Division of Research Graduate School of Business Administration The University of Michigan December 1983 Revised February 1985 Revised December 1985 ON THE NONCONCAVITY OF THROUGHPUT IN CERTAIN CLOSED QUEUEING NETWORKS Working Paper No. 356-c Kathryn E. Stecke Graduate School of Business Administration The University of Michigan Ann Arbor, Michigan Performance Evaluation, Vol. 6, No. 2, 1986, forthcoming.

ON THE NONCONCAVITY OF THROUGHPUT IN CERTAIN CLOSED QUEUEING NETWORKS Kathryn E. Stecke Graduate School of Business Administration The University of Michigan Ann Arbor, Michigan May, 1984 Revised, February, 1985 Revised, December, 1985 Keywords: Closed Queueing Networks, Flexible Manufacturing Systems, Performance Evaluation, Nonconcavity of Throughput

ABSTRACT Analytic queueing network models are being used to analyze various optimization problems such as server allocation, design and capacity issues, optimal routing, and workload allocation. The mathematical properties of the relevant performance measures, such as throughput, are important for optimization purposes and for insight into system performance. We show that for closed queueing networks of m arbitrarily-connected singleserver queues with n customers, throughput, as a function of a scaled, constrained workload, is not concave. In fact, the function appears to be strictly quasiconcave. There is a constraint on the total workload that must be allocated among the servers in the network. However, for closed networks of two singleserver queues, we prove that our scaled throughput is concave when there are two customers in the network and strictly quasiconcave when there are more than two customers. The mathematical properties of both the scaled throughput and reciprocal throughput are demonstrated graphically for closed networks of two and three singleserver queues.

1. INTRODUCTION Closed queueing network models have recently been used to analyze design issues and planning problems of both computer systems and flexible manufacturing systems. Throughput, a main performance measure of interest, can be defined as a complex, nonlinear function of several system parameters. The mathematical and qualitative properties of this function are of interest for optimization and performance evaluation purposes. For example, in the problem of maximizing throughput subject to a set of contraints, it is necessary to know if a local maximum is a global maximum. In addition, studying the qualitative properties of throughput is also useful for the analytic insight that is provided. Various versions of this problem have been reported in the computer science literature. For example, Trivedi and Kinicki [1978], Trivedi and Wagner [1979], Trivedi, Wagner, and Sigmon [1980], Trivedi and Sigmon [1981], and Kobayashi and Gerla [1983] maximize throughput in central server, single class, closed queueing networks (CQN) with a single server at each node subject to various budgetary limitations (cost constraints). The various studies optimize different parameters (decision variables) such as service rate (of a CPU, say), capacity of servers (I/O devices), device speeds, routing, and main memory size, often subject to a budget constraint. These parameters relate cost considerations to performance. All of these studies prove the convexity of reciprocal throughput in order to insure that the maximum throughput (minimum average delay or response time) is global, and not just a local optimum. To prove convexity, these studies use the results of Price [1974], who proved convexity for a particular scaled version of reciprocal throughput. However, the reciprocal of a convex function is not necessarily concave (Martos [1975]). In fact, the reciprocal of a convex function can be either quasiconcave or quasiconvex. There could be some benefits and additional

-2 -insights from investigating the mathematical properties of throughput directly. For example, Suri [1983] analyzes the sensitivity (and bounds on sensitivity) of throughput to variations in workload, as well as other properties of throughput. There have not been many studies that analyze throughput directly. Kenevan and von Mayrhauser [1984] show that throughput is a log convex function of the number of items in a closed, single class, network of an arbitrary number of single and instant servers. They also prove that reciprocal throughput is a convex function of the relative utilizations of the servers. This is a generalization of Price's [1974] proof. The following studies provide results concerning optimal solutions (workload allocations and server configurations) to problems of maximizing throughput in both singleserver and multiserver CQNs. Kobayashi and Gerla [1983] determine the optimal routing to maximize throughput in central-server, singleserver, single class CQNs. Stecke and Morin [1985] and Yao [1984] show that balancing workloads, for various scalings, maximizes throughput in singleserver, arbitrarily-connected CQNs. Shanthikumar and Stecke [1986] prove that balancing the workloads in singleserver CQNs minimizes in-process inventory under various strategies to release items to the network. For multiserver CQNs, Stecke and Solberg [1985] and Yao [19841 prove that balancing workloads per queue maximizes throughput when each queue has the same number of servers. However, Stecke and Solberg [1985] also show that when the number of servers in each queue is not the same in multiserver CQNs, the throughput is maximized by a unique unbalanced workload per server. In fact in this situation, the throughput function appears not only to not be concave, but not symmetric as well. Unbalanced optimal workload allocation ratios can be found at which the workload per server should be maintained to maximize

