Analysis of a Fork/Join Synchronization Station with Inputs from Coxian Servers in a Closed Queuing Network

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1 Analysis of a Fork/Join Synhronization Station with Inputs fro Coxian Serers in a Closed ueuing Network Ananth Krishnaurthy epartent of eision Sienes and Engineering Systes Rensselaer Polytehni Institute 8 th Street Troy NY 8 Phone: Fax: Eail: [email protected] Raan Suri Center for uik Response Manufaturing Uniersity of Wisonsin-Madison 53 Uniersity Aenue Madison WI Telephone: Fax: Eail: [email protected] and Mary Vernon epartent of Coputer Sienes Uniersity of Wisonsin-Madison W ayton Street Madison WI-5376 Telephone: Fax: Eail: [email protected] Abstrat Fork/oin stations are oonly used to odel the synhronization onstraints in queuing odels of oputer networks fabriation/assebly systes and aterial ontrol strategies for anufaturing systes. This paper presents an exat analysis of a fork/oin station in a losed queuing network with inputs fro serers with two-phase Coxian serie distributions whih odels a wide range of ariability in the input proesses. The underlying queue length and departure proesses are analyzed to deterine perforane easures suh as throughput distributions of the queue length and inter-departure ties fro the fork/oin station. The results show that for ertain paraeter settings ariability in the arrial proesses has a signifiant ipat on syste perforane. The odel is also used to study the sensitiity of perforane easures suh as throughput ean queue lengths and ariability of inter-departure ties for a wide range of input paraeters and network populations. To appear in Annals of Operations Researh-speial issue on Stohasti Models of Prodution-Inentory Systes

2 Introdution Fork/oin stations are used to odel synhronization onstraints between entities in a queuing network. The fork/oin station of interest in this paper onsists of a serer with zero serie ties and two input buffers. As soon as there is one entity in eah buffer an entity fro eah of the buffers is reoed and oined together. The oined entity exits the fork/oin station instantaneously. Subsequent to its departure the oined entity forks bak into the oponent entities whih then get routed to other parts of the network. Fork/oin stations find any appliations in queuing odels of anufaturing and oputer systes. In queuing odels of fabriation/assebly systes the assebly station is typially odeled using a fork/oin station (Harrison 973; Latouhe 98; Hopp and Sion 989; Rao and Suri 994. Reently fork/oin stations hae also been used to odel the synhronization onstraints in odels of aterial ontrol strategies for ulti-stage anufaturing systes (i Masolo et al. 996; Krishnaurthy et al.. In oputer systes analysis queuing networks with fork/oin stations hae been studied in the ontext of parallel proessing database onurreny ontrol and ouniation protools (Baelli et al. 989; Varki 999; Prabhakar et al.. To deelop effiient ethods for analyzing these networks seeral researhers hae analyzed different aspets of the perforane of fork/oin stations in isolation. The typial inputs for suh an analysis are the apaity of eah input buffer and a desription of the inter-arrial ties of the entities to eah input buffer. Perforane easures of interest inlude synhronization delays queue length distributions at the different input buffers and station throughput. For the sake of analytial tratability a aority of these efforts assue that the inter-arrial ties of the inputs to the fork/oin station hae an exponential distribution (Harrison 973; Bhat 986; Lipper and Sengupta 986; Hopp and Sion 989; So et al. 994; Takahashi et al. 996; Varki 999. Although these results are useful in any of the appliations ited aboe the inter-arrial ties hae a distribution ore different fro the exponential. Reent studies by Takahashi et al. ( and Takahashi and Takahashi ( use atrix analytial ethods to analyze fork/oin stations where the inter-arrial ties hae phase type distributions. Howeer they assue that the arrials are fro infinite populations i.e. the arrials are an uninterrupted proess and an arriing entity that finds the input buffer full is lost. Additionally the oputational oplexity arising fro the use of atrix analytial approahes sees to be a high prie to pay for analyzing fork/oin stations with inputs haing inter-arrial ties that hae a distribution ore general than the exponential. When the fork/oin station is part of a losed queuing network (suh as in odels of ulti-stage kanban systes or losed ulti-leel fabriation/assebly systes then one the queue length of an input buffer equals the size of the population that an arrie to the buffer the arrial proess shuts down teporarily. In this paper we propose an approah for analyzing suh fork/oin stations for a fairly general lass of inputs with redued oputational oplexity. More speifially we study the perforane of a fork/oin station in a losed queuing network with inputs fro serers with two-phase Coxian serie distributions. The hoie of two-phase Coxian distribution allows us to odel input proesses with a wide range of ean ( and squared oeffiient of ariation SCV [.5. We analyze the queue length proess as a ontinuous tie Marko hain to estiate the throughput and queue length distributions. Next

