Throughput Maximization for Tandem Lines with Two Stations and Flexible Servers

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1 Throghpt Maximization for Tanem Lines with Two Stations an Flexible Servers Sigrún Anraóttir an Hayriye Ayhan School of Instrial an Systems Engineering Georgia Institte of Technology Atlanta GA U.S.A. December Abstract For a Markovian qeeing network with two stations in tanem finite intermeiate bffer an M flexible servers we sty how the servers shol be assigne ynamically to stations in orer to obtain optimal long-rn average throghpt. We assme that each server can work on only one job at a time that several servers can work together on a single job an that the travel times between stations are negligible. Uner these assmptions we completely characterize the optimal policy for systems with three servers. We also provie a conjectre for the strctre of the optimal policy for systems with for or more servers that is spporte by extensive nmerical evience. Finally we evelop heristic server assignment policies for systems with three or more servers that are easy to implement robst with respect to the server capabilities an generally appear to yiel near-optimal long-rn average throghpt. 1 Introction We consier a tanem qeeing network with two stations an M servers. There is an infinite spply of jobs in front of station 1 infinite room for complete jobs after station 2 an a finite bffer of size 0 B < between stations 1 an 2. We assme that at any given time there can be at most one job in service at each station an that each server can work on at most one job. Moreover we assme that each server i {1... M} works at a eterministic rate µ ij [0 ) at each station j {1 2}. Hence server i is traine to work at station j if µ ij > 0. We assme that several servers can work together on a single job in which case their service rates are aitive. The service times of the ifferent jobs at station j {1 2} are inepenent an exponentially istribte ranom variables with rate µ(j) an service times at stations 1 an 2 are inepenent. Withot loss of generality we assme that µ(1) = µ(2) = 1. Finally we assme that the network operates ner the manfactring blocking mechanism. Or objective in this paper is to etermine the ynamic server assignment policy that maximizes the long-rn average throghpt of the qeeing system escribe above. For simplicity we assme 1

2 that the travel an setp times associate with servers moving from one station to the other one are negligible. Anraóttir Ayhan an Down [4] ientify the optimal server assignment policies for M 2. In particlar when M = 1 then any non-iling server assignment policy is optimal an when M = 2 the optimal policy involves assigning one server to work at each station in sch a way that the proct of the server rates at the stations they are assigne to is maximize with the servers only working at the other station (that they are not assigne to) when there is no work to be one at the station they are assigne to (e to blocking or starving). Conseqently this paper is focse on the sitation when M 3 so that the qeeing network has more servers than stations. We shall see that when M 3 then the optimal policy is more complicate than when M = 2 in that servers may move away from a station when there is still work to o at that station (see Sections 3 an 4 below). Mch of the existing work in the area of optimal ynamic assignment of servers to qees is focse on parallel qees. In particlar for a two-class qeeing system with one eicate server one flexible server an no exogenos arrivals Ahn Denyas an Zhang [3] characterize the server assignment policy that minimizes the expecte total holing cost incrre ntil all jobs initially present in the system have eparte. Moreover ner the heavy traffic assmption Harrison an Lopez [11] Bell an Williams [8] Williams [20] an Manelbam an Stolyar [12] evelop asymptotically optimal server assignment policies that minimize the isconte infinite-horizon holing cost for parallel qeeing systems with flexible servers an otsie arrivals. Finally ner the assmption of heavy traffic Sqillante et al. [18] se simlation to sty the performance of threshol-type policies for systems that consist of parallel qees. Most of the papers that have consiere the optimal assignment of mltiple servers to mltiple interconnecte qees focs on minimizing holing costs. In particlar for systems with two qees in tanem an no arrivals Farar [9] Panelis an Teneketzis [15] an Ahn Denyas an Zhang [2] sty how servers shol be assigne to stations to minimize the expecte total holing cost incrre ntil all jobs leave the system. Moreover osberg Varaiya an Walran [17] Hajek [10] an more recently Ahn Denyas an Lewis [1] sty the assignment of (service) effort to minimize holing costs in the two-station setting with Poisson arrivals. To the best of or knowlege Anraóttir Ayhan an Down [4 5] are the only two papers that consier the ynamic assignment of servers to maximize the long-rn average throghpt in qeeing networks with flexible servers. In particlar Anraóttir Ayhan an Down [4] characterize the optimal ynamic server assignment policy for a two-stage finite tanem qee with two servers an also present a simple server assignment heristic for finite tanem qees with an eqal nmber of servers an stations. For more general qeeing networks with infinite bffers Anraóttir Ayhan an Down [5] evelop ynamic server assignment policies that garantee a capacity arbitrarily close to the maximal capacity. Other research on ynamic server assignment policies incles the work of Ostalaza McClain 2