-3 -throughput for these networks of unbalanced multiserver queues. These allocation ratios can serve as input to more detailed workload allocation problems that are solved using more detailed (mathematical programming) models. See, for example, Berrada and Stecke [1985] and Stecke [1983, 1985a, 1985b]. We consider here a particular product form, non-central server CQN of arbitrarily-connected singleserver queues, of which the central server model is a special case. Rather than the budgetary constraints of the previous studies, we impose a constraint on the total workload in the system. The motivation for our particular CQN model is provided in the studies of optimal workload allocation and server (machine) allocation in flexible manufacturing systems (FMSs). In particular, we show, contrary to previous conjectures (Secco-Suardo [1978], Solberg [1979]), that throughput (or production rate) is not concave as a function of workload. In this paper, we show that throughput, as a function of the ratio of the "workload" (service demand) at a server to the sum of workloads is quasiconcave and not concave. Since Price [1974] and Kenevan and von Mayrhauser [1984] do not consider throughput to be a function of the same quantity (a ratio of server to total workload) and do not constrain the total workload to be allocated, their results do not necsssarily apply. However, there is evidence that the reciprocal throughput function studied here is convex, despite the particular scaling of workload and throughput. The plan of the paper is the following. In ~2, the CQN model is defined. We prove the nonconcavity results by induction in ~3. First, the concavity of throughput is proven for a closed network of two singleserver queues with two customers. Then the nonconcavity is established numerically for a network of two singleserver stations with n (greater than two) customers. Next, strict quasiconcavity of throughput is established for a CQN with two singleservers. A concave function is also quasiconcave; however, we also show in ~3 that this

-4 -scaled throughput function is not concave for m queues and n customers. For definitions of generalized concavity, see Bazaraa and Shetty [1979] or Martos [1975]. In ~4, some evidence that our scaled version of reciprocal throughput is convex is provided. If this is true, we can prove that throughput is strictly quasiconcave. ~5 concludes with a brief summary. 2. THE CLOSED QUEUEING NETWORK MODEL The product form CQN that is considered here consists of m arbitrarilyconnected singleservers, of which the central server model is a special case. There are always n items being processed in the system. The average processing time of an item at station i is ti, i = l,...,m. The routing of items among the stations is arbitrary. The routing can be described by visit frequencies, or relative arrival rates, qi, where qi can be the probability that the next server visited is i. In addition, the qi can be provided by the traffic equations, qi = y Pji q.. Details of other routing possibilities can be found in Stecke and Schmeiser [1982]. The queueing discipline can be either FCFS, infinite server, LCFS preemptresume, processor sharing (see Baskett et al. [1975]), random selection, or one developed by Kelly [1979] that allows an arbitrary distribution to be defined at each server. The service time distribution is arbitrary, except for FCFS servers, which require exponential service times. The usual measure of relative workload assigned to server i is w. (Buzen [1973], Reiser and Kobayashi [1975], Solberg [1977]) which is defined as the product of visit frequency and average processing time, or w. = qiti. These workloads are relative since the qi's need not sum to one. m For our purposes w. was scaled, where I q.t /m is the average workload j-= J per server, to provide: m X = qi /[( qj t i)/m] (1) i ii j=l