3 3 we analyze the departure proess as a sei-marko proess and derie expressions for the distribution of the inter-departure ties after deriing the state transition probabilities for the ebedded Marko hain. The reainder of this paper is organized as follows. Setion defines our odel of the fork/oin station lists the speifi inputs and outputs and desribes our analysis approah. Setion 3 proides a suary of the literature to date on the analysis of fork/oin stations. Setion 4 desribes our analysis of the queue length proess while Setion 5 desribes our analysis of the departure proess fro the fork/oin station. Setion 6 proides soe nuerial results and Setion 7 presents our onlusions. Syste esription and Oeriew of Analysis. Syste esription We desribe our odel of the fork/oin station and explain how it ould be used to represent the synhronization behaior in partiular anufaturing and oputer systes. The odel is illustrated in Figure. As shown in the figure the fork/oin station has two input buffers B and B. If an entity arriing in buffer B (B finds input buffer B (B epty it waits for the orresponding entity to arrie in input buffer B (B. As soon as there is at least one entity in eah buffer one entity is reoed fro eah buffer. The reoed entities oin together and iediately depart fro the fork/oin station. As a result the ontent of eah input buffer is redued by one. Subsequent to departure fro the fork/oin station the oined entity forks bak into two entities that are routed to separate stations with single serers. The serie ties at these serers hae two-phase Coxian distributions. At these serers eah entity queues for serie and upon opletion of serie the entity reisits the fork/oin station. There is a finite population of size K i for the entity of type i. Consequently the nuber of entities in input buffer B i and at the orresponding Coxian serer always su up to K i i and the arrial proess to buffer B i shuts down when there are K i units in buffer B i. Fork/oin stations with suh harateristis are found in odels of ulti-stage kanban systes losed ulti-leel fabriation/assebly systes and ouniation systes. We proide exaples for eah ase.. First the fork/oin station desribed aboe an represent a synhronization station before an assebly operation in a fabriation/assebly syste (Rao and Suri 994. In this ase K i ould orrespond to the fixed nuber of autoated guided ehiles (AGVs transporting oponents of type i fro the fabriation sub-network to the assebly station. Entities in buffers B and B orrespond to the fabriated parts that are to be assebled. The oin operation orresponds to the kitting operation while the fork operation orresponds to the release of free AGVs to arry the parts required for assebly. These AGVs would go bak to the fabriation sub-networks to be restoked with parts. The arrial of reloaded AGVs fro the fabriation sub-networks ould be odeled using a serer with serie ties haing a suitable two-phase Coxian distribution.

4 4 K p -p B p -p B K Figure. Fork/oin station in a losed network haing serers with two-phase Coxian serie distributions. As a seond exaple the fork/oin station odel ould represent the synhronization onstraint in a kanban ontrol syste. In odern anufaturing kanban systes are a popular for of aterial ontrol (i Masolo et al. 996; Liberopoulos and allery. Here the fork/oin station ould odel the synhronization onstraint between an upstrea and downstrea stage in a ulti-stage kanban syste. Eah entity in buffer B would orrespond to a part with an upstrea kanban attahed to it while eah entity in buffer B would orrespond to a free kanban returning fro the downstrea stage and K and K would be the nuber of kanbans in the respetie stage. uring the oin operation a part and upstrea kanban are oined with a downstrea kanban and during the fork operation the upstrea kanban is sent bak while the part and downstrea kanban are sent to the next anufaturing stage. The anufaturing proess in eah stage ould be odeled using a serer with serie ties haing a suitable two-phase Coxian distribution.

5 5 3. Finally this fork/oin station odel an also be applied to represent the synhronization behaior of parallel progras ontending for shared resoures in a parallel or distributed oputer syste. For exaple see Heidelberger and Triedi (983.. Inputs and Outputs We assue that the two-phase Coxian distribution at serer i is defined by three paraeters i i and p i with uulatie distribution funtion where for t and i ( it i t Gi ( t Ci e Cie C ( p i i i i and Ci Ci i i with i i ( Note that this iplies that wheneer the nuber of entities at buffer B is less than K i entities arrie at the rate i i λi and the SCV of the inter-arrial ties is p i i i pi i( i i( pi i for i (Altiok 996. ( p i i i i.3 Oeriew of Our Analysis For a fork/oin station haraterized as aboe our goal is to opute the throughput χ the ean queue lengths L and L at the input buffers B and B and the arginal distribution G (t of the inter-departure ties fro the fork/oin station. Correspondingly our analysis of the fork/oin station onsists of two parts: ( the analysis of the queue length proess and ( the analysis of the departure proess. We briefly suarize the analysis approah in the following paragraphs. Analysis of the queue length proess: To analyze the queue length proess we first define the state spae for the queue length proess and then analyze the queue length proess as a ontinuous tie Marko proess. We sole the ontinuous tie Marko hain to obtain the steady state probability distributions of the queue lengths at buffers B and B respetiely. Fro these we obtain different perforane easures suh as throughput and the ean queue lengths. The details are desribed in Setion 4. Analysis of the departure proess: We note that the output proess is a Marko renewal proess (isney and Kiessler 987. In analyzing this Marko renewal proess we ake use of the speial struture of the Marko proess ebedded at departure instants to obtain the transition probability atrix and stationary probabilities of the Marko hain ebedded at departure instants. Using the stationary probabilities we obtain the arginal distribution of inter-departure ties fro the fork/oin station. The details are gien in Setion 5.

