PROPERTIES OF DISCRETE EVENT SYSTEMS FROM THEIR MATHEMATICAL PROGRAMMING REPRESENTATIONS. Wai Kin Chan Lee W. Schruben

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1 Proceedngs of the 2003 Wnter Smulaton Conference S. Chck, P. J. Sá nchez, D. Ferrn, and D. J. Morrce, eds. PROPERTIES OF DISCRETE EENT SYSTEMS FROM THEIR MATHEMATICAL PROGRAMMING REPRESENTATIONS Wa Kn Chan Lee W. Schruben Department of Industral Engneerng and Operatons Research Unversty of Calforna, Berkeley 4135 Etcheverry Hall Berkeley, CA U.S.A. ABSTRACT An mportant class of dscrete event systems, tandem queueng networks, are consdered and formulated as mathematcal programmng problems where the constrants represent the system dynamcs. The dual of the mathematcal programmng formulaton s a network flow problem where the longest path equals the makespan of n jobs. Ths dual network provdes an alternatve proof of the reversblty property of tandem queueng networks under communcaton blockng. The approach extends to other systems. 1 INTRODUCTION Schruben (2000) proposed a mappng of dscrete event system models nto mathematcal programmng formulatons, where the solutons represent the system trajectores of the dscrete event systems. The dea of ths mappng s that a dscrete event system s frst modeled as an Event Graph (EG) (Schruben 1983), a powerful graphcal representaton for dscrete event systems. Then the constrants of the system are derved from the edges of the EG along wth the feasblty condtons of the state varables. It should be noted that there s another way of representng a dscrete event system as a mathematcal programmng formulaton whch s based on a Petr Net Modelng technque (Yen 1999). However, snce Event Graph Modelng (EGM) s more general than Petr Net Modelng (all Petr Net models can be transformed nto EGs, but the reverse does not hold. See Schruben 2003), our dervaton wll be based on the EG representaton. Perhaps the frst reference on modelng queueng system dynamcs as network optmzaton programs can be found n Maxwell and Wlson (1981). The objectve of ths paper s two-fold: Frst, we show how the system dynamcs of a tandem queueng network can be transformed nto a set of lnear programmng (LP) constrants by means of EGM. Second, some propertes of the system are nvestgated and proved usng the lnear programmng formulaton. Among these propertes, the reversblty property s of specal nterest. Specfcally, we study a tandem queueng system wth m consecutve stages, labeled k,..,m. In each stage, there s a sngle server and a fnte storage space for jobs. Job,..,n s processed at all stages n sequence wth servce tmes s k. It s assumed that {(s 1, s 2,,s m ), = 1,..,n} are..d. random varables. Snce the buffer spaces at each stage are lmted, a control polcy s needed to control the producton process at the stages. The most common control polces that have been consdered n the lterature are the communcaton blockng control polcy and the producton blockng control polcy (Buzacott and Shanthkumar 1993). In these two blockng schemes, a parameter a k s assocated wth each stage, representng the maxmum number of jobs at the stage at any tme of the producton process. For the sngle server system, under communcaton blockng, the server at stage k wll start processng a job whenever these three condtons are satsfed: C1) a job s avalable for processng, C2) the server at stage k s avalable, and C3) there s an empty space at the next stage. Under producton blockng, the three condtons are: P1) a job s avalable for processng, P2) the server at stage k s avalable, and P3) there s no fnshed job blocked at stage k. The only dfference between these two control polces s the thrd condton: In communcaton blockng, the server wll check the next stage for an empty buffer space before startng a servce, therefore t s blocked before servce. On the other hand, n producton blockng, the server wll check the next stage for an empty buffer space after fnshng processng a job, therefore t s blocked after servce ; t may have at most one fnshed job blocked at the stage, whch can be consdered an addtonal buffer placed at the end of the stage: ths makes producton block-

2 ng superor to communcaton blockng n terms of throughput, all else beng equal. Cheng (1993) proposed another control polcy called general blockng, whch s an extenson to Kanban blockng (Buzacott 1989). In general blockng, there are three parameters, a k, b k, and k k, assocated wth each stage. They are, respectvely, the maxmum number of jobs watng and n servce, the maxmum number of fnshed jobs allowed at the stage, and the maxmum number of jobs allowed at the stage (ncludng the jobs watng, n servce, and fnshed). To rule out trvalty, the followng condtons are necessary: k k a k 1, k k b k 0, and b k + a k k k for k = 1,..,m. Despte ts name, a system becomes more restrctve as ts number of parameters ncreases. Under general blockng, the server at stage k wll start processng a job whenever these three condtons are satsfed: G1) a job s avalable for processng, G2) the server at stage k s avalable and G3) there are strctly less than b k fnshed jobs blocked at stage k. For all blockng schemes, blockng may happen from tme to tme, complcatng the desgn and performance analyss of the system n terms of problem sze or dffculty. In order to facltate the analyss procedure, much research has been done to explore the structural propertes of the system. Of partcular nterest s the well-known reversblty property: A tandem queue s sad to be reversble f t has the same throughput as the correspondng reversed tandem queueng network (defntons n Secton 3.2). Therefore, f the system s reversble, then the system propertes of the orgnal system can be nferred form those of the reversed system, thus smplfyng the analyss process. Other motvatons for studyng the reversblty property of tandem queueng networks can be found n Chan and Schruben (2003). The reversblty property of tandem queueng networks has been establshed and proved for communcaton blockng, producton blockng (Yamazak and Sakasegawa 1978, Dattatreya 1978, and Muth 1979) as well as general blockng (Cheng 1995). All these proofs are based on the so-called actvty network, a drected graph wth arcs representng the tme requred to process jobs n the system. In ths paper, we propose another nterestng proof of the reversblty property based on the dual network graph (a flow network) derved from the dual of a set of lnear programmng constrants representng the system dynamcs. Snce our dual graph s a network flow graph, algorthms and technques n the feld of network flows can now be used n the performance analyss of queueng systems: fndng the makespan or bottleneck server can now be carred out by usng the longest path algorthm (see Secton 3.1 for detals). Another potental applcaton of our dual graph s that t can be used to prove the reversblty of mult-server tandem queues. Ths wll be presented n a subsequent paper. Maxwell and Wlson (1981) The paper s organzed as follows. In Secton 2, we derve the LP formulaton of a tandem queueng network for communcaton blockng. We provde the proof of the reversblty property n Secton 3. In Secton 4, we remark on some ongong research and possble future research drectons. 2 MAPPING EENT GRAPHS TO MATHEMATICAL PROGRAMMING FORMULATIONS In ths Secton, we derve the LP formulaton of a m-stage sngle-server tandem queue under communcaton blockng. For the dervaton of producton blockng and general blockng, see Chan and Schruben (2003). Let TQmC(G/G/R k /a k ), k,..,m, denote a m-stage tandem queue wth communcaton blockng, where stage k s a G/G/R k queue wth raw job buffer a k. Fgure 1 s an example of 3-stage sngle server tandem queue wth a communcaton blockng control polcy. Fgure 1: TQ3C(G/G/1/4, G/G/1/3, G/G/1/2) To model an m-stage tandem queue wth n jobs beng processed as an EG, the followng notaton s needed: ta k s the tme between the arrval of the -1 th job and the th job; s k s the servce tme of the th servce at stage k; A k, S k, and F k are the tme of the th Arrval event, Start event, and Fnsh event at stage k respectvely; Q k (t) s the number of jobs n front of stage k at tme t; R k (t) s the number of avalable resources at stage k at tme t; a k (t) s the number of avalable raw job spaces at stage k at tme t; C () t = lmmax { E ; t+ ε} s the rght-contnuous event E ε 0 countng functon that counts the number of tmes that event E has occurred by tme t (see Schruben 2000 for a more detaled defnton). Fgure 2 s the EG of TQ3C(G/G/1/a 1, G/G/1/a 2, G/G/1/a 3 ) where R 1 =R 2 =R 3 =1. Notce that a 1 s rrelevant because we assume that there s an nfnte nput buffer n front of the frst stage. Fgure 2: EG of TQ3C(G/G/1/a 1, G/G/1/a 2, G/G/1/a 3 )

3 2.1 Dervaton of LP Formulaton for TQ3C(.) Followng the method gven n Schruben (2000) and usng the Fnsh event tmes as the varables, we derve the LP formulaton for the TQ3C(G/G/1/a 1,G/G/1/a 2,G/G/1/a 3 ) as the followng Uncondtonal edges: In Fgure 2, there are sx uncondtonal edges, gvng us the followng sx equatons. Edge A1-A1: A1, + 1 = A1 + ta (1) Edge S1-F1: F1 = S1 + s1 (2) Edge F1-A2: F1 = A2 (3) Edge S2-F2: F2 = S2 + s2 (4) Edge F2-A3: F2 = A3 (5) Edge S3-F3: F3 = S3 + s3 (6) Condtonal edges: There are eght condtonal edges and eght state varables. The two events assocated wth the condtonal edge have only one state varable that gets ncremented and decremented between them. Here, there are eght constrants. The frst condtonal edge, A1 to S1, s assocated wth state varable Q 1 and ts correspondng constrant s derved as follows. At any tme of the smulaton, snce Q 1 s equal to the number of A1 events that have occurred mnus the number of S1 events that have occurred, t can not be less than zero at any tme t. Usng the event counts C E (t) defned earler: Q1() t = CA 1() t CS1() t 0 CA S1 CS1( S1 ) 0 CA 1( S1 ) A S 1 1 In the thrd nequalty, we have used the fact that the number of S1 events that have occurred at tme S 1 s. The last nequalty s because of the well-known relatonshp between an event and ts countng pont process, CA 1( S1 ) A1 S (Ross 1997). Combnng the last nequalty and equaton (2) gves the 1 constrant, F A + s Smlarly, the condtonal edge, F1 to S1, s assocated wth state varable R 1. Snce R 1 s equal to the number of F1 events that have occurred mnus the number of S1 events that have occurred plus 1, t cannot be less than zero at any tme t. We add 1 because t s the orgnal number of avalable resources at stage 1. R1() t + CF1() t CS1() t CF1( S1 ) CS1( S1 ) 0 C ( S ) 1 F F1 1 1, 1 S1 The thrd and fourth nequaltes are derved n a smlar manner. Combnng the last nequalty and equaton (2) gves the constrant, F F s 1 1, 1 1 Usng a smlar procedure to derve constrants for the other condtonal edges, along wth the objectve of executng all events as soon as possble (ths objectve functon ensures that the server wll work on the jobs as soon as possble), we end up wth the followng LP formulaton. TQ3C-LP1: mn st. n ( F1 + F2 + F3) = 1 F A + s,,..., n ( U ) F F s, = 2,..., n ( ) 1 1, F F s, = a + 1,..., nw ( ) 1 2, a F F s,,..., n ( U ) F F s, = 2,..., n ( ) 2 2, F F s, = a + 1,..., nw ( ) 2 3, a F F s,,..., n ( U ) F F s, = 2,..., n ( ) 3 3, where U k, k, and W k,k+1, are the correspondng dual varables for each constrant. 2.2 Extenson to m-stage Tandem Queue wth Communcaton Blockng The LP for 3-stage tandem queue wth communcaton blockng s easly extended to an LP for m-stage tandem queue. The thrd, fourth, and ffth nequaltes of TQ3C- LP1 are equvalent to the necessary and suffcent condtons C1, C2, and C3 ntroduced n Secton 1. Therefore, the m-stage formulaton should be based on these condtons, whch leads to: TQmC-LP1: mn st. m n k = 1 F k, F F s, k,.., m,,..., n ( U ) k, k 1, k k, F F s, k,.., m, = 2,..., n ( ) k, k, 1 k k, F F s, k,.., m 1, = a + 1,..., nw ( ) k, k+ 1, ak+ 1 k k+ 1 kk, + 1,

4 where F 0 = A 1,,..,n. We can also wrte the constrants as the followng: { + 1 } k+ F = max F, F, F + s k, k 1, k, 1 k 1, a k whch s the same as equaton 5.38 n Buzacott and Shanthkumar (1993). 2.3 Comparson of Communcaton Blockng and Producton Blockng Some results gven n Buzacott and Shanthkumar (1993) can be shown easly usng the lnear programmng formulatons (complete proofs are presented n Chan and Schruben 2003). 1) The departure tme under communcaton blockng s not less than the departure tme under producton blockng. Proof: Ths result s mmedate when one compares the two lnear programmngs for these systems (see Chan and Schruben 2003 for the LP of producton blockng). 2) The departure tme under producton blockng wth buffer capactes {a 1, a 2,, a m } s not less than the departure tme under communcaton blockng wth buffer capactes {a 1 +1, a 2 +1,, a m +1}. Proof: Ths s true because the LP of producton blockng wth buffer capactes {a 1, a 2,, a m } has more constrants than the LP of communcaton blockng wth buffer capactes {a 1 +1, a 2 +1,, a m +1}, all else beng equal. 3) The throughput of the communcaton blockng tandem queue wth buffer capacty {a 1, a 2,, a m } s not greater than that of the producton blockng tandem queue wth buffer capacty {a 1, a 2,, a m }, whch s, n turn, not greater than that of communcaton blockng tandem queue wth {a 1 +1, a 2 +1,, a m +1}. Proof: The proof s based on the above frst and second results. 3 REERSIBILITY OF TANDEM QUEUEING NETWORKS The well-known reversblty property of sngle-server tandem queues has been proven by many researchers usng dfferent methods. Many of the proofs are based on a characterzaton of the actvty network assocated wth the queue, see Yamazak and Sakasegawa (1978), Dattatreya (1978), Muth (1979), Lu and Buzacott (1992) and Cheng (1995). An actvty network for a tandem queue s a drected graph, representng the tme requred to process n jobs. In ths Secton, we propose a dual network graph (a flow network) that yelds not only the proof of the reversblty but also some mportant nformaton on the system, e.g. the number of jobs n each busy perod. Our dual network s a network flow model derved drectly from the dual lnear programmng formulaton, not from the actvty of the system, whch makes t fundamentally dfferent from the actvty networks n the lterature. Because of ts mathematcal form, t can also be used n the performance analyss of the queueng system. 3.1 Dual Network (Flow Network) To derve the dual network usng the 3-stage communcaton blockng tandem queue as an example, we frst take the dual of TQ3C-LP1. Wthout loss of generalty, we can assume that all servce tmes are postve (except for a dummy server who has zero servce tme, whch actually can be expressed as an empty buffer). Therefore, all the F k, k,..,m,,..,n, n the prmal are postve, forcng all the nequalty constrants n the dual to be equaltes. By examnng the structure of the dual, one can see that t s, n fact, the network flow problem of fndng the longest paths from the frst node (source) to all other nodes (snk), where each constrant represents the flow conservaton constrant of a node n the network. Snce the constrants are equalty, summng over all constrants wll yeld the n flow balance constrant for the source node, U 1 1 = m n. = Therefore, a total of m*n unts of flow (one unt flow s one Fnsh event) wll be sent from the source node to all other m*n snk nodes (one event for each snk node). The dual LP and the dual network of TQ3C- LP1(G/G/1/,G/G/1/3,G/G/1/2) are gven n the followng and n Fgure 3 respectvely: TQ3C-LP1-Dual: max st ( ) = 2 = = 2 = = 2 U U U U U U A + s U + s + sw + su + s + sw + su s U U + + W U U + + W U U W U U + W U U + + W U U + + W U U + + W U W U + W U + W U + U +

5 A 0 U 11 U 12 U 13 U 14 U 15 s s s s s 15 W 124 W 125 U 21 U 22 U 23 U 24 U 25 s s s s s 25 W 233 W 234 W 235 The flow of each arc s shown at the end of the arc,.e. W 233 s the flow from node s 31 to node s 23 and ts value represents the number of tasks (a task s fnshng processng a job at one stage, e.g. a job has three tasks n a 3- stage tandem queue) wthn the same busy perod. If a servce tme s changed, t may affect several tasks. For example, f W 233 = 4, a change n the servce tme s 23 wll affect 4 tasks, ncludng the 3 rd job tself. 3.2 Reversblty of Communcaton Blockng Tandem Queues U 31 U 32 U 33 U 34 U 35 s s s s s 35 Fgure 3: Dual Network Graph of TQ3C- LP1(G/G/1/,G/G/1/3,G/G/1/2) There are 15 constrants n the dual LP, consequently there are 15 snk nodes n the dual graph. The constrant for the source node (node A 0 ) s obtaned by summng over 5 all 15 constrants, whch s U 5. Graphcally, node = 1 1 A 0 generates 15 unts of flow and sends them out along ts 5 ncdent arcs to nodes s 11, s 12, s 13, s 14, and s 15. If jobs arrve at the system accordng to a random process,.e. {A 1,,..,5}, then the length of each of these 5 arcs equals the arrval tme of the correspondng job,.e. the length of arc (A 0, s 13 ) s A 13, the tme epoch of the 3 rd job s arrval. If all jobs are watng at stage 1 when the system starts (at tme 0), then the length of these fve arcs wll be zero. In ths case, U 12, U 13, U 14, and U 15 wll be zero and U 11 wll be 15 because the server can only serve one job at a tme and thus arcs (A 0, s 12 ), (A 0, s 13 ), (A 0, s 14 ), and (A 0, s 15 ) can be elmnated. Ths s mportant n provng the property of reversblty. The length of all other arcs equals the servce tme of the correspondng job at that stage,.e. the length of arc (s 22, s 23 ) equals s 23, the servce tme of the 3 rd job at stage 2. Ths s also true for arcs (s 13, s 23 ) and (s 31, s 23 ), meanng that all arcs that enter the same node have the same length. Hence, we can modfy the network slghtly by lettng the length of each node equal the servce tme of the correspondng job at that stage,.e. the length of node s 23 equals s 23, the servce tme of the 3 rd job at stage 2. Ths change wll facltate our proof of reversblty later. There are three types of flows n the network, U k, k, and W k,k+1,. U k s the flow from stage k to stage k+1; kz s the flow wthn stage k; and W k,k+1, s the flow from stage k+1 back to stage k. The prmal formulaton tells us that one and only one constrant regardng F k s bndng f there s no tme te. However, f there s a tme te, we can arbtrary pck one of the constrants that are blndng (e.g. always pck the server wth the smallest ndex). Ths wll not change our proof. Therefore, one and only one of U k, k, and W k,k+1, s postve. We start the proof by gvng some defntons related to reversblty. All notaton n ths Secton wth a ~ refers to the reversed system. Defnton 3.1. The throughput, TH, of a tandem queueng network s defned as the number of jobs per unt tme released from the last staton n the long run: n. TH = lm n E[ Fmn ] Defnton 3.2. Gven a tandem queueng network, the correspondng reversed tandem queueng network s obtaned by reversng the order of all statons. For communcaton blockng, snce a k s defned as the buffer between stage k-1 and stage k plus the one space for a job durng servce, ths reversal means a% k = a m + 2, k = k 2,..,m. Defnton 3.3. A tandem queueng network s sad to be reversble f t has the same throughput as ts correspondng reversed tandem queueng network. In the followng, we wll show that, under job reversal ( s% k = sm+ 1 kn, + 1, k,.., m,,.., n), F % s dentcal to F mn mn. So, f the servce tmes {(s 1, s 2,, s m ),,..,n} are..d. random varables, the completon tme of processng n jobs n the reversed queue has the same dstrbuton as the completon tme of processng n jobs n the orgnal queue (.e. Prob( F% mn x) = Prob( Fmn x) for any postve nteger n and real number x), thus the throughputs of the two systems are the same and consequently the reversblty property s proved. Now assume that, at tme 0, there are n jobs watng for processng at stage 1 and there are no jobs at the other stages. Usng the argument n Secton 3.1, we can elmnate all arcs emanatng from A 0 except the one that ends at node s 11. The dual problem s to fnd the longest paths from node s 11 to all other nodes, especally to node s 35, whch s the makespan for processng all jobs and s usually called the crtcal path. Notce that snce exactly one of U k, k, and W k,k+1, s postve, for each node n the network there s exactly one arc wth postve flow gong nto t (the others have zero flow), and the longest path from the begnnng node to the