-5 -This particular constraint on workload was chosen for many reasons associated with determining qualitative properties of optimal allocations of servers and workloads in flexible manufacturing systems —see Stecke [1985b], Stecke and Schmeiser [1982], Stecke and Morin [1985], and Stecke and Solberg [1985] for details on these studies. The state of the system is given by n = (nl,...,nm), where ni is the number of items at server i, both those waiting and in process. For all i, we m have ni c {0,1,...,n} and I n. = n. The steady-state probability of i=l 1 being in state? is p(T) = p(nl,...,nm), which has the product form solution: I nI n 2 n m P(n) - X. X x G(m,n;X) 1 2 m where: n n2 n G(m,n;X) = I X1 X2.X i=l,.,m. (2) n1+n2t+.. *nm=n ni > Throughput can be defined as a function of G(m,n;X), which in turn is a function of assigned workload, Xi. In fact, for a particular scaling of qi, the throughput, or production rate, Pr(m,n;X), is given by (Reiser and Kobayashi [1975]): n1 n2 n n +..X1 X2...Xm ~~1 m nl+.. +n =nm 2 m G(mn-;X) n. > 0 1 -n1 >0

-6 -The throughput for two singleservers and any number of items is: n1 n2 I X1 X2 nl+n2=n-1 2 Pr(2,n;X) = n n x, n~ I X 1X 2 nl+n2=n n-1 nt n-l-ni I X1 (2-X1 ) = n ~ O, 1 1 n n n-n X1 1(2-X1) n1 =0 since X+X =m = 2 (with our scaling) Xl-(2-X )n n+l - x -(2-X1 ) = x'^-cz)npl ' by dividing both numerator and denominator by (2-X1 )-X1=2(1-X1 ). (4) Throughput, as given in equation (3), is difficult to characterize analytically. However, it can be evaluated numerically using Buzen's efficient algorithm [1973]. 3. (NON)CONCAVITY OF THROUGHPUT In this section, first the concavity of throughput is proven for a closed network of two singleserver stations with two items. Then, strict quasiconcavity, but nonconcavity, of throughput is established for a network of m (> 2) singleserver queues with n (> 3) items. For a network of two singleservers with two items, from equation (4): x2-x2 x +X Pr(2,2;X) = 2. X1 -X 12 2+ Substituting X2 = 2 - X1, simplifying, and then dropping the subscript, we obtain: Pr(2,2;X) = 2 4-2X+X2

-7 -Theorem 1: Pr(2,2;X) is a concave function. Proof. Taking the first derivative of Pr(2,2;X) yields: d Pr(2,2;X) = -2(2X-2) -4(X-1) = -4(X-1) dX [4-X(2-X)]2 [4-X(2-X)]2 (X2-2X+4)2 Setting Pr'(2,2;X)=0 yields X=1. Now d2Pr(2,2;X) -4(X2-2X+4)-4(4-4X)(X-1) dx2 (X2-2X+4)3 -4X2+8X-16+16(X2-2X+1) 4(3X2-6X) 12X(X-2) (X2X+ (X2 -2X+4) 3 (X2-2X+4) -2X+4)3 Setting Pr''(2,2;X) = 0, the points of inflection are at X = 0 and 2. Note that for every X [0,1), Pr'(2,2;X) > 0, which implies that Pr(2,2;X) is increasing on [0,1); for every X (1,2], Pr'(2,2;X) < 0, which implies that Pr(2,2;X) is decreasing on (1,2].11 Theorem 2: Pr(2,n;X) is not concave for n > 3. Proof. X_ X 2 Xn-(2-X1 )n Pr(2,n;X) = 2n+l | X +Xn=2 n+ xn+l n+11 xI+X2=2 xn+I-(2-X n+I 1 2 1 1 Again the subscript is suppressed for convenience. Taking the derivative with respect to X yields: n+1 n+1 n-I n-i _XX nl nnj d Pr(2,n;X) [Xn+ -(2-X)n+l]n[n+(2-X)n][n-(2 )n](n+l)[+(2-X) dX [Xn+l -(2-X)n+1 2 which upon rearranging yields: - 2n+(2-x )2n+4nXn1 (2-X)n-1 (X- ) [Xn+1_(2_X)n+ 1]2 Evaluating at X=1, Pr'(2,n;X) = 0/0. Upon two applications of l'Hospital's rule we obtain: Pr'(2n;1) = 4n(n-1)(-2)+4n(n-1)+4n(n-1) 0 2(n+1)2 4 8(n+12 0 2(n+l) 4 8(n+l)2