6 3 Reiew of Preious Work 6 Fork/oin stations with two or ore inputs hae been used to odel synhronization onstraints in queuing odels of oputer and anufaturing systes. Harrison (973 analyzed a fork/oin station with renewal input streas and showed that when there are no apaity liits for the input buffers the fork/oin station is unstable. Howeer if the buffers are bounded then the fork/oin station is stable. Bhat (986 analyzed a fork/oin station with bounded buffers and Poisson inputs and deried expressions for the queue length distributions at the input buffers. Kashyap (965 studied a kitting proess with input queues of oponents. The kitting proess was odeled as a double-ended queue and expressions for the waiting tie distributions were deried for the ase of Poisson inputs. So et al. (994 and Takahashi et al. (996 studied the departure proess fro a fork/oin station with finite buffers and exponentially distributed interarrial ties to these buffers. They deried expressions for the arginal distribution of the interdeparture ties. Varki (999 assues a finite ustoer population in the fork/oin queuing network but restrits all serie ties to be exponentially distributed. In any appliations the inter-arrial ties hae a distribution different fro the exponential. Reent studies by Takahashi et al. ( and Takahashi and Takahashi ( use atrix analytial ethods to analyze fork/oin stations where the distribution of inter-arrial ties is of a phase type. Howeer they assue that the arrials are fro infinite populations i.e. the arrials are an uninterrupted proess and an arriing entity that finds the input buffer full is lost. When the fork/oin station is a part of a losed queuing network then one the ontents in the input buffer reah a ertain leel the arrial proess shuts down teporarily. In this paper we analyze the perforane of a fork/oin station in a losed queuing network with inputs fro serers with two-phase Coxian serie distributions. The hoie of two-phase Coxian distributions allows us to odel a wide range of input proesses naely input proesses with ean inter-arrial ties ranging oer ( and with squared oeffiient of ariation (SCV in the range of [.5. This range oers the alues typially expeted of traffi proesses in any pratial systes (Kaath et al. 988; Buzaott and Shanthikuar Analysis of the ueue Length Proess In this setion we analyze the queue length proess as a ontinuous tie Marko proess. Table suarizes the notation used in the analysis and in the rest of the paper. Let N( t and N ( t denote the nuber of units in buffers B and B respetiely at tie t. Fro the operational harateristis of the fork/oin station we note that it is not possible for both the buffers B and B to be non-epty for any finite tie. epartures our instantaneously fro the fork/oin station wheneer both buffers are non-epty. Therefore the nuber of units in both buffers at tie t an be desribed uniquely using the one-diensional rando ariable N( t N( t N ( t. N (t takes on alues K K. For instane N( t K iplies that the buffer B is epty and the buffer B has K units. If both input buffers are epty N (t.

7 7 Sybol and p K λ Table. Notation used for our analysis esription Paraeters of the two-phase Coxian distribution for the arrial proess at buffer B Size of the finite population fro whih arrials our to input buffer B Rate of arrials to buffer B when it is not shut down SCV of inter-arrial ties at buffer B when it is not shut down N (t Nuber of units in buffer B at tie t J (t Phase of pending arrial to buffer B tie t ( ( t J ( t J ( t State of the fork/oin station at tie t N( t N( t N ( t L Aerage queue length at buffer B N χ Throughput of the fork/oin station To desribe the state of the syste at any tie t we need to onsider both-the nuber of units in eah input buffer as well as the phases of the pending arrials. Then at tie t eah buffer an be in one of three distint states as defined below: J ( [ J ( ] if the pending arrial to buffer B [B ] is in phase t t if the pending arrial to buffer B [B ] is in phase if the buffer B [B ] is full i.e. has K [ K ] units and the arrial proess to the buffer has shut down. (3 With these definitions the state of the syste is opletely haraterized by J J { ( t J( t J ( t t }. Clearly J J is a ontinuous tie Marko hain that desribes the stohasti behaior of the syste. The state spae of state of J J is gien by S {( n : n K K ; ; } {( K( K( K( K} (4 Note that the Marko hain has 4(K K states. The state transition rates for the ontinuous tie Marko hain representing the queue length proess are illustrated in Figure. These transition rates are easily deried fro the paraeters of the two-phase Coxian distribution for the arrial proesses and p. For exaple if n < K transitions fro the state (n-

8 8 to the state (n our when the arrial proess for buffer B opletes stage (at the rate and stage of the arrial proess is not seleted giing a total rate of (-p. CUT ( p n- ( p n ( p ( p ( p n- n- p p ( p ( p ( p n n p p n- p n p p p Figure. State transitions diagra when K n K - 4. Transition Equations For eah ( n S let P ( n tie Marko hain ( J J denote the steady state probability that the ontinuous N is in this state. These steady state probabilities an be oputed by soling the orresponding state transition equations as follows. Corresponding to whether one of the arrial proesses to the buffers has shut down (S or both the arrial proesses are not shut down (NS we partition the state spae S into S and S where S NS

