Capacity Pooling and Cost Sharing among Independent Firms in the Presence of Congestion

Transcription

1 Capaity Pooling and Cost Sharing among Independent Firms in the Presene of Congestion Yimin Yu Saif Benjaafar Graduate Program in Industrial and Systems Engineering Department of Mehanial Engineering University of Minnesota Minneapolis, MN Yigal Gerhak Department of Industrial Engineering Tel Aviv University, Tel Aviv, Israel February 27, 2008 Abstrat We analyze the benefit of prodution/servie apaity pooling for a set of independent firms. Firms have the hoie of either operating their own prodution/servie failities or investing in a faility that is shared. Failities are modeled as queueing systems with finite servie rates. Firms deide on apaity levels (the servie rate) to minimize delay osts and apaity investment osts subjet to servie level onstraints. If firms deide to operate a shared faility they must also deide on a sheme for sharing the osts. We formulate the problem as a ooperative game and identify a ost alloation that is in the ore. The alloation rule harges every firm the ost of apaity for whih it is diretly responsible, its own delay ost, and a fration of buffer apaity ost that is onsistent with its ontribution to this ost. In settings where we relax some assumptions in our original problem, we show that apaity pooling may not always be benefiial. We also examine the impat of having one of the firms, instead of a neutral entral deision maker, be responsible for making the apaity investment deision for the pooled system. Key words: Capaity pooling, queueing systems, joint ventures, ost alloation, ooperative game theory

2 1 Introdution Capaity pooling refers to the onsolidation of apaity that resides in multiple failities into a single one. In a system without pooling, eah faility fulfills its own demand relying solely on its apaity. In a pooled system, demand is aggregated and fulfilled from a single shared faility. It has long been known that pooling is benefiial when demand is random. This benefit an be in the form of improved servie quality with the same amount of apaity or in the form of less apaity needed to provide the same quality of servie. Capaity pooling is also benefiial when there are eonomies of sale assoiated with aquiring apaity or fulfilling demand. These benefits have been shown to be true for various forms of apaity, inluding manufaturing apaity, servie apaity, and inventory. Capaity pooling has been studied mostly in situations where a single firm owns all the apaity in the system and has responsibility for serving all the demand. This firm makes the deision about whether or not to pool and how muh apaity to aquire. In this paper, we onsider a system with n independent firms, eah faing its own demand and eah having the option of either operating its own independent faility or joining some or all the other firms in a shared faility. The firms may vary in their demand levels and in their tolerane for apaity shortage. If some or all of the firms deide to pool, they must also deide on how to share the ost of the pooled faility. They must do so in a fair manner that prevents any of the firms from defeting and perhaps joining a pooled faility with other firms or staying on their own. Hene, firms that ontribute more to the ost of the pooled faility (beause of their higher usage of apaity or lower tolerane for apaity shortage) are expeted to pay a greater share of total ost. Capaity pooling among independent firms is inreasingly ommon in the manufaturing, servie, and publi setors. In manufaturing, there are numerous instanes of independent firms sharing the same prodution failities. For example, Isuzu and General Motors share the same plants to produe diesel engines; Fujitsu and AMD jointly produe flash memory; Chrysler and Chery, a Chinese ar manufaturer, plan to share assembly plants to build subompat ars; Aer, IBM, and Compaq shared the same plants in the 1990 s to assemble their own brands of omputers. In the servie setor, the sharing of servie failities is also ommon. For example, airlines share hek-in ounters, maintenane failities, and reservation systems; hospitals share expensive diagnosti equipment, laboratory failities, and in some ases medial speialists and surgeons. In the publi setor, the sharing of resoures between independent government entities is also widespread. For instane, loal governments in rural ommunities share fire and polie departments, 911 all enters, and other soial servies. In various setors, firms are inreasingly sharing infrastruture re- 1