3 an Thomas [14] McClain Thomas an Sox [13] an Zavalav McClain an Thomas [21] on ynamic line balancing. In particlar Ostalaza McClain an Thomas [14] an McClain Thomas an Sox [13] sty ynamic line balancing in tanem qees with share tasks that can be performe at either of two sccessive stations. This work was contine by Zavalav McClain an Thomas [21] who sty several server assignment policies for systems with fewer servers than stations in which all servers traine to work at a particlar station have the same capabilities at that station. Moreover assming that each server has a service rate that oes not epen on the task (s)he is working on Bartholi an Eisenstein [6] efine the bcket brigaes server assignment policy an show that ner this policy a stable partition of work will emerge yieling optimal throghpt. Finally Bartholi Eisenstein an Foley [7] show that the behavior of the bcket brigaes policy applie to systems with iscrete tasks an exponentially istribte task times resembles that of the same policy applie in the eterministic setting with infinitely ivisible jobs. The remainer of this paper is organize as follows: In Section 2 we formlate the server assignment problem consiere in this paper as a Markov ecision problem. In Section 3 we provie an optimal server assignment policy for systems with two stations an three servers. In Section 4 we present a conjectre for the strctre of an optimal server assignment policy for systems with two stations an for or more servers. Section 5 contains some reversibility reslts for tanem lines with two stations an arbitrary nmbers of servers. In Section 6 we present nmerical reslts that spport the optimality of the policy propose in Section 4 escribe some properties of this policy an sty heristic policies that appear to yiel near-optimal performance an involve groping all available servers into two or three teams. Section 7 contains some concling remarks. Finally the proof of the main reslt in this paper is given in the Appenix. 2 Problem Formlation Let Π be the set of server assignment policies ner consieration an for all π Π an t 0 let D π (t) be the nmber of epartres ner policy π by time t an let T π = lim t IE[D π (t)] t be the long-rn average throghpt corresponing to the server assignment policy π. intereste in solving the optimization problem (1) We are max T π. (2) π Π For all π Π consier the stochastic process {X π (t) : t 0} where X π (t) = 0 if there is a job to be processe at station 1 the nmber of jobs waiting to be processe between stations 1 an 2 is 0 an station 2 is starve at time t; X π (t) = s for 1 s B 1 if there are jobs to be processe at both stations 1 an 2 an in the bffer there are s 1 jobs waiting to be processe at time t; finally X π (t) = B 2 if station 1 is blocke B jobs are waiting to be processe in the 3

4 bffer an there is a job to be processe at station 2 at time t. For the remainer of this paper we assme that the class Π of server assignment policies ner consieration consists of all Markovian stationary eterministic policies corresponing to the state space S = { B 2} of the stochastic process {X π (t) : t 0}. It is clear that for all π Π {X π (t) : t 0} is a continos time Markov chain an that there exists a scalar q π M i=1 max 1 j 2 µ ij < sch that the transition rates {q π (x x )} of {X π (t)} satisfy x Sx x qπ (x x ) q π for all x S. Hence {X π (t)} is niformizable for all π Π. Let {Y π (k)} be the corresponing iscrete time Markov chain so that {Y π (k)} has state space S an transition probabilities p π (x x ) = q π (x x )/q π if x x an p π (x x) = 1 x Sx x qπ (x x )/q π for all x S. It has been shown by Anraóttir Ayhan an Down [4] that since {X π (t)} is niformizable the original optimization problem in (2) can be translate into an eqivalent (iscrete time) Markov ecision problem. More specifically let π q π (x x 1) for x {1... B 2} (x) = 0 for x = 0 be the epartre rate from state x ner policy π for all x S an π Π. Then the optimization problem (2) has the same soltion as the Markov ecision problem [ ] 1 K max lim IE π (Y π (k 1)). (3) π Π K K k=1 In other wors Anraóttir Ayhan an Down [4] showe that maximizing the steay-state throghpt of the original qeeing system is eqivalent to maximizing the steay-state epartre rate for the associate embee (iscrete time) Markov chain. In the next two sections we characterize the ynamic server assignment policies that solve the optimization problem (3) for two-stage tanem qees with M 3 servers. We consier the case when M = 3 in Section 3 an the case when M > 3 in Section 4. 3 Two Stations an Three Servers In this section we consier the special case of a tanem Markovian qeeing network with two stations an three servers. Since the nmber of possible states an actions are both finite the existence of an optimal Markovian stationary eterministic policy follows immeiately from Theorem of Pterman [16]. We assme that for all i {1 2 3} either µ i1 > 0 or µ i2 > 0. (If there exists a server i sch that µ i1 = µ i2 = 0 then the problem reces to having two servers for which the optimal policy is given in Anraóttir Ayhan an Down [4].) Withot loss of generality we also assme that there exist i k {1 2 3} sch that µ i1 > 0 an µ k2 > 0. (Note that if µ 11 = µ 21 = µ 31 = 0 or µ 12 = µ 22 = µ 32 = 0 then the throghpt is zero an any policy is optimal.) Define as { µi1 } D = arg min. i {123} µ i2 4

5 The above assmptions on the service rates garantee that µ 1 µ 2 <. Similarly efine m as { µi1 } m M = arg min. i {123}\{} µ i2 Note that if µ i2 = 0 for all i {1 2 3}\{} then M = {1 2 3}\{}. Finally efine {1 2 3}\{ m}. For reasons that become clear in Theorem 3.1 stans for pstream stans for ownstream an m stans for moving. Note that the efinitions of an m imply that µ i1 µ 2 µ 1 µ i2 0 for all i {1 2 3} an µ i1 µ m2 µ m1 µ i2 0 for all i {1 2 3}\{}. Moreover from or assmptions on the service rates an from the efinitions of m an we have that µ 2 > 0 an µ 1 > 0. For fixe m an an for all i {0 1...} efine B i2 f(i) = µ i 2 2 (µ m1µ 2 µ 1 µ m2 )(µ 2 µ m2 µ 2 ) µ j 1 (µ 2 µ m2 ) B i j2 µ B i2 i 2 1 (µ 1 µ m2 µ m1 µ 2 )(µ 1 µ m1 µ 1 ) with the convention that smmation over an empty set eqals 0. Note that f(i) 0 for i 1 f(i) 0 for i B 3. µ j 2 (µ m1 µ 1 ) i j 2 (4) Throghot or evelopments we let A s enote the set of allowable actions in state s S (see the first paragraph of the proof of Theorem 3.1 in the Appenix) an (δ) enote the policy corresponing to the ecision rle δ which is a (B 3)-imensional vector whose components δ(s) A s specify what action in A s shol be applie in state s for all s S. emark 3.1 For i = 1... B 2 let T (δi ) be the throghpt of the policy (δ i ) sch that servers m an work at station 1 for s = 0 δ i servers m an work at station 1 server works at station 2 for 1 s i 1 (s) = server works at station 1 servers an m work at station 2 for i s B 1 servers m an work at station 2 for s = B 2. (5) Then sign(f(i)) = sign(t (δi ) T (δi 1 ) ) for all i = 2... B 2. This follows with some algebra from the expression for g 0 given in the proof of Theorem 3.1 (see eqation (13)). Let S = { } s S\{0} : f(s) 0 an f(s 1) 0. The following reslt follows irectly from the fact that f(1) 0 an f(b 3) 0. 5