9 S NS {( n : n K K ; ; } and S S NS {( K ( K ( K( K} S S (5 9 Then onsidering the partition of NS S and n [ P ( n P ( n P ( n P ( n ] T P ( : we hae that in steady state for eah ( n NS S ( p ( p ( p p ( p ( p p P ( n P ( n ( p p p For ( K and S ( we hae: K S (6 ( p ( p ( p P K P ( K (7 P ( K For ( K and S ( we hae: K S ( p ( p P ( K P ( K (8 P ( K p Additionally we hae the following: ( n S P ( n (9 Soling the syste of equations 6-9 aboe for eah the steady state probabilities P n. ( 4. Aerage ueue Length and Throughput ( n S we obtain expressions for Gien the steady state probabilities of the queue length proess the aerage queue lengths L and L at the buffers B and B are obtained using equations and below. L K np ( n n K P ( K (

10 L n P ( n K P ( K ( n K Thus we hae obtained the probability distribution of the queue length proess as well as the ean queue lengths at both the input buffers. The throughput χ of the fork/oin station is oputed fro the steady state probabilities of the queue length proess as follows: χ ( P ( n P ( n ( p ( P ( n P ( n n K K n P ( K ( P ( n P ( n ( p ( P ( n P ( n ( p P ( K P ( K ( p P ( K In setion 6 we will proide soe nuerial results of our queue length analysis. First we oplete our analysis of the departure proess of the fork/oin station. 5 Analysis of the eparture Proess As noted in Setion.3 the departure proess of the fork/oin station is a Marko renewal proess (isney and Kiessler 987. To analyze this proess we first identify the states in the Marko hain ebedded at departure instants. Next we reognize that this ebedded Marko proess has a speial struture and use this inforation to identify the possible saple paths between suessie departures. By deriing the onditional probability distributions for eah saple path we obtain the transition probability atrix P of the Marko hain ebedded at the departure instants. We sole the Marko hain to obtain the stationary probability etor. Using and the distributions of the inter-arrial ties at the two buffers we obtain G (t the steady state arginal distribution of inter-departure ties fro the fork/oin station. The analysis is suarized in Figure 3. (

11 Conditional probability distributions for saple paths Regions in sei- Marko kernel Marko hain ebedded at departure instants Steady state probabilities of ebedded Marko hain Marginal distribution of inter-departure ties Figure 3. Analysis of the departure proess at the fork/oin station 5. The Sei-Marko eparture Proess A departure ours (siultaneously fro both the buffers wheneer at least one unit is aailable in eah input buffer B and B. The sequene of states of the fork/oin station at the departure instants and the orresponding sequene of departure ties desribe a pair of related rando proesses ( X. The rando proess { τ : N} where N is the set of all nonnegatie integers is a sequene of real alued rando ariables τ where τ is the tie of the th departure. Note that eah departure tie for the fork/oin station oinides with either an arrial tie in buffer B or an arrial tie in buffer B. Assoiated with the rando proess is the inreental proess T { T : N} where for > T is the tie between onseutie departures and - i.e. T τ τ and T. The states of the rando proess X X N J J N N τ { : } is etor alued and onsists of the triple where ( J J( τ and J J ( τ. τ denotes the tie instant ust after a departure at τ N( τ denotes the nuber of units at the fork/oin station and J ( τ and J ( τ respetiely denote the phases of the pending arrials to input buffers B and B ust after the th departure. We derie the distribution of the inter-departure ties by analyzing the pair of rando proesses ( X T. T depends on the present state X and the next state X. Howeer gien these states T is independent of the preious states X... X and T...T. That is the following relation holds ( X T X... X T T P ( X T X P... for all N (3 and the output proess ( X T is a Marko renewal proess. The state spae of the output proess ( X T is S R where R ( and

12 S {( K } {( n ( n K < n < } {( (( } {( ( n < n < K } n {( } In Figure 4 the set of feasible states in eah state in S all exit transitions lead to states in K (4 S are shown in gray. It an be easily erified that for S. To define these feasible states ore preisely we note that States in S S and define S S S. S are shown in dotted boxes in Figure 4. These states are not in S for the following reasons. First it is obious that neither of the buffer arrial proesses is shut down iediately S after a departure fro the fork/oin station i.e. S S. For X we reason in the NS S following anner. If K buffer B is full and its arrial proess is shut down prior to N the departure instant iplying that {( K ( K ( K } S. Siilarly for N K the arrial proess to buffer B is shut down prior to the departure instant iplying that ( K {( K ( K } S. Furtherore for K < N < the departure tie τ oinides with an arrial tie in buffer B and hene { ( N } S. Siilarly for < N < K the departure tie τ oinides with N S. Finally when N we an arrial tie in buffer B and hene {( } hae {( }. S -K -K - K - K - K -K -K -K - K - K - K -K -K -K - K - K - -K -K - K - K - X X Iplies that Iplies that X S X S and X S but X S Figure 4. Identifying the states of the Marko hain ebedded at departure instants Fro Figure 4 and equation 4 it is easy to see that S has ( K K 3 states. Sine the output proess ( X T is a Marko renewal proess it is opletely haraterized by its sei- Marko kernel where:

13 ( X X t P ( X T t X (5 3 For oneniene the eleents of the sei-marko kernel an be expressed in the Laplae transfor doain as * ( X X ds L{ ( X X dt } (6 * Setion 5. below desribes how to opute the alues of ( X X ds * ( X X ds seeral harateristis of the output proess ( T. Using X an be obtained (isney and Kiessler 987. For exaple the state transition atrix P of the underlying Marko hain X ebedded at ties τ N is obained by setting s in equation 6 aboe i.e.: P ( X X ( X J J X J J P * X X ds s ( Let Π { Π( X : X S } where Π( k k (7 X k is the steady state probability that the fork/oin station is in state X k at a departure instant. The stationary probability etor Π of the underlying Marko hain is obtained as follows: Π ΠP (8 k S ( Π X (9 k The syste of equations 8 and 9 are soled to obtain Π. Setion 5.3 desribes how to opute G (t the uulatie distribution funtion of the inter-departure ties using Π. The oputation of P and hene Π is the next step in estiating the uulatie distribution funtion G (t of the inter-departure ties. In the next setion we onstrut P using the sei- Marko kernel of the output proess (X T and use P to deterine Π. 5. Constrution of Sei-Marko Kernel and State Transition Probability Matrix P To onstrut P we note that between two suessie departure states X and X the fork/oin station isits a finite sequene of interediate states Z k k... where Z k S. Eah suh sequene orresponds to saple paths leading the fork/oin station fro departure state at tie τ to state X at tie SS X X be the set ontaining all suh sequenes of interediate states and SS ( X X denote the nuber of sequenes for the pair SS X X SS X X be the th sequene in this set. Then τ. Let ( ( X. Let ( ( X X

14 4 Z Z... Z l Z where l ( is SS ( X X an be written as the ordered sequene ( ( l( the length of the sequene ( X X Z X X SS l( Z and k... l( orrespond to the interediate states. Additionally for a gien saple path that ontains the sequene of states let t t... t l ( tl( be the ties when the fork/oin station transitions to states Z Z... Z l ( Z l( respetiely. Note t τ and t l ( τ. Figure 5 illustrates saple paths for eah of three different sequenes in SS ( X X as well as the sequene of interediate transitions in a saple path for SS ( X X. Sine the sequenes of interediate states ( X X likely we hae * SS are utually exlusie and equally SS ( X X * ( X X ds k ( Z k Z k ds l ( k Z k ( * P ( X X ( Z Z ds SS ( X X l ( k k k k s ( SS ( X X 3 SS X X X X S S ( X Z Z S Z S SS ( X X X Z 3 Z l( t τ t t t3 τ In order to opute ( X X Figure 5. Possible saple paths between departures P using equation aboe we need to identify all the sequenes X X SS( X X ( k SS ( X X SS( X X for eah k > Z Z ds * SS and then opute k ( k k s and eah Z. This ay appear to be a ubersoe task. Howeer beause arrials to buffers B and B are oposed of exponential phases the sei- Marko kernel and hene the transition atrix P has a speial struture. In other words

15 5 depending upon the alues of N J and J of states X and X the non-zero portion of P an be partitioned into 4 regions. We exploit this speial struture to identify the set of * sequenes ( X X X X and also to opute k ( Zk Zk ds s for eah Z k SS ( X X. Figure 6 illustrates these regions (labeled through 4 in P using the exaple of a fork/oin station with two input streas and with K K 5. SS for eah pair ( Regions - orrespond to transitions between states X and X in whih one of the buffers (B or B is non-epty at tie instants τ and τ. Regions -4 orrespond to the states when both the buffers B and B were epty at tie instants τ and τ. Transitions between states in a region share siilarity in the struture of the possible saple paths. We exploit this P X X. The Appendix lists the foral siilarity while deriing expressions for ( definitions as well as the expressions for P ( X X for eah pair ( X X. Here we illustrate the proedure for deriing these expressions using Region as an exaple. X P ( X X X Figure 6. Regions in the sei-marko kernel and state transition atrix P In Region we hae two sub-regions: X ( X for < N K and X ( X for > N

