Division of Research Graduate School of Business Administration The University of Michigan March 1983 Revised, December 1983 EQUIVALENT REPRESENTATIONS OF SYSTEM THROUGHPUT IN CLOSED QUEUEING NETWORK MODELS OF MULTISERVER QUEUES Working Paper No. 324-B Kathryn E. Stecke The University of Michigan and Bruce W. Schmeiser Purdue University

ABSTRACT The steady-state expected production (throughput) of many closed queueing network models of multiserver queues is a function of the number of items and facilities in the system, the number of servers at each facility, and the workload allocated to each facility. With appropriate scaling, the expected production is given six representations, each with a corresponding practical interpretation. These representations are discussed and proved equivalent. The representations both provide insight and are useful in proving some results relating production to server grouping and workload allocation decisions.

-1 - 1. INTRODUCTION Systems in which parts or customers visit several different facilities for processing can, in some cases, be adequately modeled by a multiclass or multiserver network of queues. Multiclass models can depict sequential, fixed routing for some types of parts, while multiserver models are used when there can be more than one server at some facilities. The multiclass models have been used to evaluate the performance of both computer systems (see Kleinrock [1976], Rose [1976], Baskett et al. [1975], Reiser and Kobayashi [1975, 1976], and Bard [1979]) and flexible manufacturing systems (FMSs) (see Cavaille and Dubois [1982]). An FMS consists of a set of numerically controlled machine tools, interconnected with automated material handling equipment, capable of the simultaneous and efficient manufacture of a variety of part types. Most real-time functions, such as actual machining operations, automatic tool interchange, and part movement are under the control of one or more computers (see Buzacott and Shanthikumar [1980] and Stecke [1977, 1983]). Multiserver models are well-suited for representing those FMSs where functionally similar machine tools can be pooled into a machine group (multiserver queue). These machine tools are then identically tooled so as to be able to perform the same operations during real-time control, as discussed, e.g., by Stecke and Solberg [1981a, 1981b, 1982]. Pooling machines into groups allows the machine redundancy required to automatically reroute in machine breakdown situations. Closed networks of arbitrarily connected multiserver queues are considered in this paper. The results are therefore also applicable to the central-server model. Manufacturing, rather than computer, performance terminology is used, reflecting the motivating FMS application. The correspondence with queueing terminology is direct and meaningful: expected production is throughput,

-2 - part is job or customer, part type is customer type, machine group is multiserver queue, and machine is device or server. Using the closed queueing network (CQN) model, the expected production (throughput) of an FMS is defined in ~2 as a function of several system parameters. As a result of our particular scaling of one of the parameters — the workload assigned to each machine group —several alternative interpretations of production, defined in ~3, are proven equivalent in ~4. Some of the representations are new; the equivalence of all six is new. These alternative representations help the user of queueing network models to better understand and to interpret the mathematical representation of expected production obtained from the CQN model. They also provide additional mathematical tools that can be used to discover and prove properties of optimal solutions of several associated performance optimization problems, as discussed in ~5. Finally, they can motivate further research into the modeling of FMSs and their performance. A brief summary is provided in ~6. The current research was motivated by investigations of two particular FMS production planning problems, called the grouping problem (how to best partition m machines into g machine groups) and the loading problem (how to best allocate operations and associated tooling among groups). For both problems, the objective is to maximize expected production subject to FMS technological and capacity constraints (see Stecke and Morin [1982], Stecke and Solberg [1982], Stecke [1977, 1982, 1983]). The alternative representations are useful in providing insight into understanding the mathematical definition of expected production from CQN models of FMSs. 2. THE CLOSED QUEUEING NETWORK MODEL Consider a single-class, closed system containing n parts, each of the same part type. The system consists of m machines (servers) that have been

-3 - partitioned into g machine groups, with si machines in group i. The average processing time of an operation by a machine in group i is ti, i=l,...,g. The routing is arbitrary, and can be described by the visit frequencies, or relative arrival rates, qi, where qi can be: i) the probability that the next machine visited belongs to group i; ii) the average number of times per some time period that a machine in group i is visited; or iii) the mean number of operations per part at machine group i. For example, to model realistic systems containing more than one part type, the values of qi can be determined by qi = a. rij J = the average number of operations per part at group i, where aj = production ratio of part type j relative to all part types currently being produced; aj > 0, for all j, and I a. = 1; and j J ri = number of operations on part type j performed by group i. The qi can also be solutions to the traffic equations, qi = Pjiqj. The routing might also be first-order Markovian (defined by transition probabilities, Pij), multiple-class Markovian (defined by transition probabilities for each part type k, pij(k) —see Reiser and Kobayashi [1975]), higherorder Markovian (for example, second order is defined by Pijk, which is the probability that a part previously at i, now at j, goes next to k —see Kobayashi and Reiser [1975]), or fixed routes through the system (defined by routing vectors for each part type, r(k) = (r(k,l), r(k,2),...), where r(k,j) is the index of the j'th machine group visited by a part of type k —see Kelly [1979]). All of these routing mechanisms produce the same values for certain output measures such as expected production, as does the qi. For additional routing details, see Stecke and Solberg [1981a].

-4 - The queue discipline can be FCFS, infinite server, LCFS preempt-resume, processor sharing (see Baskett et al. [1975]), random selection, or an arbitrary distribution defined at each node (Kelly [1979]). The service time distribution is arbitrary, except for FCFS machine groups, which require exponential service times. The usual measure of the relative workload assigned to group i is wi, the product of visit frequency and average processing time, or wi = qiti, i=l,...,g (Buzen [1973], Reiser and Kobayashi [1975], Solberg [1977]). These workloads are relative in that the qi's need not sum to one. The state of the system is denoted by ni = (nl,n2,...,ng), where ni is the number of parts at machine group i, both those waiting and those in process. g For all i, ni is an integer between zero and n, and I ni = n. The steadyi=l state probability of being in state n is denoted by p(n) = p(n1,n2,...,ng) which has the product form solution: 1 g p(n) =G(g,n;S,W) i=1 fi(ni)' (1) where S = (sl,s2,...,sg), W = (W1,w2,...,wg), and the normalizing constant is: G(g,n;S,W) I f (nl)f 2(n2)...f (n ) (2) nl>0 n2>0... n g g 1- 2gn + n2 +...+ng =n and ni ni < Si. n.! ni f i(n.)= n w, ni > si; i=..,g. (3) n -si si!s i 1 1

-5 - Single-machine, multiple-machine, and infinite-machine groups correspond, respectively, to si = 1, 1 < si < n, and si > n. The expected production, which is the expected number of parts produced per unit time, is an important performance measure. For a system containing n parts, it is a function of G(g,n;S,W), which in turn is a function of assigned workload, wi, and grouping, si. In fact, for a particular scaling of the qi and ti, which will be provided shortly, the production function, Pr(g,n;S,W), is given by (Reiser and Kobayashi [1975]): G(g,n-!;S,W) Pr(g,n;S,W) =;S,,n;S,W) (4) G(gn;S,W) 3. RESCALING THE WORKLOAD AND PRODUCTION FUNCTION For our purposes, wi is scaled to provide our workload measure: g g Xi q it/[( l q jt)/( X s)]. (5) j=l j=1 J Notice that the numerator is the usual definition of workload assigned to group i, and the denominator is the average workload per machine. Our particular scaling is useful for several reasons: i) For any given number of machines in the system, regardless of their grouping, the total amount of work to be allocated among groups always equals the total number of machines: g g I Xi I Si=m. i=l i=l ii) The workload is independent of any particular scaling of qi and ti; iii) Alternative workloads can be compared with a balanced workload, since a balanced workload, regardless of system size or configuration, is X1/sl = X2/S2 =... = Xg/Sg = 1. g g

-6 - iv) As shall be seen in the following section, this scaling provides a normalized, dimensionless measure of production, whose values lie between zero and one. v) As also seen in the following section, workload definition (5) allows some new, alternative, equivalent representations of expected production, which are useful in proving properties of the production function. Note that equations (1), (2), (3), and (4) remain nearly the same when Xi is substituted for wi. The only difference is that the functions fi(ni) are replaced by hi(ni), by substituting Xi for wi to reflect the chosen scaling. These hi(ni) are also valid factors of the product form solution, p(n). 4. ALTERNATIVE REPRESENTATIONS OF THE PRODUCTION FUNCTION Prior to presenting the different representations of the production function, some preliminary definitions are required. 4.1 Preliminary Notation and Definitions Def 1: Let Ii(n) denote the number of busy machines in group i when the system is in state n. For example, for a system of seven groups with two machines in each group that is in state n = (0,0,1,4,2,0,1), I1(n) = 12(n) = 6(f) = 0, I3(n) = I7() = 1, and I4(n) = I5(n) = 2. Def 2: Let U(f) denote the fraction of machines that are busy in a particular state n. That is, g g U(f) = [ I Ii(n)]/( I si). i=l i=l 1

-7 - U(n) is the ratio of busy machines when the system is in state n and the total number of machines. Referring to the above example, U(n) = 6/14 = 3/7. Def 3: Let PrI denote the expected ideal production rate of a manufacturing system, which is the expected number of parts per time unit produced if all machines are always busy. PrI is a measure of the maximum capacity of the system. Def 4: PrA(n) is the expected actual production rate when the system is in state n, which is the expected ideal production rate weighted by the utilization of machines in n. That is, PrA(n)=PrIU(n)). Def 5: Let PrA denote the expected production rate, which is the actual steady-state expected number of parts produced per time unit. That is, PrA = p(n)PrA(n), i eN g,n where Ng {n I n. = n and n. > 0 for i=l,....g N,n it PrA is the usual production function obtained by substituting wi into equation (4). 4.2 The Alternative Representations The six different characterizations of the production function are presented in the following theorem. Theorem 1. The following alternative representations of the production function, Pr(g,n;S,X), are equivalent.

-8 - g H h.(n ) nfN ni=1 1 A.,n-l. n h (ni) neN i=l g,n m B. (l/m) I k Prob{k machines are busy}; k=l C. I p(n)U(f); fieN g,n D. The steady-state probability that a randomly selected machine is busy; E. The ratio of the actual steady-state expected number of parts per time unit to the expected ideal number of parts per time unit which would be obtained if all of the machines were always busy: PrA/PrI; g E U(n)i=1 hi(n) F. g,n g E II h (n ) deN i=l 1 i g,n The first representation, A, is that defined by equation (4), using hi(ni) rather than fi(ni) Because of our particular scaling of workload as X, rather than w, this usual definition of expected production is scaled to provide the remaining alternative definitions. In particular, representations B and C are both measures of system utilization: the expected fraction of machines busy. Representation D is a measure of single machine utilization. It is the fraction of time busy, a dimensionless quantity normalized to lie between zero and one. Representation E is an efficiency measure. Finally, representation F provides a relationship similar to the-first; however, now the sums are over identical state spaces.

-9 - Notice that all representations are not defined on the same state space. The first, third, and sixth definitions include similar sums over the same state space, each sum involving ( +g ) terms. The summation of the g-l second representation, which contains many fewer terms, is over all machines, m. The fourth is a probability and the fifth provides a transformation from the normalized production efficiency measure to the actual production rate, as measured in completed parts per time unit. 4.3 Equivalence of the Six Alternative Definitions Prior to proving the equivalence of the six representations in Theorem 1, a preliminary result is required. This result mathematically defines PrI, the expected ideal production rate, which is obtained when all machines are always busy, or the maximum system capacity. g g Lemma 2. Pr = ( I s )/( qit ) i=l i=l Proof: Since qiti = workload on machine group i, in time units per part, we have: g iti = i=l total workload on all machine groups. Then g g ( qiti)/( i si) = i=l i=l average workload per machine, given that the machines are always busy, in time units per part. Inverting, we have: g g ( I si)/( I q.t.) = i=l - i=1 1 1 expected production rate of the system when all machines are always busy, in parts per time unit. PrI. 1 =

-10 - We now prove Theorem 1. The representations are proven equivalent in the following order: representation A is equivalent to E; E to C; C to B; C to D; and finally C to F. Proof of Theorem 1: i) Representation A is equivalent to E: I g,n-l g h hi(ni) i=l (by Definition A) I gnN g,n g h hi(ni) i=l g g g ni y [ In ( y sj / q jtj) fi(ni)] fEN i=l j=l j=l g,n-l - - g gg ni I [ H ( I sj / I qjtj) fi(ni)] nefN i=l j=l j=l g,n (by substitution) (by simplification) g g n-l g ( I sj / I qjtj) I [ n fi( j=l j=l iENgn- i=l g,n-1 - g g n g ( I sj / I qjtj) I [ n fi(ni) j=1 j=l fEN i=l g,n g g = PrA/ ( I si / X qiti) i=1 i=l = PrA/PrI which is definition E. ni)] (by equations (3), (2), and (4)) (by Lemma 2) ii) Representation E is equivalent to C: X p(n)U(n) g p(,n Y p(5)U(ff)Pr n EN I n,g Pr (by Definition C)

-11 - - p(n)PrA(n) neN g,n PrI = PrA/PrI which is definition E. Il iii) Representation C is equivalent to B: C p(n)U(n) neN g,n (by Definition 4) (by Definition 5) (by Definition C) m = I p(n) (k/m) k=l nsEN nf {n U(n)=k/m} m -= p(k machines are busy)(k/m) k=l which is definition B. I iv) Representation C is equivalent to D: Prob{a randomly selected machine is busy} (by Definition D) g Si -= X Prob{machine k in machine group i is selected} ) i=l k=l Prob{machine k of group i is busy| machine k of group i is selected} g Si g = I (1 / i si) Prob{machine k in group i is busy} i=l k=l i=l g g = X E{number of busy machines in group i}/( X si) i=l i=l g g = ( F [ p(n)Ii(n)])/( x si) i=l nEN i=l g,n (by Definition 1)

-12 - g g = ( X p(n) I Ii(n))/(. si) A EN i=l i=l = I p(n)U(n) g,n which is dfinitin C. which is definition C. l1 (by Definition 2) v) Representation C is equivalent to F: E p(n)U(n) nii eN g,n -1 g = [G (g,n;S,X) N hi(ni)]U(i) neN i=l g,n g I U(n) H hi(ni) fnN i=l - g,n (by Definition C) (by Definition of p(n), equation (1)) (by equation (2)) g E n hi(ni) n N i=l g,n which is definition F. II The proof of Theorem 1 is now complete. The alternative representations are used in the following section to explore and demonstrate additional properties of optimal workloads and the production function. 5. APPLICATIONS OF THE ALTERNATIVE REPRESENTATIONS Some properties of the production function are developed in this section by using the alternative definitions. ~5.1 contains some preliminary results for networks of multiserver queues that are based on the equivalent definitions. These results are used in ~5.2 to prove some results for balanced workloads and their optimality in closed networks of singleserver queues.

-13 - 5.1 Preliminary Results for Networks of Multiserver Queues The following results are valid for systems consisting of groups of machines with s. machines in each group i. 1 Corollary 3. For any grouping S, the expected number of busy machines is given by: g Pr(g,n;S,X) I s.. i=l 1 Proof: The result follows from Representation B of Theorem 1. The next theorem defines the number of states utilizing k out of m machines when n parts are in the system. Theorem 4. For any S, the number of states, n, such that U(n) = k/m is ()(n- ), for k = 0,1,2,...,m. Proof: Consider any state, n, such that U(n) = k/m; that is, there are k out of m machines busy and m-k machines idle. The number of ways to select k machines out of m machines is (k). This is the number of states utilizing k of m machines with k parts in the system. Furthermore, the number of ways to distribute n indistinguishable parts among k busy machines is (n-l)(n-2)...(n-k-+) _ (n-l)! = (n-l (k-l)(k-2)...1- (n-k)!(k-l)! n-k) Therefore, the number of states utilizing k out of m machines with n parts in the system is m)( (n-1 ) I k n-k As a direct consequence of applying Theorem 4 to representation C, we have:

-14 - Corollary 5. If p(n) = pk for all n such that U(n) = k/m, then Pr(g,n;S,X) = X Pk(k) (nk=l kk n-k Proof: The result follows directly from Theorem 4 and representation C. | Corollary 5 may also be seen to be true by noting that under the assumptions of Corollary 5, p (n-lk = Prob{k machines are busy}, Pk(k) n-k and applying representation B. Corollary 5 is required in the proof of Theorem 9. 5.2 Results for Closed Networks of Singleserver Queues The results in this section consider systems having only one machine in each group. In this case, sl =... = s = 1, and S = (1,...,1) = t. An equal, balanced workload on each machine implies X = X =... = X =1, which is i /2 g represented by X = (X1, X2,...,Xg) = t. Theorem 6. Pr(g,n;t,l) = n/(n+g-l). Proof: Pr(g,n;t,t) = G(g,n-l;t,f)/G(g,n;t,t) (by equation (4)) The result follows directly by substituting X = t into equations (3) and (4) and simplifying. I The following corollary to Theorem 6 states that production for a balanced workload monotonically increases in n, and approaches one in the limit as n approaches infinity. Since Pr(g,n;S,X) is a probability, by representation D of Theorem 1, then balancing the workload is optimal for the limiting case of an infinite number of parts in the system. Corollary 7. For g > 1, d[Pr(g,n;t,t)]/dn > 0 and lim Pr(g,n;t,t) = 1. Hence balancing is optimal for n =.n+ balancing is optimal for n =.