-8 - Therefore, X=1 is a critical point. Taking the second derivative with respect to X and rearranging we obtain: d Pr(2,n;X) 2[X3n-(2-X)3n+Xn-2(2-X) 2n-(X3-(8n+2)X2+(-4n +8n)X+4n -4n) dX2 [Xn+l-(2-X)n+1]3 X2n- (2-X) n-2 (X3+(8n-4)X2+(-4n2-24n+4)X+4n2+20n)] [Xn+-(2-X) n+] 3 The throughput function is now demonstrated graphically to be: i) convex for X1 e [O,X'], X' < 1; ii) concave for X1 e [X',X"], X" > 1; and iii) convex for X e [X",2]. Then there are three points of inflection: at X', 1, and X". Moreover, X' and X" are symmetric about the point X=1. The points of inflection of Pr(2,n;X) can be found by setting the numerator of the second derivative of the nonlinear equation (5) equal to zero and solving for the roots. Two different IMSL (International Mathematical and Statistical Library [1979]) routines, called ZREAL1 (see Muller [1956] and Leavenworth [1960]) and ZREAL2, were used to find the roots. Both were used as a check on accuracy and to help note any numerical or roundoff problems. Each routine finds N real zeros of a function F(Y). The routines were set up to search for 5; each always found only three roots, including and symmetric about X=1. There are two convergence criteria necessary. X. is a root if: i) JF(Yi)J < EPS, and?m+l Ym i i -NSIG ii) 1 N< 10 1

-9 - In the program, EPS (epsilon) was set equal to 1.E-8 and NSIG=5. The roots remained the same for both ESP=1.E-5 and 1.E-8, which implies that sufficient accuracy was attained. Both routines found the same roots, as seen both in Table I and graphically in Figure 1. The graph and points of inflection show that Pr(2,n;X) is not concave on [0,2].11 TABLE I Points of Inflection and Approximation for n = 2, 3, 4, 5, 10, and 99 ZREAL1 ZREAL2 2n-3 2n+l n = 2 X=0, 1, X=0, 1, and 2 and 2.42265.42265 3 =.4286 n = 3 1.0 1.0 7 1.57735 1.57735.55452.55452 5 =.5555 n = 4 1.0 1.0 9 1.44548 1.44548.62943.62943 7 =.6366 n = 1.0 1.0 11 1.37057 1.37057.71563.71563 11 =.7333 n = 7 1.0 1.0 15 1.28437 1.28437.78377.78377 17 =.8095 n = 10 1.0 1.0 21 1.21623 1.21623.89339.89339 47 =.9215 n = 25 1.0 1.0 51 1.10661 1.10661.9649.964903 195 =.9799 n = 99 1.0 1.0 199 1.0351 1.035097___ However, both Table I quasiconcave with a global and Figure maximum at 1 indicate that throughput is strictly X=1. Theorem 3: For n > 2, Pr(2,n;X) is a interval X. e [0,2], i = 1, 2. 1 strictly quasiconcave function on the Proof. For each n > 2, there are three critical points, one at X. = 1 that 1

-10 - 0) 0L o O 0.9 I O. c: 0.8 - z 0 - 0.6 D a. 0.5 0 0.5 1.0 WORKLOAD ON MACHINE I Figure 1 1.5 2.0 Graph of Pr(2,n;X), n=2-5, 10, and 99: Maxima (X) and Points of Inflection (o).

-11 - gives the maximum Pr(2,n;X), and the remaining two symmetric about the line Xi = 1. The function is increasing for X1 e [0,1) and decreasing for X1 (1,2]. Therefore, throughput is quasiconcave for X1 e [0,2] with the unique global maximum at X1 = X2 = 1. | The last column of Table 1, labeled (2n-3)/(2n+l), shows the results of the attempt to provide a simple function that would closely approximate the values of the roots of Pr(2,n;X), or the points of inflection. We now show that throughput is not concave. Theorem 4: Pr(m,n;X) is not concave for any m > 2 and n > 3. Proof. Consider the throughput function for any m or n: Pr(m,n;X) = n n2 n 2 m x X x 1 2 m n n2 n E, x1 X2 * m n +...+n =n 1 m Evaluating Pr(m,n;X) along any hyperplane such that Xi=0 for m-2 of the i, say, for i = 1,2,...,m-2, we have: Pr(m,n;X) = n +n I n-l m-1 m n M+n =n m-1 m n n Xm-1X m n m m-1 m m-1 m in-i in

