEN IBM J. RES. DEVELOP. successive departures from a non-empty queue. The exact dependence of the system parameters on these matrices is discussed in detail in the third section. Let T , ( k = 1 , 2, . . ., a; Tkl < T , < Tk+l) be successive instants of termination of the service quanta, and STk be the state of the system at instant T,; then P i j Prob [STk+ , = j l S , = i], 0 5 (i , j) 5 N ; ~ , ~ = P r o b [ ( T , , , T , ) = ( t , + j t , ) I S ~ , = i ] , 0 5 i 5 N ; 1 5 j ; cij =Prob [ ( T , , T,) = ( j t s + Nt , ) IS 'k = i , Q Tk = 11, 0 5 ( i , j ) 5 N; where QTk = 0 if the server departs from an empty queue at instant T,, and QT, = 1 otherwise. The quantities N I t , and t , are as defined in the Introduction. The state transition probabilities p i j may be written as 1 p i j = Prob [ST,+l = j J S T , = i, Qr, = r] r=O X Prob [ QT, = rJ ST, = i] . Let Pr = [pz] ; pLj = Prob [ S = jl STk = i. QT, = r] ; 5.. Tk+ 1 Prob [ Q , = 01 ST, = i] for i = j 0 otherwise; = Z = [5,,], 0 5 ( i? j ) 5 N ; then, P = Z . P O + ( I Z ) . P I , (1 ) where I is the unit matrix. It is clear that P is both irreducible and aperiodic; hence? there exists a unique vector solution to the steady state equation: p = p . p , where p = { p , } ; p i = Prob [ S T , = i ] ; 0 5 i 5 N ; k + m. From Eq. ( 1 ) the steady state equation may be written as p . [Z . PO + (I Z) . P'] = p. (2) In Appendix A we show that the matrices Po and P' may be obtained in recursive form. Still, vector p cannot be obtained directly from Eq. (2) because the matrix Z is also a function of p, as can be seen from the definition of [ i j . The functional dependence, however, may be obtained through the following relation: Prob [ ST, = i + 1 I QTk = 11 = Prob [ ST, = iJ QTk = 03. t 'k I + Figure 1 Schematic diagram of a loop model. Because the message lengths are geometrically distributed with mean R , Prob [Q,, = 01 = 1 / R . Consequently, Prob [ S T , = i + 1 , Q , = 11 Prob [ Q , = 11 Prob [ Qrk = 03 Prob [ S T , = i, e, = 01 = ( R 1 ) Prob [ S T k = i, Q,, 01. On the other hand, Prob [S. = i + 1, Q , = 13 /x + Prob [ S T , = i + 1 , Qrk = 01 = pi+l. Combining these equations yields the recursive relation Prob [ S T , = i + 1 , QT, = 01 pi+l ( R 1 ) Prob [ST, = i, Qrk = 01. Observe that Prob [ST, = 0, Q , = 13 = 0; therefore, Prob [ ST, = 0, QTk = 01 = po. Then the recursive relation generates the closed form solution below: