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HD2{ hJDi lli&x,. ALFRED P. WORKING PAPER SLOAN SCHOOL OF MANAGEMENT MULTIPRODUCT QUEUEING NETWORKS WITH DETERMINISTIC ROUTING: DECOMPOSITION APPROACH AND THE NOTION OF INTERFERENCE by Gabriel R. * Bitran and * Devanath Tirupati WP # 1764-86 March 1986 MASSACHUSETTS INSTITUTE OF TECHNOLOGY 50 MEMORIAL DRIVE CAMBRIDGE, MASSACHUSETTS 02139 MULTIPRODUCT QUEUEING NETWORKS WITH DECOMPOSITION APPROACH DETERMINISTIC ROUTING: AND THE NOTION OF INTERFERENCE by R. Bitran and * Devanath Tirupati Gabriel WP # 1764-86 * Massachusetts Institute of Technology March 1986 I Multiproduct Queueinq Networks with Deterministic Routing: Decomposition Approach and the Notion of Interference Gabriel R. Bitran and Devanath Tirupati Massachusetts Institute of Technology, Cambridge, Mass. ABSTRACT Queueing networks have been used to model the performance of a variety of complex systems, however, exact results exist for a limited class of networks. The used extensively to analyse networks of queues is methodology that has been the decomposition approach. In this paper, we consider open queueing networks with multiple product classes, deterministic routings and general We examine the decomposition method arrival and that provides estimates of key parameters with an accuracy that it settings. service distributions. Recognizing previously ignored. this weakness, we is for such systems not acceptable in enriched the approach by modelling a We consider interference among products and describe its effect and show many practical phenomenon on the variance of the departure streams. The recognition of this effect has significantly improved the performance of this methodology. We provide extensive experimental results manufacturer of semiconductor devices. based on the data of a 1.0 Introduction; Queueing networks have been used to model the performance of a variety of complex systems such as production job shops, flexible manufacturing systems, computers, urban service systems, communication networks etc. However, exact results exist for the limited class of Jackson type For more general networks several approaches networks (Jackson 1957,1963 and Kelly 1975). leading to approximate solutions have been proposed in the literature. Reiser and Kobayashi (1974) and Kuehn (1976, 1979) were among the first proponents of the parametric decomposition approach, which was later used by Shantikumar and Buzacott (1981) for single product networks and by Whitt This approach generalizes the notion of independence and product form solutions of (1983a,b). Jackson type networks to more general models In this method, the squared (see section 2). coefficients of variation (scvs) of interarrival intervals at each station are computed approximately. The performance measures such as mean number of jobs, queue lengths at each station are estimated based on these scvs. In this paper, we consider an open queueing network with multiple product classes, We deterministic routing for jobs in each product class and general arrival and service distributions. such a class and show that the approximations do examine the parametric decomposition approach for not perform well in certain instances with large number of products. Our analysis shows that, under the assumption that the interdeparture intervals are independent and identically distributed (iid), the squared coefficient of variation (scv) of interdeparture intervals for each product stream leaving a station can be expressed as the sum of two terms. The first term reflects the influence of congestion and the service at the station while the second term represents the products in the network. approach fact that to a of a product stream mentioned above. all presence of other This result can be interpreted as a generalization of the decomposition multiproduct network with deterministic routing. The generalization comes from the we recognize aggregate effect of the is the interference among products ie. at a given station, the scv of the departures distorted by the presence of other products. This In order to compute is reflected in the second term this interference effect for a product at a given station, other products into a single product. Therefore, the two product model is we an important building block in our methodology. Since it is difficult to determine the interference efi"ect of the other products exactly, we obtain two approximations that are based on the following assumptions about the behavior of the aggregate product. The i) arrivals of the aggregate product follow a Poisson process. This assumption by the results of Franken (1963) and Cinlar (1972) and number of a large of independent renewal processes relies is inspired on the notion that superposition can be approximated by a Poisson process. ii) The interarrival intervals of the aggregate product have an Erlang distribution. approximation than is intended for cases with moderate number of products with an scv of less interarrival intervals. .5 for Both approximations are easy the demand due to each product tends to compute and are asymptotically exact when the proportion of to zero Our computational results demonstrate that these are quite robust and provide reasonably accurate estimates for other distributions with the The paper organized as follows. is In section 2 briefly describe the parametric decomposition approach. based on (i) above to compute the scvs deterministic routing for we provide a review In section 3 4. these approximations in a network context. The of the literature and we develop an approximation to the new detailed study of the Erlang approximation and the and 6 respectively. In section 7 we examine The network chosen the production facility of a semiconductor company. Finally, is based on real we summarize life data and models the results in section 8. Review of the Literature: The research on network of queues has focussed primarily on performance the purpose of a brief review of the literature, i) we evaluation. For classify the research as follows: Exact Analysis. ii) Approximation Methods. iii) is scv. departure streams for networks with multiple products and related computational results are discussed in sections 5 2.1 same Computational experiments demonstrating the improvement due approximation are described in section 2.0 This Simulation and related techniques. Exact Analysis : Exact results exist for Markovian systems. A seminal contribution in this area the paper by Jackson (1963) which provides the results for equilibrium probability distribution of the number of jobs for a variety of systems that are referred to as restricted class of networks can be described in the following framework. Jacksonian networks. This 1) The system consists of N stations or Jobs which need processing at any machine centers. station form a queue at that stage. 2) A first come first serve discipline is observed at each station. 3) The where service time at each station follows k; is the total The system 4) number of jobs 5) The number of jobs at station an exponential distribution with parameter pd.kj) i (in queue and in service). can be described by an N-dimensional vector state at station arrival process i, is i= (ki,k2,...,kN) where k, is the 1,2,...,N K Poisson with parameter A(K) where the total is number of jobs in the system. Job route 6) is modeled as a random walk i.e. the next step on the job route depends only on the current station and not on the past processing history. Thus, the routing behavior can be described by a matrix R. R= where {rij,i€[0,N].j€[l.N + l]} ro,i is the probability that an arriving job visits station rjj is the probability that a job visits station 'i,N + 1 and is is equilibrium distribution, if it exists, is Probability that the system = being processed at station i, 1^ ij i. quite general enough to model arbitrary networks. Jackson provides sufficient conditions for the existence of = first. the probabilty that a job leaves the system from station The routing behavior n(k) j aifter i an equilibrium of a product form is in distribution. The main result is that the i.e. state k in equilibrium Br(k)ei(ki)e2(k2)...GN(kN) where k = state vector (ki,k2,...kN) r(.) 9i(.) = = a function of the number of jobs in the system. a function that depends on the nature of station probability distribution at station i i. 0; is proportional to the equilibrium assuming the arrivals at the station follow a Poisson process. andB = a normalizing constant. This result implies that to consider each station individually. compute the equlibrium probability of a given state k, we can Kelly (1975) extended Jackson's result to networks with more general routings and priority behavior. Specifically he considered a multiple product network with the following characteristics. 1) 2) Job arrivals follow independent Poisson processes. The service time at each station service time at a given station is is homogeneous and has an exponential independent of product type. distribution. The The routing behavior 3) each class can be described by a routing matrix as in a Jackson for network. This permits deterministic routing for some or 4) It is possible to define on job in position Ofcourse, the 5) 5(i,nj) is PjS i a variety of queue disciplines. at station together j The nj jobs. The product classes. Pj(i,nj) is service rate for this job would then be a function which defines another type of priority. system state of the the proportion of effort expended pj Pj(i,nj). sum to one. arrival at station j with nj jobs 6) with all is would occupy position i 6(i,nj) is the probability that a in the queue. Again, the 5's described by specifying the number sum new to one. of jobs at each station and the product type in each position of the queue. Kelly showed that even in this more general network, the equilibrium probability distribution = n(k) is of the product form B. r(k).Ai(ki). A2(k2)... where k = ki = Ai(.) a vector describing the product type of jobs at station = A function that depends on the = a function that depends on is i, state vector. station i, a normalizing constant. The product form time AN(kN) state vector, r(.) andB = i.e. result holds in the case of random order service discipline. not homogeneous, the result holds with a preemptive resume, last In particular it does not hold for first come first come When the service first serve discipline. serve policy and non-homogeneous service times. Similar results are provided for closed and mixed networks by Baskett et al (1975), Gordon and Newell (1967). The reader (1985) for referred to survey papers by Lemoine (1977) and Disney and Konig is more references and details of this approach. While the product form results are interesting and useful, they are practice due to the very large state space. mean Also, in values of queue lengths, waiting times etc. many and not difficult to implement in cases, the parameters of interest are the n(k). Reiser and Lavenberg (1980) consider closed networks for which the product form results hold. For such networks, they have developed a procedure (mean value analysis) to compute The main n(k). with result is values without evaluating the state probabilities a recursive relationship describing the performance measures of the system K jobs as a function of those with In mean summary, the exact K-1 jobs. results encountered in the literature basically rely on the following assumptions; 1) Exponential distribution 2) Homogeneous for service times service requirement at all stations. 3) Priority discipline independent of customer class. Poisson arrivals. 4) However, these are overly restrictive not extend to more general networks, it many in practical situations. Since the exact results do has led to the development of approximation schemes described in the subsequent sections. 2.2 Approximate Methods: The lack of success in obtaining exact solutions for general motivated research in developing approximations to evaluate performance measures . networks has These may be broadly described under the following four categories, i) Diffusion approximation, ii) Mean value analysis, iii) Operational analysis, iv) Decomposition methods. The first mentioned here three approaches are not directly related to the for completeness. We do work in this paper and are not describe these in any detail, but provide references for the interested reader. Diffusion Approximations are motivated by heavy traffic limit theorems and are based on asymptotic method for approximating point processes. The work by Iglehart and Whitt (1970), Newell (1971), Gaver and Shedler (1973), Kobayashi (1974), Gelenbe (1975), Harrison and Reiman (1981) reperesentative of this type of analysis. is Mean Value Anlysis (1980) and is is a heuristic approach similar to the work by Reiser and Lavenberg intended for closed networks. Schweitzer (1979) proposed an approximation that has been extensively tested by Bard (1979). The resulting Schweitzer-Bard algorithm requires solution of a non-linear system to estimate the performance of closed network. Buzen (1976) was among the first to use operational analysis computer systems. approach focuses on directly measurable quantities and testable assumptions. distribution free and relies (1978) provide a tutorial This The analysis is on flow balance and homogeneous service principles. Denning and Buzen on the approach. This paper contains a detailed list of references on the subject. Decomposition Methods are essentially attempts to generalize the notion of independence and product form results for Jackson type networks to more general systems. This approach relies on two notions - (typically first a) the nodes can be treated as being stochastically independent and b) two parameter mean and variance) approximations provide resonably accurate results. The approach was proposed by Reiser and Kobayashi (1974) and has been used by several researchers in developing approximations. Sevick et al. (1977), Chandy and Sauer (1978), Kuehn (1979), Shantikumar and Buzacott (1981), Buzacott and Shantikumar (1985) and Whitt (1983a) have all used a similar approach. Essentially the method involves three steps- Step 1: Analysis of interaction between stations. Step 2: Decomposition of the network into subsystems of individual stations and their analysis. Step 3: Recomposition of the results Step 1 is critical to the to obtain the network performance. decomposition approach and we describe in detail using the it models of Shantikumar and Buzacott (1981) and Whitt (1983a,b). These are representative of the method and illustrate the approach. Also, we have used these as a bench mark proposed in this paper. The interaction between the stations a composite of three basic processes compare the approximations analysed by looking at the network as - a) superposition or merging, b) flow through a queue or a station, and c) is to splitting or decomposition. The first process typically represents the arrivals at a station while the third describes the departures from a station. The splitting and the merging processes model the product routing in the network. processes. We first describe the approximations used by Shantikumar and Buzacott for the three basic They considered networks with a single product, Poisson arrivals and general service times. a) and cai The merging process considers superposition of arrivals at a station (say station be the arrival rate and the scv of interarrival time of the flow from station external arrivals). The resulting mixture is i to approximately described with an arrival rate j). Let \,j j (i = for \j and scv caj as follows: A 1 = AT ca ^ y i caj and (1) The flow process describes approximately csj are the scv of the interarrival and service time at station is = J c) number the scv of the departure stream from a station. If j with a utilization pj, the scv of approximately given by cd p^ J + (1-p^) cs J J ca (2) J In the decomposition process a product stream with arrival rate \j of substreams, each stream representing the flow from a station network. The routing i, l\^ca b) the departures a (X = then the substream is i assumed is to be Markovian. If p, is = p \. j to is split into other stations in the the probability that a job would follow path characterized by the following parameters: \ and scv of cdj = p cd Expression mean 1/A (3) implicitly +1-P cd (3) assumes that interdeparture intervals of the product stream are and scv of cd. Note that caJi = iid with cdjj. interactions between the stations are reflected by the scv of the arrivals at each station. The Shantikumar and Buzacott consider single product networks with Poisson arrivals and Markovian Combining the approximations routing and general service time distribution. for the three basic processes described above leads to the following linear systems. N = Ar„ + a•.-'"0.." y ica- y a (1-pV. ca. =Ar- + Pj into the utilization at station j caj = scv of arrivals at station j csj = scv of service at station j = The r (p^r cs ^J J.' J + 1 - )],i=l,2,...N r J' (5) ' routing matrix Note that station. a system = {r^ j} Y (4) J=l net arrival rate at station j R= J.I J J.' = Qj J =l J=i where \ = arrival rate ,i=l,2,...N a r -^— J (4) is the standard flow balance equations to determine the net arrivals at each internal flows Xij are given by airij. (5) is the approximation to compute the coefficient of arrivals at each station. In step 2, each station is analysed based on the partial information obtained in step Shantikumar and Buzacott examine M/G/1 and G/G/1 approximations performance. interarrival to 1. evaluate the station approximation requires only the mean and scv of the In particular, the G/G/1 and service times. Finally in step 3 these results are synthesized and performance measures for the network are estimated. This procedure is mean queue straight forward for In developing the queueing network analyser approach similar to the lengths, and lead times. (QNA) Whitt (1983a) has essentially used an one described above in analysing the interactions between the nodes. The superposition or the merging process has been modified as follows: = wca + ch J where chj is the modified scv of the merged stream, function that depends on the station utilization follows. l—w (la) J pj and caj is Xjj. determined by In the QNA the (1) and weight w w is is a weighting determined as ) w -1 = l+4(l-p )^v-l) J wt This modification leads to 1) is ^^ kj J motivated by the observation that neither the asymptotic method (which nor the stationary interval approach (see Whitt 1982 for details) by itself perform well over wide range of ca for approximating the superposition process, (la) above both approaches and is is a hybrid approximation of based on the work of Whitt (1982) and Albin (1981,1982). Whitt also refines the approximation of the queue process to provide for multiple servers and uses the approximation (3) for the splitting process. Estimation of station performance measures in step 2 are also modified to provide for multiple servers at each station. use the first QNA in the deterministic routing for each product. is briefly is is formed by appropriately done with the aggregate data. The aggregation described below. m = number of products Afc = Hfc = number of operations for product k "k,i arrival rate of product k = lH(x) to analyse networks with multiple products and to an aggregate product First, combining the product data and the analysis Let However, the approximations continue two moments of the arrival and service distributions. These approximations are used procedure same station visited by product k at step = The external 1 if xc i H and Ootherwise. arrival rates and the flow rates within the network are obtained as m k=i m k t=U = The routing matrix r^j is i defined by X _ ~ 'i/ N ''ij y + *— 0,1 (A. X, k,i k=\ The mean and the scv of the service time at each station 8 is computed by follows. m At H\%n(k,[):-^,rJ^ X. = k=ll=l m * II\\/K/+^^^«*'^^"*,/=^^ 2. ^ xlcs J where Xj k and csj k are the +1)= *=1/=1 J mean and n. e m scv of the service time for product j at step k. Both Shantikumar and Buzacott and Whitt report encouraging results based on experiments However, the experiments are approximations with simulation. comparing their respective The one exception (except for one case) based on a single product. is all the case with two products presented in Whitt (1983b). However, in the manufacturing context, the problem with more than two products is more common and relevant and provides the motivation 2.3 Simulation Methods: for the analysis in this paper. In the absence of (exact) analytic results except for very restricted cases, discrete event monte-carlo simulation has been an obvious alternative to evaluate large queueing networks. This approach permits use of more elaborate assumptions that are closer main drawback the computational requirement. is The process is time consuming and except in very small examples only a limited number of alternatives can be examined. Recent developments in This perturbation analysis suggest considerable promise in reducing the computational needs. technique, developed by Ho et al. (1979) provides a derivatives of performance supporting the technique method measures with respect is to to estimate, in to the decision some extent experimental. SCV of Departures from a Station one single simulation run, parameters. The evidence Recent work by Suri and Zazanis (1984) and Zazanis and Suri (1984) examine some related theoretical 3.0 The to reality. issues. with Multiple Products- Characterization of Interference in the presence of multiple products The basis determination of of the decomposition mean and approach described scv of arrivals at each station. It in the may previous section is the be noted that specification of product arrival rates and routings determine the means exactly and hence the quality of the results depend on the to the scvs. In the case of multiple product estimation of scvs of each product stream. Poisson distribution, the use of splitting process perform well in some instances (this conjecture is networks with deterministic routing, this reduces We observe that (3) to if the arrivals do not follow a describe the scv of each product may not supported by the computations described in sections 4 and we In this section, 6). characterize, for each product, the scvs of departures from a single server station processing multiple products. In the process, We product due to the presence of other products. discussion in this section will aggregating all two product case Our analysis is described below The iii) refer to this distortion as interference effect. clear that this interference effect it may and in detail is relies be estimated by on the following assumptions. FCFS. intervals from the station are iid. The product arrivals are independent. The interarrival times for each product are These assumptions are consistent with the decomposition approach, approximation, (i) (ii) iid. is clearly an together with the single server assumption implies that the sequence of job arrivals (by product type) In this section, is identical to that of departures. we consider a single server station processing multiple products characterize, approximately, the scv of departures for each product. generalizes the approximation for the splitting process given by stations in a network are now described by The (3). result is and we analogous to the approximations for the superposition (1) and queue Notation: p = number of products station utilization cs = scvof service time Xi = X arrival rate of product = = ca, i arrival rate at the station = S X, scv of interarrival time of product Xi = interarrival time of product Pi = Xi / d; = interdeparture interval for product = cdi = d nlj n; (random variable) i (random variable) scv of interdeparture intervals of product inter departure interval nli + i from the station (random variable) intervals from the station = number ofjobs of the aggregate product = cni i X = scvof interdeparture cd i that arrive during an interarrival time of product 1 = scvof ni E(.) = expected value of (.) V(.) = variance of (.) 10 and The interactions between processes and this generalization. m= The product of interest and the aggregation of the other products. - priority discipline at the station The interdeparture ii) identify the distortion in the scv of a given other products. Thus, in our methodology, the determination of interference reduces to the analysis of the i) make we i (2) Note that the random variable assumption d, is the sum of n, random variables which are iid by (ii). Hence, E(di) = E(ni) E(d) = and V(d,) since, cdi we = = get, cdi = E(d)/pi E(ni) V(d) + l/(piA) 1/Xi V(ni) (E(d))2 V(d,)/(E(di))2 = V(d)/(E(ni)(E(d))2) = pi The = cd + en, + V(n,)/(E(ni))2 (6) scv of the departures of each product stream is characterized approximately by interesting to note that (6) expresses the scv of the departure stream as the term can be considered as reflecting the first effect of the captures the interference due to other products. This The expression explicitly recognise the latter effect. multiple products. (3) for It reduces to (3) if Poisson (ca b) the arrivals at the station are is It is of two terms. The queue process while the second term which does not in contrast to that given by (3) in (6) may be interpreted as a generalization of either a) the routing = sum (6). is Markovian with probability p; or Also, note that, in the case of single product and 1). deterministic routing (6) reduces to cd. We observe that (6) is not easy to implement, since it is difficult to evaluate en; in the general In the remainder of the paper, case. we propose and test approximations to determine en,. These approximations are based on assumptions about the behavior of the aggregate product. approximation 3.1 Poisson for the In this approximation aggregate product we assume that the arrivals of the aggregate product follow a Poisson process. This is motivated by the notion that superposition of a large processes may be approximated by a Poisson process and and Cinlar (1972). In subscript from i what we derive expressions follows, X,, the interarrival is time for product number of independent renewal inspired by the results of for cn^ and cdi. For simplicity, we omit the i. Let fx(x) be the probability density function of the interarrival interval for product and let E(X) = First, we derive Pr[nli = n] = i, 1 / \i The arrivals of products other than i follow a Poisson process with parameter A(l-pi) the probability distribution of nli probability that nli takes value n Pr[nl^= n] = Pdnl ^n\X = x]/'Jx)dx ^ , ' (X(l-p)xf since =n\X = x]= Pr[nl ' exp{— X(l n\ 11 Franken (1963) —p )x) ' (Ml CO we have = Pr[nl -p)xf = n] -p •exp(-\(l )x)f^x)dx i! It can be shown that £(nl y = ) nPdnl =n] = a-p)/p n= and Next, we compute £(n = ) 1/p E(nli2) as follows: £(al^)= y T nVKnl_ = n]= n(n- l)PKnl, = n] ^ + •^- I n= - and ^ n(n-l)Pr[nl^ = ^ -2 n= itcan be shown that n(n— / n l)Pr[rtl = + =n] = ( An approximation for cdj is )(1 — p T/p. +ca ) ' (£(nl )) en I I (1-P. = I + ca ' r> = (l-p) p + =p^V(n) '^i I (1 -p )ca I given by cd = p cd + Note that approximation +ca (1-p) = £(nl^)- )= V(nl), Since Vin (1 [d-pP (1 n II ) Vv-Wdx ^ ' = I and V(nl -p)fc (Ax(l ^"-2)' (1-p) hence E(nl^) 1 (A(l-p)xf~2 - = n] nPAn\_^n\ n= fi=0 (7) reduces to (3) (I —p when ) + (1-p.) p '^i ^i ca (7) I the arrivals are Poisson (or ca; = 1) and thus it can be considered to be a generalization of the approximations used by Whitt and Buzacott and Shantikumar. However, approaches zero, (3) it differs from (3) would suggest that even qualitatively cd; As the following proposition demonstrates, in the general case. tends to one, where (7) is (7) exact in the limit. 12 For example, as indicates that cdi approaches pi ca;. Proposition: Consider a single server station with multiple products, arrival rate A, utilization p Assume given service distribution. interarrival times. Then as pi that the arrivals to the station can be -> 0, cdj -cai for all Outline of Proof: Let Wj be the waiting (queue WjS are identically distributed for all modelled as a G/G/1 queue. Oa j. + w Let and a approximated by iid i. Note service) time for job j. that, in equilibrium, the be the equlibrium waiting time. The station can be An upper bound on V(w), the variance of w is given by Oa^ + 2 ot^, where and Ob are the standard deviations of the interarrival and service times respectively. Let V{w)* be the upper bound on V(w). (For details see Kleinrock, 1976). Consider an interdeparture interval for product wi and W2 be the corresponding waiting times Then, »|12 d. = x. - w, + w. we have, V(x) - 4 for the and V(d.) ( V(.w^-w^ ^ 4 V(w) ^ 4 Viw)' and Since \Cov{x. ^'^ Viw)* [V(x.) Note that V(w)* does not depend on Let i. <. ] x, be an interarrival interval for product and two successive jobs of this product. = V{x) + V{w,-wJ 12^ - 2Cou(x.,w,-wJ il2^ i ,w^-w^\ ^[Vix) V(w^-w^f^ 5 2[V{xWiw)' V{d) <. V{x) + 4 V(u>)* + 4 [V{x) V{w) Also note that in equilibrium, E(di) pi. i = = E(xi) f'^ \'^ l/Xj = l/(Xpi). Hence, ca.