-12 - n-l n n-l-n "1 m-1 X (m-X n) m-=0 1 _-1 n n n-n rn m-i rnm-1 X 1 (m-X ) rm-1 n n X (m-X ) m-1 m-1 n+l n+i X -1 (m-X )n+ n xn m-1 m m- l 1, nsince Xi+X2+...+X X +X = m. Xn+l Xn+1 12 m m-1 m m-1 m But this is the same form as Pr(2,n;X), which has already been shown to be not concave in Theorem 2.1 The following figures help to further demonstrate and clarify the behaviour of throughput. Figures 2 and 3 are different views of a 3-dimensional graph of Pr(3,5;X). (Numerous other plots of Pr(3,n;X) for many values of n are very similar in form to these.) Both figures show the function over its entire range of relevant workload values: since there are three singleserver queues in the closed network, our scaling ensures that X1 + X2 + X3 = 3, with each X. e [0,3]. The quasiconcavity can be seen as the function dips near the extreme boundary points. Figure 4 appears to demonstrate some bizarre behaviour of the throughput function, particularly outside the dashed box. The function appears to change direction. However, the function is well-behaved within the dashed lines, which define the relevant range for our scaled workload. Figure 5 interestingly demonstrates the non-symmetry of a one-dimensional slice of Pr(3,n;X) over a range of n, despite the symmetry of the entire function. Figure 5 also shows the strict quasiconcavity of throughput as a function of workload.

c'T ' ['0]3 X ' (X'S'c)ad Jo qdeio z ai3nS-F. (XDS'f)8d ~05 C P C -C I

-14 - PR(3,NXr) -- N=5 Figure 3 Graph of Pr(3,5;X), X1,2 e [0,3]. 1,2

-15 - 4.0 3.0 2.0 1.0 0.0 -- -1.0 -2.0 -2.0 Figure 4 Contour map of Pr(3,5;X).

-16 - 1.000 >hz l-i z I — CC m cr ci CrI z D LJ FI: z CE a — f —I FCI ZD 0 -c_) as].880.780.640.520 -.400 0.833 1.667 2.500 WORKLOAD PER MACHINE: X1I THREE GROUPS OF SIZES 1,1 AND 1 Figure 5 A one-dimensional slice of Pr(3,n;X) as a function of server 1 for n=4,...,14 and. '.

-17 - Figures 6 and 7 also show the strict quasiconcavity of the production function. When contrasted with Figures 2 and 3, the behavior is seen to exaggerate as n increases. In this example, n doubled, from five to ten customers. The closed network is more congested. 4. RECIPROCAL THROUGHPUT For certain singleserver queueing networks, reciprocal throughput has been shown to be convex. (For example, see Price [1974].) However, Price does not consider throughput to be a function of the same quantities that are considered in this paper and does not consider the same total workload constraint. Hence, his results do not necessarily apply to our scaled versions of workload and reciprocal throughput. However, although we have not formally proven convexity, we can offer some computational evidence that our scaled reciprocal throughput function is also convex. In particular, Figure 8 demonstrates convexity for a closed network of two singleserver queues, for a variety of n, ranging from n = 2 up to 99. Also, Figure 9 demonstrates the convexity for a closed network of three singleserver queues with the number of customers, n, equal to 5. Graphs for other values of n are similar. Finally, Figure 10 provides some values, via a contour graph, for the reciprocal throughput function, Pr(3,5;X)-1, over the relevant range of scaled workload, X. ~ [0,3], for i = 1, 2, and 3. These figures and many other similar graphs provide evidence that reciprocal throughput is convex. If this observation is true, we can use some previous results in mathematical programming to prove directly that throughput is strictly quasiconcave.

-18 - d m 0 -d v PR(3,10X) Figure 6 Pr(3,10;X), X12 e[0,3].

-19 - 0 a PR(3,N1) Figure 7 -- N=10 Pr(3,10;X), X1 2 e[0,3].,2'

-20 -PR (2,n; X)1 CMJ = 2 X: X = 0.0 TO 2.0 Figure 8 Pr(2,n;X), for a variety of n.