6 Proposition 3.1 S. We are now reay to state the theorem that characterizes the optimal server assignment policy. Theorem 3.1 Define s S an let δ (s) = δ s (s) for all s S (see eqation (5)). Then (δ ) is optimal in the class of Markovian stationary eterministic policies. Moreover this is the niqe optimal policy if S = {s }. emark 3.2 Theorem 3.1 shows that in the optimal policy the pstream server works at the pstream station 1 nless that station is blocke the ownstream server works at the ownstream station 2 nless that station is starve an the moving server m works at the pstream station 1 when the nmber of jobs in the bffer is small an then moves to the ownstream station 2 when the nmber of jobs in the bffer has become sfficiently large. Note that the efinitions of m an imply that the server whose service rate at the pstream/ownstream station is relatively the largest (relative to the server s rate at the other station) shol be assigne to the pstream/ownstream station an the server whose service rates at the pstream an ownstream stations are relatively the most balance shol move between the two stations epening on the content of the bffer. Note also that the optimal policy in the case when M = 2 is essentially the above policy withot a moving server see Anraóttir Ayhan an Down [4]. The proof of Theorem 3.1 is presente in the Appenix. We now present a proposition which illstrates some properties of S. Throghot or evelopments #A enotes the carinality of any set A. Proposition 3.2 (i) If #D = 1 an #M = 1 then S has at most two elements an if S has two elements then these are two consective states. (ii) If #D = 1 an #M = 2 then S = {B 2}. (iii) If #D = 2 then #M = 1 an S = {1}. (iv) If #D = 3 then #M = 2 an S = S\{0}. Proof: Note first that for all s S \ {0} f(s) f(s 1) = (µ m1 µ 2 µ 1 µ m2 )(µ 2 µ m2 µ 2 )µ s 2 2 (µ 1 µ m2 µ m1 µ 2 )(µ 1 µ m1 µ 1 )µ B s1 1 Hence f(s) is non-increasing in s S \ {0}. B s1 µ m2 s 2 µ m1 µ j 1 (µ 2 µ m2 ) B s j1 µ B s2 1 µ j 2 (µ m1 µ 1 ) s j 2 µ s 1 0. (6) (i) It follows from #D = 1 an #M = 1 that µ m1 µ 2 µ 1 µ m2 > 0 an µ 1 µ m2 µ m1 µ 2 > 0. Then eqation (6) implies that f(s) is (strictly) ecreasing in s. Let s = min S. If s = B 2 6 2

7 then there is nothing to prove so assme that s {1... B 1}. From the efinition of s we have that f(s 1) 0. If f(s 1) < 0 then f(s) < 0 for all s s 1 an hence S = {s }. On the other han if f(s 1) = 0 then f(s) < 0 for all s s 2 an hence S = {s s 1}. (ii) It follows from #D = 1 an #M = 2 that µ m1 µ 2 µ 1 µ m2 > 0 an µ 1 µ m2 µ 2 µ m1 = 0. Then eqation (4) implies that f(s) > 0 for all s = 1... B 2 an f(b 3) = 0. Ths s = B 2 is the only s S\{0} sch that f(s) 0 an f(s 1) 0. (iii) Since #D = 2 we have µ 1 µ 2 = µ m1 µ m2 < µ 1 µ 2. Hence #M = 1 µ m1 µ 2 µ 1 µ m2 = 0 an µ 1 µ m2 µ m1 µ 2 > 0. Then eqation (4) implies that f(1) = 0 an f(s) < 0 for all s 2. Ths s = 1 is the only s S\{0} sch that f(s) 0 an f(s 1) 0. (iv) Since #D = 3 we have µ 1 µ 2 = µ m1 µ m2 = µ 1 µ 2. Hence #M = 2 µ m1 µ 2 µ 1 µ m2 = 0 an µ 1 µ m2 µ m1 µ 2 = 0. Then eqation (4) implies that f(s) = 0 for all s 0 so that S = S\{0}. emark 3.3 It immeiately follows from Proposition 3.2 that in orer to have S = {s } an hence to have a niqe optimal policy it is necessary to have #D 2 an sfficient to have either #D = 2 or #D = 1 an #M = 2. Moreover note that when either #D = 2 or #D = 1 an #M = 2 then the optimal policy is niqe even thogh the servers m an are not niqely efine in these cases. In particlar when #D = 2 then D = { m} an Theorem 3.1 an Proposition 3.2 imply that servers an m are both at station 2 in all states s { B 2} (since S = {s } = {1} in this case). Similarly when #D = 1 an #M = 2 then M = {m } an Theorem 3.1 an Proposition 3.2 imply that servers m an are both at station 1 in all states s { B 1} (since S = {s } = {B 2} in this case). 4 Two Stations an More than Three Servers In this section we provie a conjectre for the strctre of the optimal server assignment policy for systems with M > 3 servers an two stations. Let L = {1... M}. Withot loss of generality we again assme (as in Section 3) that for all i {1... M} either µ i1 > 0 or µ i2 > 0 (becase if there exists a server i sch that µ i1 = µ i2 = 0 then the problem reces to having M 1 servers). Moreover we again assme that there exist i k {1... M} sch that µ i1 > 0 an µ k2 > 0 (becase if µ 11 = = µ M1 = 0 or µ 12 = = µ M2 = 0 then the maximal throghpt is zero an any policy is optimal). Let { µi1 } l 1 L 1 = arg min (7) L µ i2 7