6 last node s contnuous and unque (ths can be easly seen f one starts from the last node, gong backward to the frst). To show that the queue s reversble, we reverse the order of all stages and let the jobs enter the system n the reversed order. Ths s same as reversng the drecton of all arrows and lettng the jobs enter the system from the last node, node s 35. It s obvous that ths reversal does not change the structure of the network. The reversed dual problem s now to fnd the longest paths form node s 35 to all other nodes. Snce the structure of network does not change, the longest path from node s 35 to node s 11 remans unchanged. However the longest paths from s 35 to other nodes n the reversed network are not the same as the longest paths startng from s 11 n the orgnal network. For example, the length of the path from node s 35 to node s 23 n the reversed network s not equal to the length of the path from node s 11 to node s 23 n the orgnal network. But these dfferences do not affect the makespan of processng n jobs. The longest path also reveals the bottleneck servers. The fact that the structure of the reversed dual network s the same as that of the orgnal dual network can be verfed by reversng the order of stages and the servce tmes n formulaton TP3C-LP1 (the reversed queue s denoted as TQ3CR-LP1(G/G/1/,G/G/1/3,G/G/1/2), where the R stands for Reversed ), takng the dual of t, and drawng the dual network graph from that reversed formulaton; ths s also a tandem queue and can be expressed as TQ3C- LP1(G/G/1/,G/G/1/2,G/G/1/3). Fgure 4 shows that structure remans unchanged. Therefore, we have F % mn = F, whch mples the tandem queue s mn reversble. A 0 U 11 s s s s s 31 W 123 W 124 W 125 U 21 U 22 U 23 U 24 U 25 s s s s s 21 W 234 W 235 U 31 U 32 U 33 U 34 U 35 s s s s s 11 Fgure 4: Dual Network Graph of TQ3CR- LP1(G/G/1/,G/G/1/3,G/G/1/2) 3.3 Reversblty of Producton Blockng and General Blockng Tandem Queues Cheng (1995) gves a suffcent condton for the reversblty of a general blockng tandem queue, whch s the nterchangeablty of a k and b k (b k = a k ). Ths condton s restrctve. In Chan and Schruben (2003), the authors examne the reversblty of producton blockng and general blockng tandem queues and provde a more relaxed condton for the reversblty of a general blockng tandem queue, whch s stated as the followng: Condton: A general blockng tandem queue s reversble f t satsfes the followng condton: a k b k = Z, k,..,m, where Z s an arbtrary nteger. Ths condton mples Cheng (1995) s condton when Z s 0. The ntutve nterpretaton of ths condton s that, as long as the buffer sze between stage k-1 and stage k n the reversed queue s same as the buffer sze between stage m+1-k and stage m+2-k n the orgnal queue, the blockng scenaro s the same and thus the queue s reversble. 4 CONCLUSION AND EXTENSION Although we study only sngle-server tandem queues here, t s possble to extend our LP method to do the performance analyss of mult-server tandem queues. It s a potental tool for provng the reversblty of mult-server tandem queues. Another possble use of our methodology s to prove the reversblty of closed tandem queueng networks. Whle we focus our attenton on tandem queueng networks, the method gven n Secton 2 can also be used n dervng the mathematcal programmng formulaton for other dscrete event systems. An mportant mplcaton of the prmal formulatons gven n Secton 2 s that they all have a specal structural property: they are totally unmodular. Ths property ensures that the dual s a network problem and the soluton s nteger. The mathematcal programmng representaton of dscrete event systems makes t possble for us to use the technques or algorthms of mathematcal programmng for the analyss of dscrete event systems;.e. when dong a smulaton of a tandem queue, one may solve the mathematcal programmng problem, fnd the bottleneck server, and do senstvty analyss usng the dual varables. ACKNOWLEDGMENTS UC Berkeley graduate students Theresa Roeder and Debbe Pederson provded helpful suggestons and crtcsms. Ths work was supported n part by the Semconductor Research Corporaton and the Natonal Scence Foundaton.