-21 - I nL lY Pr(3,5;X)1 Figure 9 A three-dimensional graph of Pr(3,5;X), for Xi e [0,3]. ~

-22 - 3.0.0 '..0 2.0 ~ --- —-E. 0 O.O 0.0 0 2 3.0 Figure 10 1Contou

-23 - In particular, the following result that is stated as Theorem 5 can be proven in several ways. One proof can use 3.39 on page 116 of Bazaraa and Shetty [1979]. Another can use Variant H of Table 3.4 on page 63 of Martos [1975]. Our proof shall use the latter. Theorem 5: Throughput is strictly quasiconcave, if reciprocal throughput is convex. f(x) Proof. Variant H of Martos [1975] can be stated: A function g- is g(x) strictly quasiconcave, if f(x) is concave and nonnegative and g(x) is convex and positive. Let f(x) = 1 and g(x) = reciprocal throughput. Then f(x) is clearly concave and nonnegative for all x, and g(x) is convex by assumption and positive. Hence throughput is strictly quasiconcave. | We note that without our particular scaling of workload (expressed via the constraint: X1 + X2 +...+ X = m), our reciprocal throughput function given 1 / m by equation (3) (Pr(m,n;X)-l) can be shown to be convex. One proof would mimic that found in Kobayashi and Gerla [1983]. 5. SUMMARY We have attempted to provide some mathematical and qualitative insights into a particularly useful scaled version of both throughput itself and reciprocal throughput as functions of a particular scaled workload measure. As a result, throughput is also scaled. In fact, it represents a probability: all values lie between zero and one. (See Stecke and Schmeiser [1982].) To our knowledge, such properties concerning the generalized concavity, of throughput in particular, have not previously been investigated. This is,

-24 - in part, because reciprocal throughput has been easier to get a handle on and is also better behaved. If this particular, scaled, reciprocal throughput function is formally proyen to be convex, Theorem 5 is required to characterize throughput itself directly, since the reciprocal of a convex function can be either quasiconcave or quasiconvex. (Recall Table 3.4 on page 63 of Martos [1975].) Then throughput is strictly quasiconcave. Of course, if additional, general information can be discovered about either function, some of that information can be useful, for example, for performance evaluation or optimization purposes or general insights into the behaviour of the functions. ACKNOWLEDGEMENT This research was supported in part by NSF Grant No. ECS 8406407 as well as by a Grant from the Graduate School of Business Administration of The University of Michigan.

-25 - REFERENCES BASKETT, FOREST, CHANDY, K. MANI, MUNTZ, RICHARD R. and PALACIOS, FERNANDO G., "Open, Closed, and Mixed Networks of Queues with Different Classes of Customers," Journal of the Association for Computing Machinery, Vol. 22, No. 2, pp. 248-260 (April 1975). BAZARAA, MOKHTAR S. and SHETTY, C. M., Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, New York NY (1979). BERRADA, MOHAMMED and STECKE, KATHRYN E., "A Branch and Bound Approach for Machine Load Balancing in Flexible Manufacturing Systems," Working Paper No. 329-c, Division of Research, Graduate School of Business Administration, The University of Michigan, Ann Arbor MI (October 1985). BUZEN, JEFFREY P., "Computational Algorithms for Closed Queueing Networks with Exponential Servers," Communications of the Association for Computing Machinery, Vol. 16, No. 9, pp. 527-531 (September 1973). IMSL Reference Manual, IMSL, Inc., Houston TX (1979). KELLY, FRANK P., Reversibility and Stochastic Networks, John Wiley & Sons, New York NY (1979). KENEVAN, JAMES R. and VON MAYRHAUSER, ANNELIESE K., "Convexity and Concavity Properties of Analytic Queueing Models for Computer Systems," Performance '84, E. Gelenbe (Editor), pp. 361-375, Elsevier Science Publishers B.V. (North-Holland) Amsterdam (1984). KOBAYASHI, HIROSHI and GERLA, MARIO, "Optimal Routing in Closed Queueing Networks," ACM Transactions on Computer Systems, Vol. 1, No. 4, pp. 294-310 (November 1983). LEAVENWORTH, B., "Algorithms 25: Real Zeros of an Arbitrary Function," Communications of the Association for Computing Machinery, Vol. 13 (1960). MARTOS, BELA, Nonlinear Programming: Theory and Methods, North-Holland, Amsterdam (1975). MULLER, D. E., "A Method for Solving Algebraic Equations Using an Automatic Computer," Mathematical Tables and Aids to Computation, Vol. 10, pp. 208 -215 (1956). PRICE, THOMAS GORDON, "Probability Models of Multiprogrammed Computer Systems," Ph.D. dissertation, Department of Electrical Engineering, Stanford University, Stanford CA (December 1974). REISER, MARTIN and KOBAYASHI, HISASHI, "Horner's Role for the Evaluation of General Closed Queueing Networks," Communications of the Association for Computing Machinery, Vol. 18, No. 10, pp. 592-593 (October 1975).