8 an for 2 j M let l j L j = arg min L\{l 1...l j 1 } { µi1 µ i2 }. (8) Then we conjectre that there exist 1 = s 1 s 2 s 3 s M 1 s M = B 2 sch that the server assignment policy (δ ) given as servers l 1... l M work at station 1 for s = 0 servers l 2... l M work at station 1 server l 1 works at station 2 for 1 s s 2 1 servers l 3... l M work at station 1 servers l 1 l 2 work at station 2 for s 2 s s 3 1 δ (s) = servers l 4... l M work at station 1 servers l 1 l 2 l 3 work at station 2 for s 3 s s 4 1. server l M works at station 1 servers l 1... l M 1 work at station 2 for s M 1 s B 1 servers l 1... l M work at station 2 for s = B 2 (9) is optimal. It is clear from eqations (7) an (8) that it is easy to etermine l 1... l M for any particlar problem. Moreover once l 1... l M are etermine then one can compte s 2... s M 1 by consiering all the possibilities an choosing the one that provies the best throghpt. This procere reqires mch less effort than etermining the optimal policy withot knowing this strctre which is conjectre to be optimal. In particlar we now nee to compte the throghpt of ( B M M 2 1 ) policies rather than consiering all 3 M(B3) Markovian stationary eterministic policies (or all 2 M(B1) non-iling policies). Note that the optimal policy for M = 3 specifie in Section 3 an the optimal policy for M = 2 given by Anraóttir Ayhan an Down [4] agree with the conjectre given above (when M = 2 server l 1 shol be at station 2 an server l 2 at station 1 for all 1 s B 1). Moreover the extensive nmerical examples given in Section 6 emonstrate that the conjectre policy also appears to be optimal for systems with M > eversibility of Two Station Tanem Lines with Flexible Servers Sppose that the original (forwar) line has two stations an M 1 servers an consier the reverse line in which station 2 is followe by station 1 (note that we have not relabele the stations or change the size of the bffer). Sppose that the original an reverse lines operate ner the Markovian stationary eterministic server assignment policies π an π respectively. Let δ an δ be the ecision rles associate with the policies π an π respectively (so that δ an δ specify what servers are assigne to stations 1 an 2 respectively as a fnction of the state s S of the two systems). Let {X π (t)} be the Markov chain moel for the reverse line corresponing to the moel {X π (t)} specifie in Section 2 for the original line (for example X π (t) = s {1... B 1} if at time t the reverse line has jobs to be processe at both stations an s 1 jobs waiting to be processe in the bffer) an let T π be the long-rn average throghpt ner policy π in the reverse line (note that T π an T π may epen on the initial states of the Markov chains {Xπ (t)} 8

9 an {X π (t)} respectively if these Markov chains have more than one recrrent eqivalence class). Throghot this section we will assme that δ (s) = δ(b 2 s) for all s S. We have Proposition 5.1 If B 2 X π (0) belongs to the same recrrent eqivalence class of the Markov chain {X π (t)} as X π (0) then T π = T π. Proof: For all s S let κ π 1 (s) an κπ 2 (s) enote the sets of servers assigne to stations 1 an 2 respectively in state s of the original line ner the policy π. It is clear that the stochastic process {X π (t)} is a birth-eath process with state space S. For all s S let λ π (s) an γ π (s) enote the birth an eath rates in state s respectively. Then λ π (B 2) = γ π (0) = 0 an λ π (s) = γ π (s) = µ i1 for s = 0... B 1 i κ π 1 (s) µ i2 for s = 1... B 2. i κ π 2 (s) Moreover {X π (t)} is also a birth-eath process with state space S. For all s S let λπ (s) an γ π (s) enote the birth an eath rates in state s of the reverse line respectively. Then the assmption that δ (s) = δ(b 2 s) for all s S implies that λ π (B 2) = γ π (0) = 0 an an hence that λ π (s) = γ π (s) = µ i2 for s = 0... B 1 i κ π 2 (B2 s) µ i1 for s = 1... B 2 i κ π 1 (B2 s) λ π (s) = γ π (B 2 s) an γ π (s) = λ π (B 2 s) for all s S. Now sppose that X π (0) E π where E π S is a recrrent eqivalence class of the Markov chain {X π (t)}. Then E π = {s S : B 2 s E π } is a recrrent eqivalence class of the Markov chain {X π (t)} with Xπ (0) Eπ. Let {p π (s) : s E π } be the stationary istribtion for {X π (t)} on E π an let {p π (s) : s E π } be the stationary istribtion for {X π (t)} on Eπ. It is clear that p π (s) = p π (B 2 s) for all s E π. Therefore X π (0) Eπ an X π (0) E π imply that T π = s E π γ π (s)p π (s) = an the proof is complete. s E π λ π (B 2 s)p π (B 2 s) = s E π λ π (s)p π (s) = T π Let Π F Π be the class of all non-iling threshol policies for the forwar system so that for all π Π F an i = 1... M there exists t π i {1... B 2} sch that server i is at the pstream station 1 in all states s < t π i an at the ownstream station 2 in all states s t π i. For all π Π F 9