16 6 Consider P ( X X for the first sub-region where X ( and X for < N K. In this ase P ( X X is the probability of the eent that in buffer B nd exatly arrials and only phase of a two-phase arrial are opleted before the opletion of phase of the arrial in buffer B. Sine arrials to buffers B and B X we an write: are independent and oposed of exponential phases for ( ( X X P X P(Exatly arrials to B opletes before phase of arrial to B opletes nd P(Phase of a two-phase arrial to buffer B opletes before phase of the arrial to B opletes nd P(Phase of the arrial to B opletes before phase of arrial to buffer B ( Noting that eah of the arrials in B ould be oposed of one or two exponential phases and thus there are possible sequenes in SS ( X X equation leads to the P X X : following expression for ( P ( X X ( p p N p (3 Using arguents siilar to the aboe it an be shown that for this sub-region: * ( X X ds ( p s p s s N p s s (4 * Fro ( X X ds the eleents of the sei-marko kernel orresponding to this subregion an be obtained. Using an approah siilar to that desribed aboe we an derie the expressions for * P ( X X and ( X X ds when X ( and X for * > N in Region. The expressions for P ( X X and ( X X ds for X are deried in a siilar anner. The full set of expressions for the eah other pair ( X eleents of P is listed in the Appendix.

17 5.3 Marginal istribution of Inter-departure Ties 7 Using P deried in the preious setion we sole the syste of equations 8 and 9 for the stationary probability etor Π. We use Π to obtain the arginal distribution of the interdeparture ties. Note that sine P is independent of the inter-departure ties are identially distributed. Let G (t be the distribution funtion of the inter-departure ties i.e. G ( t P T t for any. G (t an be written in ters of Π and the distribution funtion { } for inter-arrial ties at input buffers and naely as follows: G (t Π G ( t Π G ( t Π ( G ( t G ( t t t Π ( G ( t( e Π ( G ( t( e where Π Π( Π S X Π( S X S S {((( } { n : < n K ; } {( } S S ( K ( n : K n < ; ( K t t C e G ( t C e { } { } S (5 Π : The steady state probability that a departure results in a non-epty buffer B. Π : The steady state probability that a departure results in a non-epty buffer B. Fro G (t we obtain the ean inter-departure ties. χ C C C C Π C C Π Π( C C C C CC CC CC CC Π C C Π( ( ( ( ( ( E of inter- Siilarly fro G (t we an derie expressions for the seond oent departure ties and hene ties. (6 the squared oeffiient of ariation (SCV of inter-departure Note that the expression for the arginal distribution G (t is also gien by G (t Π (tu where U is a olun etor with all eleents equal to. Furtherore the oint distribution of k

18 8 suessie inter-departure ties fro the fork/oin station P{ T t T t... T k t k } is equal to Π (t (t (t k U for any (isney and Kiessler Nuerial Results In this setion we present soe nuerial exaples to illustrate the usefulness of our analysis. First we opare the results of our analysis of a fork/oin station in a network haing serers with two-phase Coxian serie distributions against the results that assue serers with exponential serie distributions. Seond we study the sensitiity of perforane easures suh as throughput ean queue lengths and ariability of aerage inter-departure ties to the paraeters of the input proesses and to the ustoer populations. In the latter ase we inestigate the ipat of ( ean rates of the input proesses ( λ λ ( different SCVs of the input proesses ( and (3 network populations K and K on perforane easures suh as throughput rate aerage queue lengths at the input buffers and SCV of inter-departure ties. Figure 7 opares the alues of K and K required to obtain a target throughput fro a fork/oin station with input proesses haing inter-arrial ties with the sae ean but with different SCVs. An appliation where suh insights would be useful is losed loop fabriation assebly systes. In suh systes it would be neessary to deide the nuber of fixed pallets in the networks supplying the fork/oin station so as to guarantee a required leel of throughput. As seen in Figure 7 the throughput inreases onotonially with K and K. Howeer as the SCVs of the input proesses inrease the alues of K and K required to obtain a gien throughput inrease signifiantly. As disussed in Kaath et al. (988 and Buzaott and Shanthikuar (993 in pratial anufaturing systes we ould find SCVs ranging fro to 4.. KK KK λ λ Throughput Figure 7. Ipat of SCV on K and K required to obtain a gien throughput χ

19 9 For exaple for a throughput requireent of.88 as SCV inreases fro.5 to 4. the required alues of K and K ore than triple. For losed loop fabriation assebly systes this would iply signifiant inestent in pallets to buffer against ariability of input proesses. Conersely ignoring the ipat of ariability in the input proesses (for exaple by using the nuber of pallets needed for the ase of Poisson input proesses would result in signifiantly lower throughput than the target alue if the SCV is atually equal to 4.. This iplies that analyzing fork/oin stations for inputs ore general than the Poisson proess has iportant pratial ipliations. eparture SCV SCV SCV.6SCV.6. SCV4SCV SCV4SCV K KK K Figure 8. Ipat of K K and input SCVs on departure SCVs (λ λ Figure 8 shows the SCV of the inter-departure ties for a fork/oin station with input proesses haing rates equal to but different SCVs. As the alues of K and K inrease the SCV of the inter-departure ties tends to the aerage of the SCVs of the input proesses. Therefore while additional pallets help iproe the throughput perforane fro the fork/oin station they would not help in reduing the ariability of the output proess. In ost appliations one would expet that through reasonable initial design of the syste the rates of the input proesses at the fork/oin synhronization station would be alost equal. Howeer in reality ibalanes in arrial rates ould our and it is useful to predit the perforane ipat of suh ibalanes. Figures 9 and indiate that for ertain paraeter settings throughput ean queue length and SCV of inter-departure ties are ery sensitie to ibalane in input rates.