7 REFERENCES Buzacott, John A., Queueng models of Kanban and MRP controlled producton systems, Engneerng cost and Producton Economcs, 17(1989), Buzacott, John A. and Shanthkumar, J. George, Stochastc Models of Manufacturng Systems, New Jersey: Prentce Hall, Chan, Wa Kn and Schruben Lee W., Reversblty of Tandem Queueng Networks From Ther Mathematcal Programmng Representatons, Techncal Report, IEOR Department, UC Berkeley, CA, Cheng, Dnah W., Tandem Queues wth General Blockng: A Unfed Model and Comparson Results, J. Dscrete Event Dynamc Systems: Theory and Applcatons, 2 (1993), Cheng, Dnah W., Lne Reversblty of Tandem Queues wth General Blockng, Management Sc. 41(1995), Dattatreya, Eswarahall Sundararajan, Tandem Queueng Systems wth Blockng, Ph.D. dssertaton, IEOR Department, Unversty of Calforna, Berkeley, CA, January 31, Lu, Xao-Gao and Buzacott, John A., The Reversblty of cyclc queues, Operatons Research Letters, 11(1992), Maxwell, W. L. and R. C. Wlson (1981). "Dynamc Network Flow Modelng of Fxed Path Materal Handlng Systems." IIE Transactons 13(1): Muth, Egnhard J., The Reversblty Property of Producton Lnes, Management Sc., 25(1979), Ross, Sheldon M., Introducton to Probablty Models, Sx Edton, 1997, Academc Press. Schruben, Lee W., Smulaton Modelng wth Event Graphs, Communcatons of the Assocaton of Computng Machnery, 26(11), 1983, Schruben, Lee W., Mathematcal Programmng Models of Dscrete Event System Dynamcs, 2000 Wnter Smulaton Conference Proceedngs, IEEE. Part vol.1, 2000, pp vol.1. Schruben, D and Schruben, Lee W., Graphcal Smulaton Modelng usng SIGMA, Custom Smulaton, Schruben, Lee W., Condtonal Parametrc Petr Nets and ther Mappng to Smulaton Event Graphs, submtted to Europe Smulaton and Modelng Conference, Italy Oct Yamazak, Genj, Sakasegawa, Hrotaka and Kawashma, Takesh, Producton Rate Estmated wth Flow-Shop Reversblty, Bulletn of the JSME, 21(1978), Yamazak, Genj, Sakasegawa, Hrotaka and Shanthkumar, J. George, On Optmal Arrangement of Statons n a Tandem Queueng System wth Blockng, Management Scence, 38.1(1992), Yen, H.-C., Integer Lnear Programmng and the Analyss of Some Petr Net Problems, Theory Comput. Systems 32, (1999) AUTHOR BIOGRAPHIES WAI KIN CHAN s a PhD student n the Industral Engneerng and Operatons Research Department at the Unversty of Calforna, Berkeley. Hs research nterests nclude desgns of smulaton experments, queueng theory, and computatonal optmzaton. Hs e-mal s <kn@eor.berkeley.edu> and hs web address s < LEE W. SCHRUBEN s the Charman of the Department of Industral Engneerng and Operatons Research at the Unversty of Calforna, Berkeley. He joned the faculty at UC Berkeley after spendng 25 years as a member of the School of Operatons Research and Industral Engneerng at Cornell where he held the Andrew S. Schultz Endowed Professorshp. Hs research s prmarly on dscrete event smulaton modelng methodology and n the desgn and statstcal analyss of smulaton optmzaton experments. Hs e-mal s <schruben@eor.berkeley.edu>