-26 - SECCO-SUARDO, G., "Optimization of a Closed Network of Queues," Report No. ESL-FR-834-3, Electronic Systems Laboratory, M.I.T., Cambridge MA (1978). SHANTHIKUMAR, J. GEORGE and STECKE, KATHRYN E., "Reducing Work-In-Process Inventory in Certain Classes of Flexible Manufacturing Systems," European Journal of Operational Research (1986), forthcoming. SOLBERG, JAMES J., "A Mathematical Model of Computerized Manufacturing Systems," Proceedings of the International Conference on Production Research, Tokyo, Japan (August 1977). SOLBERG, JAMES J., "Stochastic Modeling of Large Scale Transportation Networks," Report No. DOT-ATC-79-2, School of Industrial Engineering, Purdue University, West Lafayette IN (1979). STECKE, KATHRYN E., "Formulation and Solution of Nonlinear Integer Production Planning Problems for Flexible Manufacturing Systems," Management Science, Vol. 29, No. 3, pp. 273-288 (March 1983). STECKE, KATHRYN E., "Useful Models to Address FMS Operating Problems," Proceedings of the Advances in Production Management Systems Conference, J. Browne and E. Szelke (Editors), pp. 271-283, Budapest, Hungary, Elsevier Science Publishers B.V. (North-Holland) Amsterdam (1985a). STECKE, KATHRYN E., "A Hierarchical Approach to Solving Machine Grouping and Loading Problems for Flexible Manufacturing Systems," European Journal of Operational Research, (1985b), forthcoming. STECKE, KATHRYN E. and MORIN, THOMAS L., "The Optimality of Balancing Workloads in Certain Types of Flexible Manufacturing Systems," European Journal of Operational Research, Vol. 20, No. 1, pp. 68-82 (April 1985). STECKE, KATHRYN E. and SCHMEISER, BRUCE W., "Equivalent Representations of System Throughput in Closed Queueing Network Models of Multiserver Queues," Working Paper No. 324, Division of Research, Graduate School of Business Administration, The University of Michigan, Ann Arbor MI (December 1982). STECKE, KATHRYN E. and SOLBERG, JAMES J., "The Optimality of Unbalancing Both Workloads and Machine Group Sizes in Closed Queueing Networks of Multiserver Queues," Operations Research, Vol. 33, No. 4, pp. 882-910 (JulyAugust 1985). SURI, RAJAN, "Robustness of Queueing Network Formulas," Journal of the Association for Computing Machinery, Vol. 30, No. 3, pp. 564-594 (July 1983). TRIVEDI, KISHOR S. and KINICKI, ROBERT E., "A Mathematical Model for Computer System Configuration Planning," Performance of Computer Installations, D. Ferrari (Editor), North-Holland, Amsterdam (1978).

-27 -TRIVEDI, KISHOR S. and SIGMON, TIMOTHY M., "Optimal Design of Linear Storage Hierarchies," Journal of the Association for Computing Machinery, Vol. 28, No. 2, pp. 270-288 (April 1981). TRIVEDI, KISHOR S. and WAGNER, ROBERT A., "A Decision Model for Closed Queueing Networks," IEEE Transactions on Software Engineering, Vol. 5, No. 4, pp. 328-332 (July 1979). TRIVEDI, KISHOR S., WAGNER, ROBERT A. and SIGMON, TIMOTHY, M., "Optimal Selection of CPU Speed, Device Capabilities and File Assignments," Journal of the Association for Computing Machinery, Vol. 27, No. 3, pp. 457-473 (July 1980). YAO, DAVID D. W., "Some Properties of the Throughput Function of Closed Networks of Queues," Technical Report, Department of Industrial Engineering, Columbia University, New York NY (1984).