20 . K K 6 Throughput Ratio of arrial rates SCVSCV.5 SCVSCV. SCVSCV4. 4 SCV4. SCV.5 4 SCV4. 4 SCV. ( λ λ with λ Figure 9. Ipat of ibalaned input rates on throughput In Figure 9 we see that as would be expeted the throughput fro the fork/oin is priarily influened by the arrial rate of the slower of the two input proesses. In Figure we see that substantial queues are obsered at the buffer of the input proess with a higher rate of arrials. 6 K K 6 5 L ( Ratio of arrial rates λ λ SCVSCV.5 SCVSCV. SCVSCV4. 4 SCV.5SCV4. 4. SCV4. 4 SCV.5.5 with λ Figure. Ipat of ibalaned input rates on ean queue length at buffer B

21 5 K KK6 K 6 eparture SCV SCV SCVSCV.5 SCVSCV. SCVSCV4. 4 SCV4. 4 SCV.5 SCV4. 4 SCV Ratio of arrial rates Rho( λ λ with λ Figure. Ipat of ibalaned input rates on SCV of inter-departure ties In Figure we see that the SCV of the departure proess fro the fork/oin station is influened by the SCV of the slower of the two input proesses. Howeer the agnitude of the influene depends on the ratio of the two arrial rates. Finally fro all the figures it is lear that the SCVs of the input proesses hae a signifiant ipat on the perforane of the fork/oin station priarily when the input proesses hae nearly equal rates. As disussed earlier this would often be the ase by design hene prediting the ipat of SCVs sees iportant. Next we disuss a typial senario where the analysis presented in this paper ould be used to ake useful design deisions. We onsider a fork/oin station with input proesses haing different SCVs and arginally ibalaned inter-arrial ties. The input proess to buffer B has inter-arrial ties with ean. ( λ.9 and SCV equal to 4. while input proess to buffer B has inter-arrial ties with ean.9 ( λ. and SCV equal to.5. For a gien target throughput we onsider the ipat of hoosing different alues of K and K on the output proess. For a losed loop fabriation assebly systes this would translate into deiding whether to inest in additional pallets of one type or the other. For exaple Figure indiates that the sae target throughput of.88 an be obtained by setting K K or by setting K 8 and K. Howeer as seen fro Figure 3 setting K K results in higher ariability of the inter-departure ties. This ould be detriental to the downstrea work enters or ould require larger buffers downstrea. Een setting K 8 and leaing K would not redue the ariability of inter-departure ties. To derease the SCV of the

22 inter-departure ties one would hae to not only to set a high alue of K but also a low alue of K. This large ibalane in K and K setting ay not be intuitie to syste designers yet our odel points out its benefits. 5 K K K K6 6 K8 K 8 K K λ.9 λ. 4.5 K K Throughput χ Figure. K and K required for a gien throughput requireent 4 λ.9 λ. K 4.5 K 6 K 8 K K K6 K8 K 3 eparture SCV 5 K 5 5 Figure 3. Ipat of K and K on SCV of inter-departure ties

23 3 Before we onlude we oent on the oputational oplexity of our analysis. The transition probability atrix used for oputing the queue length distribution has a size 4(K K. For the departure proess we onsider the Marko hain ebedded at departure instants. By analyzing the orresponding transition atrix of size (K K -3 the throughput and arginal distribution of inter-departure ties fro the fork/oin station are oputed. The oputation effort in onstruting this transition atrix is further siplified by the fat that regardless of the paraeters of the Coxian distributions and the alues of K and K the non-zero portion of the atrix an be partitioned into 4 regions. The oputational tie required for estiating the throughput distributions of queue lengths and inter-departure ties was less than 3 seonds on a Pentiu III 3MHz personal oputer while using MATLAB. Next we ontrast the oputational oplexity of our analysis with siilar works in the literature. So et al. (994 and Takahashi et al. (996 analyze a fork/oin station with Poisson inputs fro infinite populations and finite buffers. In their analysis they assue that arrials that find the buffers full are lost. The oputations in these works inole transition atries of size (K K and (K K - for the queue length and departure proesses respetiely. Additionally beause of the eory-less property of the exponential distribution the analysis presented in So et al. (994 and Takahashi et al. (996 is also appliable for a fork/oin stations where the inputs to eah buffer is fro a losed queuing network with an exponential serer. Our analysis is a first step towards analyzing fork/oin stations with arrials fro finite populations haing inter-arrial ties with distributions that are ore general than the exponential. The hoie of two-phase Coxian distribution allows us to odel input proesses with a wide range of ean ( and squared oeffiient of ariation SCV [.5 and our oputations inole transition atries of size 4(K K and (K K -3 for the queue length and departure proesses respetiely. Takahashi et al. ( and Takahashi and Takahashi ( analyze fork/oin stations with inter-arrial ties haing phase type distributions (with p phases and arrials fro infinite populations. Their oputations inole transition atries of size p (K K and p (K K - for the queue length and departure proesses respetiely. While a siilar analysis for the ase of finite populations would be ery useful a ery oon pratial otiation for onsidering ore general input proesses is to study the ipat of ariability in the input proesses in addition to those of the eans and analyze their ipat on deisions regarding systes design. Our analysis for the ase of Coxian inputs perits suh a study without signifiant inrease in oputational effort. 7 Conlusion Models for analyzing fork/oin stations an be useful in the analysis of pratial anufaturing and oputer systes as well as for deeloping effiient deoposition ethods for queuing network odels of suh systes. This paper analyzes a fork/oin station in a losed queuing network with inputs fro serers haing two-phase Coxian serie distributions. Using an exat analysis we show that when the Coxian serers goerning the input proesses hae alost equal rates ariabilities in the arrial proesses an hae signifiant ipat on throughput queue lengths and ariability in the inter-departure ties. Suh insights are not aailable ia odels that assue exponential interarrial ties sine ariability annot be odeled separately fro