-15 - Proof: d[Pr(g,n;1,1)]/dn = (g-l)/(g+n-1)2, which is positive for all g > 1. Also, n1, Pr(g,n;t,t) = _ -- = 1, by Z'Hospital's Rule. H ' 'A n+g-1 Theorem 6 and Corollary 7, in terms of a different workload scaling, can be found in Buzacott and Shanthikumar [1980]. Additional information on the optimality of balanced (unbalanced) workloads in certain networks of multiserver queues can be found in Stecke and Morin [1982] (Stecke and Solberg [1982]). The following theorem provides another result for balanced workloads. Theorem 8. If S = t, then X = t if and only probable. Proof: (Sufficient): Assume that X1= 2 = X = X = 1. g if each state, n, is equally For all inN, g,n p(n) = G(g,n;t,X) 1 X. ni -11 1 1=1 (equation (1)) where G(g,n;1,X) = n~N g i Xini i=l g,n Then, evaluating at X = I, p(n) = ( I 1)1 g,n -1 = g-n -1 = (g- ), for all neN g-1 I g,n Therefore each state is equally likely. (Necessary): Assume that p(n) = l/(n+gl), for all nN+ g-1 for all g,n

-16 - Then Agt, —i g Tn+g-l G(g,n;t,X)1 n Xi = 1/(n ), for all ieN = i1 g-' g,n i=1 g+n-l1 This provides ( g ) equations in g unknowns. Choose the g states, ni, where all n parts are at machine i; i=l,...,g. Then we have the g equations: n = G(g,n;,X)/( n+g) = C, i=,.,g i O)/g-1 Taking the single, positive, real root of Xn = C yields: 1 Xi = [G(gn;tX)/( g-)]ng = Constant = K, i=l,...,g. (6) Since X. = g (by definition) i=l = Kg, (by equation (6)) then Xi = 1, i=l,'.',g~. I Corollary 5, Theorem 6, and Theorem 1 are now used to prove Theorem 9, which claims that if all states, i, of a system of single machines are equally probable in steady-state, then the expected production achieved is the same as that for a balanced workload. Note that Theorem 9 actually follows directly from Theorem 8. However, additional insight can be obtained from the following direct proof of Theorem 9 that uses the alternative definitions. Theorem 9. If p~ii)= l(nfg-1 Theorem 9. If p(n) = l/( g ) for every state n and S = t, then Pr(g,n;t,X) = Pr(g,n;t,t). 1'

-17 - Proof: Pr(g,n;, X) = p(n)U(n) niN g,n = Y (n~-l ) U(i) i:eN g,n (n+g-1- (g (n-I) k k=i g-i k n-k g & —L (by Representation C of Theorem 1) (by assumption) (by Corollary 5 and m=g) (n+g- )-1 g-l ( l)(nl) k-1 n-k k=l (nig-g-1 n+g-2 g-1 n-1 n n+g-l = Pr(g,n;, f) (by Theorem 6). 11 The following Lemma provides a result on the unweighted (by p(n)) sum of the utilizations of all states, U(n). Lemma 10. Proof: I U(n) neN g,n = (n+m-2 m-2), if S t ur-1 Equating the numerators of representations A and F of Theorem 1 yields I U(n) neN g, n g H hi(ni) = G(g,n-1;S,X), i=l for all S and X. particular, when X1- =.. = 1 + n. If S = 1, then hi(n.i) = Xi, for all i. In 1 1 i '0 = X = 1 g I U(n) = G(g,n-1;t,t) n~N g,n = (n+g-2) g-1 (by substitution in equations and (3) and using Theorem 9) (2) 11

-18 - 6. SUMMARY As a result of a particular scaling of workload in a closed network of multiserver queues, six equivalent representations of an also scaled production function emerged. The new representations are useful to provide insight into understanding the production function. Some of the equivalent definitions of expected production were used to derive some additional relationships that are potentially useful for future research. In addition, some properties of balanced systems that may also prove useful in the study of both manufacturing and computer system performance are presented. ACKNOWLEDGMENTS The research reported here has been partially supported by the National Science Foundation under GRANT No. APR75 15256 and also by a grant from Ford Motor Company. The authors gratefully thank James J. Solberg of Purdue University for many useful discussions.

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