10 let l1 π... lπ M be sch that {lπ 1... lπ M } = {1... M} an 1 sπ 1 sπ M B 2 where s π i = t π li π for i = 1... M (see eqation (9)). Hence lπ 1 is the first server to move from station 1 to station 2 an lm π is the last server to o so ner the policy π Π F. Let Π Π an t π i lπ i an s π i where π Π an i = 1... M be efine in the same manner for the reverse system. We have Proposition 5.2 (i) π Π F if an only if π Π. (ii) For all i = 1... M an s = 1... B 2 t π i = s if an only if t π i = B 3 s. (iii) For all i = 1... M li π = l π M1 i. (iv) For all i = 1... M an s = 1... B 2 s π i = s if an only if s π M1 i = B 3 s. Proof: We have that t π i = s if an only if δ(s ) has server i at station 1 for all s s 1 an δ(s ) has server i at station 2 for all s s. The efinition of π now implies that t π i = s if an only if δ (B 2 s ) has server i at station 1 for all s s 1 an δ (B 2 s ) has server i at station 2 for all s s. This proves parts (i) an (ii) of the proposition. Moreover since t π i 1 t π i 2 implies that t π i1 = B 3 t π i 1 B 3 t π i 2 = t π i2 it is clear that the orer in which the servers move from the pstream station to the ownstream station is reverse ner π an part (iii) of the proposition follows. Finally from (ii) an (iii) we have s π i = t π li π = s if an only if s π M1 i = t π l π = t π l π = B 3 s for all i = 1... M an s = 1... B 2 proving that part M1 i i (iv) of the proposition hols. We now present several corollaries that follow from Propositions 5.1 an 5.2. ecall that a policy is optimal if it leas to the maximal throghpt regarless of the initial state of the nerlying Markov chain see eqations (2) an (3). Corollary 5.1 The policy π is optimal in the forwar system if an only if the policy π is optimal in the reverse system. Proof: If π is not optimal in the reverse system then there exists a policy π with ecision rle δ sch that T π > T π at least for some initial states Xπ (0) an Xπ (0). Let π (with ecision rle δ ) be the policy in the forwar system sch that δ (s) = δ (B 2 s) for all s S. Proposition 5.1 now implies that if X π (0) an B 2 X π (0) belong to the same eqivalence class of {X π (t)} an also X π (0) an B 2 X π (0) belong to the same eqivalence class of {Xπ (t)} then T π = T π > T π = T π which contraicts or assmption that π is an optimal policy for the forwar system. Corollary 5.2 If the service rates µ ij where i = 1... M an j = 1 2 are rawn inepenently from a certain istribtion then the probability that π is optimal in the forwar system is eqal to the probability that π is optimal in the reverse system. 10

11 Let µ be the M 2 matrix containing the service rates µ ij where i = 1... M an j = 1 2 an let µ be the M 2 matrix containing the two colmns of µ in the reverse orer (corresponing to reversing the orer of the two stations). For all µ an π Π let T π (µ) be the throghpt of the system operate ner the server assignment policy π when the service rates µ ij where i = 1... M an j = 1 2 are given by µ see eqation (1). Moreover for all µ let Π (µ) Π F Π be the set of policies conjectre to be optimal in Section 4 for the forwar system with service rates µ. Assme that we pick a policy π (µ) Π (µ) at ranom. For all µ i = 1... M an s = 1... B 2 let N(µ) be the nmber of policies in Π (µ) let N is (µ) be the nmber of policies in Π (µ) sch that s i = s an let s i (µ) be the vale of s i corresponing to the policy π (µ) see eqation (9) (note that s i = () sπ i ). We have Corollary 5.3 Sppose that the service rates µ ij where i = 1... M an j = 1 2 are rawn inepenently from a certain istribtion an that we choose a policy π (µ) from Π (µ) at ranom. Then IP{s i (µ) = s} = IP{s M1 i (µ) = B 3 s} an IE[s i (µ)] IE[s M1 i (µ)] = B 3 for all i = 1... M an s = 1... B 2. Proof: Parts (i) an (iv) of Proposition 5.2 an the fact that π (µ) is chosen from Π (µ) at ranom imply that IP{s i (µ) = s} = IE[IP{s i (µ) = s µ}] = IE[N is (µ)/n(µ)] = IE[N M1 ib3 s (µ )/N(µ )] = IP{s M1 i(µ ) = B 3 s} = IP{s M1 i(µ) = B 3 s} for all i = 1... M an s = 1... B 2 where the last step follows from the fact that µ an µ are ientically istribte. Moreover we have IE[s i (µ)] = = B2 s=1 B2 s=1 sip{s i (µ) = s} = for all i = 1... M an the proof is complete. B2 s=1 sip{s M1 i(µ) = B 3 s} (B 3 s)ip{s M1 i(µ) = s} = B 3 IE[s M1 i(µ)] We now consier server assignment policies that involve groping several servers into teams where all servers in a team will move together between the two stations in the system. The following corollary clearly follows from Propositions 5.1 an 5.2. Corollary 5.4 If the servers in a set C {1... M} are a team in an optimal policy for the forwar system then they are also a team in an optimal policy for the reverse system. For all µ let l 1 (µ)... l M (µ) be the orere servers when the service rates are given by µ ij for i = 1... M an j = 1 2 see eqations (7) an (8) (if eqations (7) an (8) o not specify l 1 (µ)... l M (µ) niqely then ties can be broken arbitrarily as long as this is one consistently 11

12 in the forwar an reverse systems so that l i (µ) = l M1 i (µ ) for all i = 1... M). K = 1... M let For all N K = {(n 1... n K ) IN K : n 1... n K 1 an n 1 n K = M}. (10) Moreover for all µ K {1... M} an (n 1... n K ) N K let Π (n 1...n K ) (µ) Π be the set of all policies with K teams where each team k {1... K} consists of servers l Σ k 1 j=1 n j1 (µ)... l Σ k j=1 n j (µ) (an hence has n k servers) all servers in each team k {1... K} are at station 1 in all states s < s k (µ) an at station 2 in all states s s k (µ) an the switch points s 1 (µ)... s K (µ) satisfy 1 s 1 (µ) s K (µ) B 2. For example the first team to switch from station 1 to station 2 consists of servers l 1 (µ)... l n1 (µ) an the last team to switch consists of servers l M nk 1(µ)... l M (µ). For all µ an (n 1... n K ) N K let π (n 1...n K ) (µ) Π (n 1...n K ) (µ) be any policy with the team strctre escribe above an with the switch point s (n 1...n K ) k (µ) of each team k {1... K} chosen optimally (if there are mltiple sets of optimal switch points s 1 (µ) s K (µ) then we choose a policy π (n 1...n K ) (µ) arbitrarily from all policies with the prescribe team strctre an optimal switch points). Let π (n 1...n K ) (µ) be the corresponing policy in the reverse system. Then we have the following corollary: Corollary 5.5 Sppose that K {1... M} (n 1... n K ) N K the service rates µ ij where i = 1... M an j = 1 2 are rawn inepenently from a certain istribtion an the Markov chains {X π(n 1...n K ) () (t)} an {X π(n K...n 1 ) () (t)} have only one recrrent eqivalence class. Then IP{π (n 1...n K ) (µ) optimal} = IP{π (n K...n 1 ) (µ) optimal}; IP{T π(n 1...n K ) () (µ) T πn () (µ) n N K } = IP{T π(n K...n 1 ) () (µ) T πn () (µ) n N K }; IE[T π(n 1...n K ) () (µ)] = IE[T π(n K...n 1 ) () (µ)]. Proof: From eqations (7) an (8) an parts (ii) an (iii) of Proposition 5.2 it is clear that π (n 1...n K ) (µ) is eqivalent to π (n K...n 1 ) (µ ) (the only possible ifference being that if there are mltiple optimal sets of switch points then we pick one sch set arbitrarily in each of π (n 1...n K ) (µ) an π (n K...n 1 ) (µ )). Hence Proposition 5.1 an the fact that µ an µ are ientically istribte imply that IP{π (n 1...n K ) (µ) optimal} = IP{π (n 1...n K ) (µ) optimal} = IP{π (n K...n 1 ) (µ ) optimal} = IP{π (n K...n 1 ) (µ) optimal}. The other two conclsions of the corollary can be prove in a similar manner. 6 Nmerical eslts In this section we provie nmerical reslts for systems with two stations an M 3 servers. In Section 6.1 we first investigate whether the policy escribe in Section 4 is optimal for systems 12

13 with M > 3 an then present some interesting featres of this policy. In Section 6.2 we evelop heristic policies for tanem qees with two stations that grop the M 3 available servers into two or three teams an compare the throghpt of these heristics with the optimal throghpt. 6.1 Conjectre Optimal Policy In this section we iscss two sets of nmerical experiments aime at etermining whether the policy conjectre to be optimal in Section 4 is in fact optimal for M > 3 an also at nerstaning the behavior of that policy (recall that the conjectre policy is known to be optimal for M 3 see Theorem 3.1 an Anraóttir Ayhan an Down [4]). In the first set of nmerical experiments we consier systems with two stations M { } servers an B { } bffers between the two stations where the service rate µ ij of each server i {1... M} at each station j {1 2} is rawn inepenently from a niform istribtion with range [0 100]. For systems with M 5 we generate sets of service rates µ ij where i = 1... M an j = 1 2 for each bffer size B. On the other han for M {7 10} an each choice of B we obtain or nmerical reslts from an 1000 ranomly generate systems respectively. In the secon set of nmerical experiments we consier systems with 3 4 or 5 servers where the service rates µ ij for i = 1... M an j = 1 2 take on all combinations of the vales for systems with 3 or 4 servers an all combinations of the vales for systems with 5 servers. The size B of the bffer between stations 1 an 2 again satisfies B { }. For each system with M > 3 consiere in the two sets of nmerical experiments escribe in the previos paragraph we compte the throghpt of the policy that is conjectre to be optimal in Section 4 as well as the throghpt of the optimal policy (which is obtaine by sing the policy iteration algorithm for commnicating Markov chains as escribe in the Appenix). (We se a smaller nmber of systems for M {7 10} an ranomly generate service rates an also for M = 5 an eterministic service rates becase etermining the optimal policy reqires a consierable amont of effort for systems with large nmbers of servers.) The throghpt of the conjectre optimal policy is always eqal to the optimal throghpt which implies that for all of the systems consiere in the two sets of nmerical experiments iscsse in this section the policy escribe in Section 4 is inee an optimal policy. These extensive nmerical reslts emonstrate that the policy escribe in Section 4 appears to be optimal for systems with M > 3 (at least with high probability). We now sty the behavior of the conjectre optimal policy in a more etaile manner. ecall that s i where i {1... M} enotes the state where the ith orere server (i.e. server l i see Section 4) moves from station 1 to station 2 accoring to the conjectre optimal policy (so that 1 s 1 s 2 s M 1 s M = B 2). We o not have simple expressions for compting the switch points s 2... s M 1 even when M = 3 see the efinition of the set S in Section 3. For each choice of M B an either ranom or eterministic service rates let s i be the average s i vale over the total nmber of systems consiere for i = 1... M. In orer to obtain a better 13

14 nerstaning of when servers l 2... l M 1 move from station 1 to station 2 we consier the ratio of the average switch points s 2... s M 1 to the total nmber of states (B 3) in the nmerical experiments escribe above. In the interest of space we isplay these ratios only for systems with for or five servers an ranomly generate service rates in Table 1. Bffer M = 4 M = 5 Size s 2 B3 s 3 B3 s 2 B3 s 3 B3 s 4 B Table 1: atio of the average switch points to the nmber of states for ranomly generate service rates. an The nmerical reslts given in Table 1 are consistent with Corollary 5.3 in that s i B 3 s M1 i B 3 s (M1)/2 1 for all i = 1... M 0.5 when M is o. B 3 Moreover for i M/2 the ratios s i /(B 3) are ecreasing an for i (M 2)/2 the ratios s i /(B 3) are increasing. Similar reslts were obtaine for M {7 10} an ranomly generate service rates an for M {3 4 5} an eterministic service rates. More specifically if we consier M = 7 with B = 20 an ranomly generate service rates then we obtain s 2 B s 3 B s 4 B s 5 B s 6 B Similarly when M = 10 B = 20 an the service rates are ranomly generate then we have s 2 B s 3 B s 4 B s 5 B s 6 B s 7 B s 8 B s 9 B Since the ratio s i /(B 3) is very close to 0 or 1 for several servers i {2... M 1} these nmerical reslts sggest that policies that grop several servers into teams that move together 14

15 between stations 1 an 2 may yiel goo performance at least for systems with two stations large nmber of servers M an large bffer B between the two stations. We next investigate how many teams the conjectre optimal policy has in the nmerical experiments escribe above (for example there are two teams if there exists i {2... M} sch that s i = s 1 = 1 for all i < i an s i = s M = B 2 for all i i an there are M teams if 1 = s 1 < s 2 < < s M 1 < s M = B 2). Note that the total nmber of teams cannot excee B 2 the total nmber of possible switch points. For each choice of M B an either ranom or eterministic service rates let r i enote the ratio of the nmber of systems where the conjectre optimal policy has i teams to the total nmber of systems generate for i = 2... M (since 1 = s 1 < s M = B 2 the policy conjectre to be optimal in Section 4 cannot have fewer than two teams). Hence r i estimates the probability of having i teams in the conjectre optimal policy. Table 2 shows the average nmber of teams M i=2 ir i as a fnction of the nmber of servers M an bffer size B for the two sets of nmerical experiments escribe previosly an Tables 3 an 4 isplay the vales of r 2 r 3 an r M for varios nmbers of servers M an bffer sizes B for the first an secon sets of nmerical experiments respectively. Bffer anom Service ates Deterministic Service ates Size M = 3 M = 4 M = 5 M = 7 M = 10 M = 3 M = 4 M = Table 2: Average nmber of teams. As expecte Table 2 shows that the average nmber of teams increases both with the nmber of servers M an with the bffer size B. However the growth rate is rather slow so that the average nmber of teams is significantly smaller than the maximm possible nmber of teams (i.e. min{m B 2}) for large M an B. Moreover Table 2 shows that for given vales of M an B the average nmber of teams in the ranom an eterministic cases are qite similar with the averages being slightly larger when the service rates are generate at ranom rather than eterministically (this may be e to the fact that we se a larger range of possible vales when the service rates are generate at ranom rather than eterministically leaing to larger ifferences between the 15

16 Bffer M = 3 M = 4 M = 5 M = 7 M = 10 Size r 2 r 3 r M r 2 r 3 r M r 2 r 3 r M r 2 r 3 r M r 2 r 3 r M Table 3: Team probabilities for ranomly generate service rates. capabilities of the ifferent servers at the two stations in the system). Similarly Tables 3 an 4 show that the probability of having two teams ecreases as the bffer size B increases for all M 3. Moreover the probability of having three teams ecreases with the bffer size for all M 5 (looking only at B 1 so that it is possible to have three teams). When M = 3 r 3 increases as the bffer size increases which is reasonable since in this case r 3 = r M ; when M = 4 r 3 first increases an then ecreases with B. Finally r M increases with the bffer size in all cases. Note however that for fixe B Tables 3 an 4 show that r M ecreases as M increases (in fact when M = 10 in Table 3 then r M = 0 for all B { }). Together with Table 2 this sggests that the conjectre optimal policy is likely to have some servers grope into teams at least for large nmbers of servers M. 6.2 Heristic Server Assignment Policies with Two or Three Teams The nmerical reslts given in Section 6.1 sggest that the optimal server assignment policy for systems with two stations in tanem an M servers has the strctre escribe in Section 4. However this policy may be ifficlt to implement in practice when M is large. In this section we consier policies in which the servers are grope into two or three teams an then the teams are assigne to stations in the manner fon to be optimal for systems with two or three servers see Anraóttir Ayhan an Down [4] an Section 3. Or goal is to evelop server assignment heristics that are easily implementable an also robst with respect to the server capabilities in that their average throghpt as the service rates vary is near-optimal. We first orer the servers as is one in Section 4 see eqations (7) an (8). Then we consier all ways of groping the orere servers into two or three teams. More specifically for all (n M n) N 2 (see eqation (10)) we consier sing the server assignment policy of Anraóttir Ayhan an Down [4] with servers l 1... l n in one team an servers l n1... l M in the other team. Similarly for all (n 1 n 2 M n 1 n 2 ) N 3 we consier sing the server assignment policy fon to be 16

17 Bffer M = 3 M = 4 M = 5 Size r 2 r 3 r M r 2 r 3 r M r 2 r 3 r M Table 4: Team probabilities for eterministic service rates. optimal in Section 3 with servers l 1... l n1 in the first team servers l n l n1 n 2 in the secon team an servers l n1 n l M in the thir team. Note that in all of these server assignment heristics the orer of the servers (an hence the composition of the teams) epens on the service rates bt the size of the teams oes not. In orer to evalate an compare the performance of these policies we perform three sets of nmerical experiments in which the service rates are rawn inepenently from niform istribtions with ranges [40 60] [20 80] an [0 100] respectively. Note that these three niform istribtions all have a common mean bt ifferent variances. Hence these istribtions are chosen to moel sitations where the capabilities of the servers at the two stations ten to be qite similar qite ifferent an very ifferent respectively. In all cases we consier systems with M = servers an B = bffers. For each M 6 an each choice of B we generate sets of service rates µ ij where i = 1... M an j = 1 2. On the other han for M = an 10 we generate an 500 sets of service rates respectively for each vale of B (as in Section 6.1 we generate fewer sets of service rates for systems with large nmbers of servers M becase of the excessive amont of comptational effort reqire for etermining the optimal policy for systems with many servers). In the two team setting the nmerical experiments escribe in the previos paragraph sggest that the two team heristic with the nmber of servers in the two teams iffering at most by one performs the best on average in that it yiels the highest average throghpt an also leas to the highest throghpt among all the two team policies that we consier more often than any other two team policy. Ths if M is even then or two team heristic assigns servers l 1... l M/2 to station 2 nless station 2 is starve an servers l (M2)/2... l M to station 1 nless station 1 is blocke. All servers work at station 2 (station 1) when station 1 (station 2) is blocke (starve). On the other han if M is o then either servers l 1... l (M 1)/2 form the ownstream team an 17