24 4 the ean arrial rates. Our hoie of the two-phase Coxian distribution allows us to analyze ore general input proesses without uh added oputational oplexity. We also study the sensitiity of throughput queue lengths and ariability in the inter-departure ties for a wide range of input paraeters and network populations. In soe ases our results point out behaior that would not be intuitiely obious and thus our odel ould proide aluable insights to syste designers. The insights here ould be used diretly in the design of siple systes ontaining a fork/oin station. In addition when the fork/oin station is part of a larger network inforation about the inter-departure ties ould help to haraterize the output proess whih ight in turn be the input proess for other stations in the network. Finally we note that the analysis here is also useful in extending deoposition ethods for perforane analysis of queuing networks. Suh ethods require effiient "two-oent" approxiations to haraterize the perforane of stations in the network. Howeer suh approxiations were preiously not aailable for fork/oin stations. The analysis and insights in this paper hae been used in the deelopent and alidation of two-oent approxiations for fork/oin stations (Krishnaurthy et al.. These approxiations are enabling the extension of deoposition ethods to oplex queuing networks with fork/oin stations (Krishnaurthy.

25 Appendix We list the expressions for P ( X X for eah pair ( X X in Table A below. To siplify the notation in the table we define the following quantities: 5 a b a b ( p p ( p p a b a b ( p p ( p p Additionally we define: and s a b s a b N N s H ( r for r s i i r i

26 6 Region 3 Table A. Expressions for P ( X X for eah pair ( X X X P ( X X X < N K for N N K a s for b H ( s /s bb H ( s /s- for N for < N K Sae as aboe for > N for N N a s b H s /s bb H s /s- for N for > N Sae as aboe N < N K as ( ( for < N K for 3 b b bs b bh s /s- bb H s /s- for N for < N K 3 Sae as aboe for < N K for N s N > N ba s ( ( ( for > N for N 3 b b s ( ( ( b b H s /s - bb H s /s - for N for > N K 3 Sae as aboe for > N for N N s for < N K for N N K 3 b ss ( ( b ( bh s /s b bh s /s- for N for < N K 3 Sae as aboe for > N for N N 3 b s s b b H s /s bb H s /s- for N for > N 3 Sae as aboe ( ( (

27 Region Table A. Expressions for P ( X X for eah pair ( X X ontd. X P ( X X X for < N K for N N K a s b s ( bh s/s- ( bb H s /s- for N for < N K Sae as aboe for > N for N N a s b s b H s /s- bb H s /s- for N for > N Sae as aboe for N K ( ( ( K ( -b s b H ( s /s bb H ( s /s- K ( s b H ( s /s b b H ( s /s for N ( -b - for N K ( K b ss b b s ( ( b ( bh s /s- bb H s /s- for N ( K b ss b b s ( ( ( b b H s /s- bb H s /s- for N K ( K b ss ( ( b ( bh s /s bb H s /s- for N ( K b s s b b H s /s bb H s /s- for N K ( K ( -b s b s ( ( bh s /s- bb H s /s- for N ( K ( -b s b s b ( ( H s /s- bb H s /s- 7 ( ( (

28 Region Table A. Expressions for P ( X X for eah pair ( X X ontd. X P ( X X X for < N K s for > N s ( K s ( ( ( ( ( K s ( for < N K b s ( for > N b s ( ( b s b ( b b ( ( b s ( ( ( s s b ( ( ( b b b bs ( ( b s ( ( ( s s b ( ( s s b s bs bb bb ( ( ( b s s b b s s bb b ( b s s b b s s b b b ( ( 8

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