-4X[ca. V(u))*] Since X and V(w)* are constant, as As (7) is in the number of products p ' pi -* -^ cd. -^ 0, cdi -> we can p. p, to number If the number be small and thus the approximation of products. The formula has other of products processed at each station ignore the interaction between stations and analyse each station independently. mean and variance assume that ]' caj. in the implications as well for queueing networks. large, + A\^V{w) p^ + ^\[ca.V{w) we can expect increases, some sense asymptotically exact ca of each product stream would be preserved throughout the network at every station visited by a product, the first two moments would be the is The and we could same as those at the time of the external arrivals into the system. We make two remarks about the quality of approximation (7). First, we can expect that the approximation will perform well as the arrivals are close to Poisson and the performance will deteriorate with increasing deviations from the Poisson process. approximation will be good section 4.0 we describe for small values of results of computations to pi (and large number Second, we can expect the of products). In the following examine the goodness of this approximation. Computational Results In this section we report the results of computational experiments comparing the various approximations. In the absence of any exact results, we use simulation as a bench the approximations. This approach is common enough 13 in the analysis of mark to evaluate queueing networks. For example see Fraker Whitt (1983b, 1984) Kuehn (1971), etc. Shimshak (1979), The experiments were designed We from a single station processing multiple products. Shantikumar and Buzacott (1981), (1979), examine the scv of the departure streams to considered the following four factors in the design of the test cases. 1) Number of products (and the fraction Four levels were considered p): Number of for this factor. products were set at 2,3,5 and 10. In each case the arrival rates and distributions of interarrival times were identical for from 2) fraction of demand due each product to (p) ranged 0.1 to 0.5. The distribution of interarrival time: These were assumed The Erlang parameter was 3) The the products. all Station Utilization: set at 2,3 Two and 4 be to with Erlang distribution. iid to yield 3 levels for this factor. levels of station congestion, p = 0.6 and 0.9 were These were tested. considered representative of moderate and high utilizations in manufacturing settings. 4) Service time distribution: The service times were assumed The parameter of the Thus, in the fact that in distribution was and set at 2 to be iid with Erlang distribution. 3. 48 problems were simulated. The choice of Erlang distribution was motivated by all, many manufacturing environments much smaller than that of a Poisson process used in the reported literature some other distributions in the to It may is The Erlang scvs are typically smaller than one. i.e. family provides a range of scvs in this domain. the variability in process times and job releases be noted that the Erlang distribution has been examine cases with scvs than one. less experiments reported in section we consider In addition, 6. To evaluate the performance of the approximations, we compare the results with the estimates provided by the following alternatives. 1) Approximation (7) together with interference introduced in this paper (1) is 2) Simulation. 3) Application of approximations (la), (1983a) and described earlier. 2. and This approach, which uses the notion of (2). referred to as (2) and (3) to The routing matrix We refer to this as the aggregation approach INTl in subsequent discussions. the aggregate product defined by Whitt for the aggregate product was given in section and the approximation is denoted by AGPl in the rest of the paper. 4) Application of AGPl. This is approximations (1), (2) and (3) for a generalization of the approach by the aggregate product which Shantikumar and Buzacott multiple products and we denote this approximation by Figures 1 to is defined as in networks with AGP2. and 2 describe the behavior of the approximations as a function of the fraction p two data sets specified by compare the estimates simulation as p tends ca and cs and are representative of the results obtained. p, of cdi and clearly bring out the to zero. fact that In contrast, the approximations 14 INTl converges The for figures to that given by the AGPl and AGP2 which are based on random routing provide estimates that approach the value one. represents substantial improvement over both AGPl and AGP2 INTl requires an estimate departures for scv of station The same procedure parameter. is used for AGP2 use while (la) and difTerences in the estimates of cd; arise from the use of (3) and be observed that INTl in the estimates for cd,. We cd. may It (7) for (1) (2) and Note that estimate this (2) to AGPl. are used for The the splitting process. Tables 1-4 describe the results in detail and compare the estimates of cd; for the three approximations with the simulation value. In each case, by symmetry, the cd, values are the same for all the products and the simulation value reported in the tables represent the mean value obtained in the experiments. These results support the analysis of the previous section and permit the following conclusions. 1) The computational experiments suggest AGP2 in the cases is .39 the two product case is a distinct improvement over is 0.81 and .75 is AGPl and 0.333 in table .976 and .935 compared to the simulation value of .36. within 10%. The results are less dramatic but which - INTl For example, consider the ten product case when ca tested. The AGPl and AGP2 estimates are INTl estimate that by AGPl and AGP2 and .625 of still 4. The significant for INTl compared to .545 given by simulation. 2) The approximations deteriorate as the consider the ten product case in table for .5). The error 2. is less than 6% when ca is .5 The errors become progressively worse as the ca value decreases - error .333 and 3) arrivals diverge from the Poisson process. For example, 15% when ca takes As expected the (estimate of .545 of 10% when ca is the value .25. quality of the approximation improves with the number of products (smaller Pi). 4) The approximations over estimate the value of cd;. The above conclusions should be tempered by the fact that the measures of interest are mean queue lengths, number of jobs, and waiting times at the stations and not the Hence, it is necessary to examine the impact of errors in cd on the queue length estimates at the subsequent stations. Note that the cd; of the departure stream represents the arrivals at subsequent stations and should be interpreted as ca at the next station. approximations Lq = to p2/(2(l-p))(ca observe that the two moment L = p + Lq + cs)g, L = mean number of jobs f(ca,cs,p) We evaluate the queue lengths are of the form where Lq = the mean queue length andg = coefficient cd itself. = at the station at the station exp[-2(l-p)(l-ca)2/(3p(ca This approximation is due to + cs))] Kraemer and Langenbach-Belz Whitt and Shantikumar and Buzacott. The latter, in fact, (1976) and has been used by both have used two other approximations. The choice of the appropriate formula depends on the values of ca and 15 cs. The function g is not very sensitive to either ca or cs. Thus, if would result in an error of magnitude (x/2)% in the estimate of in the estimate of ca the estimate of L would be even lower. occur when the value is illustrate this effect, we consider a departure streams from station mean number table 5 be the 1 at stations 2-6 of the station 1 in table 6. The of jobs Observe that, sets of Note that the We now examine parameters are described in by symmetry, the mean number of jobs The table provides, computed from an error is to All the in figure 3. in addition to result. AGPl, AGP2 and INTl figures in the parenthesis in We also provide an estimate free estimate of the scv of the departures illustrate the performance of the approximation to lengths. 12% error in the estimate of and AGP2. For example, note that 1, mean number the error in the The corresponding cd;. of jobs is 4.2% AGP2 and AGPl the arrivals are Poisson (cai= (1984) indicate that 5.0 in case 2 AGPl overestimate the parameter by 52.3% and 57.9% 1 for all AGPl and AGP2 perform well. in figures for case 2 are 9.7% and The M/M/1 approximation performs poorly with an error when from estimate 18.2% respectively. These result provide additional evidence of the improvement by INTl over respectively. will result reported in table 6 represents the average of the results in the table are encouraging. For case contrast to the in When unlikely to be large. at stations 2-6. M/M/1 and M/G/1 approximations. The by simulation. The intent mean queue the The two measure devations of the estimates from the simulation mean number is network shown and need a second operation 1 and the simulation estimates, those resulting from The error note that large errors in the estimate of cd; correspond to the experiments described earlier. values observed at these stations. the table to six station, five product of jobs at stations 2-6 for two cases. and the results are given same encouraging Lq. error and would not affect the queue length estimates substantially. products are first processed at station the It is small and the impact on queue length estimates the cdi are large, the errors are small To x% ca and cs are of comparable magnitude, the impact of an i), of 124%. The reader should the computational results presented in Whitt In that case INTl, AGPl and AGP2 are the same. Erlang Approximation In this section we processing two products. (whose en; is to As explained earlier, the two products represent the product of interest be determined) and the aggregation of additional approximations the fact that study, in detail, the scv of departure streams from a single server station INT2 and INT3 INTl does not perform very this section are based Erlang distribution. to all the other products. estimate the interfernce well for small values of ca,. effect. We propose two These are motivated by The approximations proposed in on the assumption that the interarrival times of the two products have an The choice of the distributions with scv smaller than one Erlang and is due to the fact that it this is of interest in provides a family of many manufacturing settings. Further, this assumption facilitates the analysis and leads to computationally viable approximations. The computational results of the following section demonstrate that the scvs are not sensitive to the 16 ^ specific distributions but depend on the Arrivals at the station belong to two product classes the other products. ii) which follows, we assume that also. In the analysis used with other distributions i) two moments only and hence the approximation can be first . The arrivals in each group are independent of each other. The interarrival times each product are independent and have an Erlang distribution. The for parameters of the distribution are \i, ki and \2. the aggregate product). Note that ca, is l/k, for iii) As before, product of interest and the aggregation of - we assume ^^2 for products 1 = 1 i and and 2 respectively. (Product 2 is 2. that interdeparture intervals from the station are iid with a scv of cd. An estimate of cdj is given by (6) , cd, ^— p = let = pjcd , i=l,2 + cnj We describe below,the two approximations INT2 and INT3 to estimate en,. 5.1 Approximation INT2 In evaluating cni product 1 (2), parameters we assume (cn2), the arrivals of jobs of product 2 (1) follow an independent Erlang ditribution with A2, k2 (Ai, ki). We are, in effect ignoring the between successive interarrival intervals of product approximation now that during any interarrival interval of for the first arrival of (1) However, note that, an 1 (2). product 2(1) only and derive the probability distribution of nlj the interarrival time of product dependence in the arrivals of product 2 number it is this is exact for subsequent arrivals. We of jobs of product 2 that arrive during an 1. Let fxi(t) and fx2(t) be the probability density functions of interarrival intervals for products 1 and 2 respectively. -1 * ^ *, ^^i^'^ t = 'i'^rri]i^^^-v^''-« *2-^ ^X2W = Property 1: Let X ^'^][tT^^^(-V) -'"O be a random variable with an Erlang distribution with parameters X and cumulative distribution of X is k. The given by the following: F^it) = 1 - *-Va<)' i e-^, t^O = Property 2 The probability density function for n arrivals (n-fold convolution) of product 2 with parameters X2 and nk2. Hence, 17 is Erlang — The probability distribution of nli is obtained as follows PKnlj-n]= = n\Xl =x] = Prinl But, Pr[nl^ Pr{n\ = n\Xl =x]f^^(x)dx ^^n\Xl=x]- Pdnl ^ n + 1 |X1 =x] j J "*2-Va2x)' Pr[nlj^n|Xl=x] = and rj^it)dt= 1 y —p exp{-x^) - , (n Pr[il. /lence, = '• + m^-l^^^^. —— ^ = H-^1 =^] = exp(. '. — \^) i = nk. (n + l)*„-l.- anrf =n]= Pr[nl Jo > ^^, (n + l)*2-l I . 1 {k,-l)\ 1 It «^-l)! — = = n] is nonnegative and it 77 7; exp{-{\^ (1-g)' 1 1 it for given values of the , where q = (8) \ + ^2 is a proper probability mass function. can be shown that these probabilities sum parameters by using c^x + \^x} dx rt*„ easy to obtain closed form expression for V(nli), the variance of nli. compute «cp(-^,^) (^l-D! X can be verified that the distribution obtained for nli Clearly, Pr[nli ^i i\ y =n] = hence, Pr{nl ^ , —r~ ^P^-^o^^ Jo = n*2 -1 * ,j (A^) 2 p (8) directly. ^^^ it is Since ni is to one. It is not straight forward to nli + 1, we have the following. E(ni) cni = = E(nli) + andV(ni) = V(nli) 1 V(nli)/[E(nli)+l]2 Hence cni can be computed from the parameters of the arrival for the 5.2 distributions. The coefficient second product can be computed in an analogous fashion. Approximation INT3 : In this approximation we assume instance of random incidence in the arrival stream of product first arrival of product 2 that the arrival of product 2. during an interarrival interval of product 18 The 1 is distribution of the time is an till the 1 then given by the following "2 i-l J fx2M^ = 7 I = '^2 The ^2 1 1 V exp(-A^) x^ , '• distribution for the subsequent arrivals is Erlang with parameters from INT2 in the assumptions regarding the distribution of time differs distribution of nl 1 can be derived as in the previous case. *2 It j-\ till A2, k2. Thus INT3 the first arrival. The can be shown that (i + k-\)\ =01= P7{nl 1 (A. -!)!*„ ;=i 1 = n*2+7-l (l+k^-\)\ Pr{n\=n]— 1 and (1-q)' (^,-1)!*, ' 2 1 j = a for > (9) t={n-\.)k +j \ Again, we do not have expressions in closed form for variance of nli and ni. For specified parameters, cn^ can be computed directly from We make a few remarks about INT2 and INT3. similar and for low values of p, Second, the approximation is time is INT2 is likely to we do not expect any only in respect of the during an interarrival interval results. (9). is large (or p very small. This will happen most often This hope is significant difference in the two estimates. first arrival. small), two approximations are very Hence, if the number of jobs arriving we may expect INT2 and INT3 to give good perform poorly when the number of jobs arriving during an interarrival applicability of the two approximations to be when the fraction p; is close to one. more general than the case for We also expect the which they were derived. based on the fact that two moment approximations provide reasonable results in most cases. Hence, scv. is First, the we expect INT2 and INT3 to perform well We empirically examine some of these issues in the for other arrival distributions with the same computational experiments described in the next section. 6.0 Computational experiments In this section to test we empirically Erlang approximations: test the goodness of approximations INT2 and INT3 and report the results of the computational experiments. estimates of cdj by INT2 and INT3 with In the first set of experiments we compare the other approximations and simulation results for a two product, one station system. In the design of these experiments the following factors were considered. i) The fraction p; section 4, p; is the proportion of demand at the station due to product was the same for all the products (1/m). In the present case, i. In the experiments of we consider nine Pi, 0.1 to 0.9 in steps of 0.1 to analyse impactof changing the product proportion. ii) The arrival Job arrivals distribution: in each product class are independent of each other. The scvs of the interarrival times for each product are set at .333. with the same scv - a) levels of Erlang distribution with k = 3, 19 b) We consider three distributions Uniform distribution over [0,a], and c) Beta distribution with parameters 2 and approximations to distributions As iii) Station utilization: iv) Service distribution In all 108 approximations for is The 6. intent is to examine the robustness of the with the same scv. in section 4, assumed we consider two levels of congestion, to be Erlang with parameters 2 and problem sets were simulated. p, shown because they are very in the cd estimates by INT2 and INT3 give comparable (p) and service distribution otherhand, for larger values of The detailed expectations. i) p, close to this value. INTl, INT2 and INT3 over results and appear INT3 seems The 10% increasing INT3 consistently For low values of and lead This is the consistent with our to the following conclusions. The estimates of cd are in the test cases. On While the performance of INT2 deteriorates a typically little with gives good estimates of cd. The scv of the departure streams ii) figures demonstrate the perform better than the others. results are furnished in tables 7-10 of the simulated values. p;, to The The (cs). results with other AGPl and AGP2. be the best method. to INT2 and INT3 are very good approximations within .9. 3. simulation values in the figures correspond to the Erlang arrival case. improvements and .6 Figures 4 and 5 describe the behavior of the two data sets defined by station utilization distributions are not p= is cds for the three distributions tested not sensitive to the arrival distributions. is less than 5% in most The range of the cases. This supports our conjecture that the approximations are robust and should give reasonable results for distributions with the same scvs. We now examine the behavior of INT2 and INT3 for the multiproduct case. The test problems are those described in section The The 4. cdi estimates given by INT2 and INT3 are displayed in table 11. table also provides the simulation values for comparison. results, case. it is From figures and 2 and the tabulated 1 obvious that INT2 and INT3 perform better than the others even for the multiproduct The value of p, is not greater than 0.5 and both suggest that with increasing number of products, explained by the fact that as the number INT2 and INT3 INT2 performs give good results. These results better than INT3. of products increases, the aggregate product departs from an Erlang process, and the corresponding scv of the interarrival time tends situations, the random incidence considered We modelling the interference phenomenon. infinity, within INTl 5% (the will in error is to increase. In such INT3 becomes gradually counter productive in would expect that as the number of products tends outperform both INT2 and INT3. maximum This might be The errors to in the cd estimates are typically 12%) which would suggest that the errors in the queue length estimates are likely to be very small. Finally to see the impact of the Erlang approximation on the queue lengths, five product, six station example of section and the simulation results are displayed 4. The INT2 and INT3 estimates in table 12. 20 The error in the for we consider mean number INT2 estimate for the the of jobs two cases are 1.51 and 2.13% compared to 27.7 and 52.3% given by The corresponding figures for the INT3 estimates are 7.0 AGP2, and 30.67 and 57.9% given by AGPl. 1.51 and 1.12% respectively. A Network Example To recapitulate, the intent of the approximations presented in this paper is to evaluate performance measures in a queueing network and in this section we report the results of an experiment in this context. The network models the production facility of a for this set of experiments semiconductor company. The is based on real facility is station 2 more than once. We data and represented by 13 machine stationsandprocessesjobsof 10 product families. The routing for jobs in each family The network characteristics are described, life is deterministic. Note that product families in detail, in the appendix. visit considered the following factors in designing the problems for this experiment. Arrival Distribution: The interarrival intervals for each product family were assumed to be 1) iid. For each product the arrival distribution was randomly assigned from Erlang with parameters The same 2) 2,3,4, five alternatives Station utilization was utilization at each station We were considered for the distribution of service time. assumed to be uniformly distributed between 0.65 and 0.95. Again, the was determined in a random manner. Summary presented in table 13. In the table we compare the estimates of results for the ten problems are mean number by approximations INTl, INT2 and INT3 with those given by AGPl, term. Overall, improvements that can be obtained by INT2 seems to For each time was randomly assigned. generated ten problems for this experiment. results demonstrate the - exponential and uniform distribution. station, the distribution of service 3) five alternatives of jobs in the network AGP2 and simulation. The explicitly recognizing the interference be the best procedure, while in most cases INT3 provides errors between those of INTl and INT2. These results are consistent with those in tables 1-4 and 11. The average error with INT2 and INTl are 1.78% and 5.95% respectively. In contrast, the average error with AGP2 and AGPl are 23.67 and 27.39% respectively. aggregation approach (AGPl and is AGP2) The errors in the are typically abovel9.45% in the problems tested. Case 10 an exception with errors of 12.5 and 15.26%. This problem corresponds arrivals of five of the ten products follow a Poisson process. the two cases in which approximation The estimates at the AGPl and AGP2. within 10% INTl It is still (86% of the observations). is 21 (error 3.76%). mean number The corresponding figures at nearly 60% of jobs was for the other AGP2 and AGPl respectively. may be noted that, with of the stations. The maximum station level statistics in table 14. more than 20% INT2 substantially better than those given by approximations were 93 (71.5%), 14 (10.77%) and 9 (6.92%) for INTl, the aggregation approach, the error case in which the also not surprising that this is one of In the ten test problems, the error in the estimate of We present the summary results for the to the (error 1.47%) performs better than station level are not as good but are at 112 stations estimates by the It error is 64 and 63%for AGPl and AGP2 In contrast, the respectively. maximum error with INT2 is 18.8%. We make two remarks about the use of the Eriang approximations INT2 and INT3 estimates. First, note that for a given en, pi, in these can be estimated for discrete values of arrival scvs. Since the scvs in the network are continous variables, we obtain en; by linear interpolation in such cases. Secondly, since the cniS need to be consistent with these scvs, an iterative procedure determine them. In the test problems, typically three to four iterations is required to were required. 8.0 Conclusions; In this paper we have examined the use of the parametric decomposition approach to the analysis of multiproduct queueing networks with deterministic routings. the approximation for the disaggregation or the splitting process based on quite poor. sum as the The analysis in this of two terms. The time while the second term is We have demonstrated that Markovian routings can be paper shows that the scv of product departures can be approximated first term represents the influence of station congestion and service the interference term due to the presence of other products. interference term explicitly recognizes the presence of multiple products This and can be considered as a generalization of the one used by Shantikumar and Buzacott, Whitt and others in the decomposition approach. Since the estimation of the interference effect some tested three approximations that are, in (INTl) evaluate exactly, we proposed and sense, asymptotically exact. may be approximated by a Poisson process. The computational performs reasonably well over the range of parameters tested. INT2 and INT3 are based on an Eriang for arrivals with scvs much smaller than one However, INT2 seems Our computational increases. substantially superior results MMl approximation (for approximations. This in the estimate of the results compared is the aggregation approach. 0.5). and are intended While INT2 and gives better estimates at high values of p perform better than INT3 as the number of products to show that approximations INTl, INT2 and INT3 give aggregation approach (AGPl and to the demonstrated by the mean number The other approximations, example, scvs smaller than INT3 show that INTl of the approximation distribution approximation for the arrivals give comparable results for low values of p, (greater than 0.5). and first results The quality deteriorates a little as the scv of arrivals diverge substantially from one. INT3 The based on the notion that the aggregation of the arrival streams of a large number of is products is difficult to of jobs is The experiments five product, six station 13% by INT2 compared 2 AGP2) and the MGl example. The error to errors in excess of 50% with also suggest that the results are robust with respect to the arrival distributions and the critical factor is the scv. This is in the same spirit as the two moment approximations in the decomposition approach. The potential of the approximations proposed in this paper is demonstrated by the network experiments with representative data from a semiconductor manufacturing company. 22 The application of INT2, INT3 and INTl simulation figures. This result in errors of over In table 15 is give estimates that are typically within 5 and a substantial improvement over the approximations 10% of the AGPl and AGP2 that 20% in most cases. we summarize our recommendations regarding the application of INTl, INT2 and We INT3. These are based on the computational results which support the preceeding analysis. recommend the use of INTl when greater than 0.5, 0.5), INT2 INT3 may be used or we recommend use the scv of arrivals of INT3. routings are representative of arrivals and service times is, and scvs smaller than one are Finally, for is greater than low values of p. .5. When the scv of arrivals is not For larger values of (greater than we observe that multiple products and deterministic many manufacturing environments. Also, the variability in job according to our experience, usually less than that of a Poisson process typical. This enhances the jjotential for the approximations proposed in this paper. Acknowledgements: The authors are helpful grateful to Professors Hirofumi Matsuo and Stephen C. Graves for their comments on an earlier version of this paper. References Albin, S.L., "Approximating queues with superposition arrival processes," Dept. of IE and PhD dissertation, OR, Columbia Univ., 1981. 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Suri, "Comparison of perturbation analysis with conventional sensitivity estimates for regenerative stochastic systems," Dec 1984. 25 Utilization = scv of arrivals scv of service # 0.9 = 0.333 = 0.5 of products Utilization = 0.6 scv of arrivals = 0.5 scv of service = 0.333 # of products 4 Figure 3: The 5 Product. 6 Station Example > *- PRODUCT 1 PRODUCT 2 PRODUCTS PRODUCT PRODUCTS Table 1: Comparison of Approximations Estimation of cd Station Utilization p = 0.9 scvof service = 0.333 #of products - Table 3: Comparison of Approximations Estimation of cd Station Utilization, p = 0.6 = 0.3333 scv of service #of products - Table 5; Parameters for the 6 Station. 5 Product Example Table 7: Comparison of Approximations Varying Station Utilization, p = 0.6 scv of service = 0.333 Pi • pi Table 9: Comparison of Approximations Varying Station Utilization, p scv of service Pi = 0.5 = 0.9 p; Table 11: Performance of Approximations INT2 and INT3 with Multiple Products Estimation of cd, the scv of departure streams #of producu Table 13: Comparison of the Approximations for the Network Experiment L = Mean number ofjobs in the network. e = % absolute error in the estimate of L relative to the simulation value, c is the average of e for the ten problems. Cas€# APPENDIX: Data Base Number of Stations = 13. Number of products = 10 Product Routing Characteristics: Product for the Network Example ifiASEMtNT Date OCT. ^ 3 '-'^ f^-^ viBa pR , jtfH (J^ ,-> JUL i>« •ill 8 199' ««i m. 2 8 1^33 i<Aft Lib-26-67 „n ?06D L18BA»;« QO'^ °^2 ^°'^