18 servers l (M1)/2... l M form the pstream team or servers l 1... l (M1)/2 form the ownstream team an servers l (M3)/2... l M form the pstream team (note that Corollary 5.5 shows that on the average these two server assignment heristics behave in the same manner). In the three team setting the best average performance is obtaine by forming the teams in sch a way that the size of the moving team is smaller than the sizes of the pstream an ownstream teams an the sizes of the pstream an ownstream teams iffer at most by one (as in the two team setting this approach to forming the teams maximizes both the average performance an also the probability of achieving the best performance among all three team policies ner consieration). When M is o this translates into having servers l 1... l (M 1)/2 as the ownstream team server l (M1)/2 as the moving team an servers l (M3)/2... l M as the pstream team. However when M is even then there are two cases: If M {4 6} then either servers l 1... l M/2 form the ownstream team server l (M2)/2 forms the moving team an servers l (M4)/2... l M form the pstream team or servers l 1... l (M 2)/2 form the ownstream team server l M/2 forms the moving team an servers l (M2)/2... l M form the pstream team (note that the average performance of these two teams is the same by Corollary 5.5). On the other han if M { } then servers l 1... l (M 2)/2 form the ownstream team servers l M/2 an l (M2)/2 form the moving team an servers l (M4)/2... l M form the pstream team. We now compare or two an three team heristics with other server assignment policies incling the optimal policy (etermine by sing the policy iteration algorithm for commnicating Markov chains see the Appenix) the best two team policy the best three team policy an a benchmark policy namely the teamwork policy of Van Oyen Gel an Hopp [19] (where all servers work in a single team that will follow each job from the first to the last station an only starts work on a new job once all work on the previos job has been complete). For each ranomly generate choice of µ (see Section 5) the best two team policy is the one that yiels the highest throghpt among all two team policies sch that servers l 1 (µ)... l n (µ) are primarily assigne to station 2 an servers l n1 (µ)... l M (µ) are primarily assigne to station 1 where n {1... M 1}. Similarly for each choice of µ the best three team policy is the one that yiels the highest throghpt among all three team policies sch that servers l 1 (µ)... l k1 (µ) form the ownstream team servers l k1 1(µ)... l k2 (µ) form the moving team an servers l k2 1(µ)... l M (µ) form the pstream team where k 1 {1... M 1} an k 2 {k 1... M 1}. (In other wors in the best two an three team policies the team sizes are allowe to epen on µ.) Althogh we perform nine sets of nmerical experiments (one for each combination of three choices of istribtion for the service rates an three bffer sizes) in the interest of space we only show reslts from for sets of nmerical experiments here. For all a b I with a b let U[a b] enote the niform istribtion with range [a b]. Tables 5 throgh 8 show 95% confience intervals for the average throghpt vales of the policies escribe in the previos three paragraphs when the service rates are rawn from either the U[40 60] or U[0 100] istribtion an the bffer size B satisfies B {5 20}. Since we have two alternative ways of forming the teams for the two team 18

19 heristic when M {3 5 7} we arbitrarily choose the one that has servers l 1... l (M 1)/2 in the ownstream team an servers l (M1)/2... l M in the pstream team. Similarly for the three team heristic an M {4 6} we assign servers l 1... l M/2 to the ownstream team server l (M2)/2 to the moving team an servers l (M4)/2... l M to the pstream team. Nmber of Teamwork Two Team Best Two Three Team Best Three Optimal Servers Policy Heristic Team Policy Heristic Team Policy Policy ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.64 Table 5: Throghpt vales for systems with U[40 60]-istribte service rates an B = 5. Nmber of Teamwork Two Team Best Two Three Team Best Three Optimal Servers Policy Heristic Team Policy Heristic Team Policy Policy ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.64 Table 6: Throghpt vales for systems with U[40 60]-istribte service rates an B = 20. As expecte Tables 5 throgh 8 show that the throghpts achieve by all the policies consiere in this section appear to increase as the nmber of servers M increases. Similarly the throghpts of all the policies except for the teamwork policy appear to increase with both the nmber of bffers B an the variability in the service rates µ ij where i = 1... M an j = 1 2. On the other han the throghpt of the teamwork policy is by efinition insensitive to B an it appears to ecrease slightly as the variability in the service rates increases. The fact that the 19

20 Nmber of Teamwork Two Team Best Two Three Team Best Three Optimal Servers Policy Heristic Team Policy Heristic Team Policy Policy ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 3.46 Table 7: Throghpt vales for systems with U[0 100]-istribte service rates an B = 5. throghpts of all the policies except for the teamwork policy increase with the variability in the service rates is reasonable becase only the teamwork policy is nable to take avantage of this variability by assigning servers primarily to tasks that they are goo at (i.e. have a high service rate at). Tables 5 throgh 8 also show that the average behavior of or two team heristic is in all cases very close to that of the best two team policy an also that the average behavior of or three team heristic is always very similar to the average performance of the best three team policy. Both the two an three team heristics perform significantly better than the teamwork policy especially when the service rates are highly variable with the three team heristic showing slightly better average performance than the two team heristic. Finally the average performance of the three team heristic is always very close to that of the optimal policy (it is eqal to the average performance of the optimal policy when M = 3 as preicte by Theorem 3.1). These observations sggest that both of or server assignment heristics are likely to yiel very goo performance in practice an that the behavior of the three team heristic is sally near-optimal. Moreover althogh or heristics are esigne to be both easily implementable an also robst with respect to the service rates (in that the sizes of the teams o not epen on the service rates) or nmerical reslts inicate that there is very little room for obtaining improve average performance throgh the se of more complex policies or policies that epen more heavily on the service rates. 7 Conclsion For Markovian qeeing systems with two stations in tanem finite intermeiate bffer an three flexible an collaborative servers we have completely specifie how servers shol be assigne to stations in orer to achieve maximal long-rn average throghpt. Moreover we have provie 20