3 soures suh as teleommuniation networks, omputer servies, in-house all enters, and failities for handling bak-offie operations. Capaity pooling among independent firms raises several important questions. For example, is pooling always benefiial to all firms? Does it always lead to a redution in total apaity in the system? How should apaity osts be shared among the different firms? Is pooling among all the firms the best arrangement or would pooling among smaller subsets of the firms be more benefiial to partiular firms? How should priorities for apaity usage be set among the firms and how do these priorities affet ost sharing? How are the answers to the above questions affeted by whether an impartial deision maker hooses apaity levels or one of the firms selfishly makes this deision? In this paper, we address these and other related questions. Our treatment is inspired by the above examples of apaity sharing in pratie. However, the setting we study is speifi: we onsider appliations where failities an be modeled as queueing systems. Demand for eah firm onsists of an independent stream of ustomers (or orders) that arrive ontinuously over time with random inter-arrival times. Customers are proessed at eah faility one at a time with stohasti servie time. The apaity at eah faility is determined by the rate at whih ustomers an be proessed. Beause ustomers are proessed one unit at a time and beause ustomer arrivals and proessing times are random, ongestion arises and ustomers an experiene delay prior to proessing (if a ustomer arrives and finds the servie faility busy, the ustomer must wait for servie). Eah firm an install and operate its own faility where its ustomers are proessed. We refer to this senario as the distributed system. Firms make deisions about how muh servie apaity to aquire in order to minimize two types of osts, delay ost due to ustomers spending time at the faility prior to ompleting servie and apaity investment ost, subjet to a onstraint on the amount of delay ustomers experiene. If the firms hoose to olletively operate a shared faility, then they must selet the apaity of the faility and deide on a sheme to share the orresponding ost. We refer to this senario as the pooled system. Our hoie of modeling framework is in part motivated by the fat that types of manufaturing and servie failities an be viewed as queueing systems. Suh a view is indeed prevalent in the literature. Despite this prevalene, the literature on apaity sharing in queueing systems is limited and few treatments exist on how the assoiated ooperation among independent firms an be modeled. Although we restrit our analysis in this paper to simple models of queueing systems (for the most part, single server systems where apaity orresponds to a servie rate), the models are suffiiently rih to apture many of the important issues that arise when apaity is shared among independent firms. Moreover, the models we onsider have been widely studied in settings with a 2

4 single firm. Therefore, it is perhaps appropriate to enrih that literature with models with multiple firms. The ontributions of this paper are as follows. We provide a framework for modeling apaity pooling in queueing systems with independent firms. To our knowledge, our paper is among the first to model the issue of ooperation and apaity sharing in a queueing ontext. We formulate apaity pooling as a ooperative game and show that for systems where failities are modeled as M/M/1 queues (queues with Poisson arrivals and exponential servie times) the ore of the game is non-empty. That is, there exists a ost alloation rule for whih all the firms are better off than under any other alternative pooling arrangement. We identify a simple and easy to implement alloation rule with desirable properties that is in the ore. The alloation rule harges every firm the ost of apaity for whih it is diretly responsible, its own delay ost, and a fration of buffer apaity ost that is onsistent with its ontribution to this ost. This alloation rule is perhaps onsistent with those observed in pratie. We extend our treatment beyond the M/M/1 queue framework and identify settings where pooling is not optimal. We also explore the impat of assigning priorities for apaity usage among firms. We use these results to draw various managerial insights. We study the effet of who makes the deision regarding apaity (a neutral deision maker or one of the firms) on the desirability of pooling and the role of alloation rules. More generally, we view our paper as ontributing toward a more omprehensive examination of the issue of ooperation in queueing systems, whether it arises in servies, manufaturing, or elsewhere. We also view it as a ontribution, in the form of a potentially rih appliation domain, to the literature on ooperative games. The rest of the paper is organized as follows. In Setion 2, we provide a brief review of related literature. In Setions 3 and 4, we analyze the distributed and pooled systems respetively in settings where servie failities are modeled as single server queues with Poisson demand and exponentially distributed servie times. In Setion 5, we disuss ost alloation among the firms in a pooled system and desribe an alloation rule that is in the ore. In Setion 6, we relax some of the assumptions in the original model and show that this an affet the benefit of pooling. In Setion 7, we desribe settings where firms are assigned priorities based on their delay osts. In 3

5 Setion 8, we onsider the ase where one of the firms makes the apaity deision in the pooled system instead of an impartial deision maker. In Setion 9, we provide a summary of the main results and offer onluding omments. 2 Related Literature There is a rih literature on apaity pooling in queueing systems, with appliations ranging from manufaturing and servie operations to teleommuniations systems to omputer networks. This literature an be lassified broadly as relating to either the pooling of servie rates or the pooling of servers. Server rate pooling refers to the onsolidation of multiple servers into a single one with a faster rate (e.g., N servers, eah with servie rate µ and demand rate λ, are replaed by a single server with servie rate Nµ and demand rate Nλ). Server pooling on the other hand refers to plaing multiple servers in a single faility from whih all demand streams are served (e.g., N single server queues are replaed by a single multi-server queue with N servers and a demand rate Nλ). Kleinrok (1976) disusses various examples of both types of pooling. Stidham (1970) onsiders a design problem where the deision variables are the number of parallel servers and the servie rate of eah server. Smith and Whitt (1981) and Benjaafar (1995) show that server pooling, when the number of servers is exogenously determined, is benefiial as long as all ustomers have idential servie time distributions. Buzaott (1996) onsiders the pooling of N servers in series, with eah server dediated to one task, into N parallel servers, with eah server arrying out all the tasks. Mandelbaum and Reiman (1998) onsider the pooling of general Jakson networks into single server queues with phase-type servie time distributions. Several reent papers onsider partial forms of pooling, where eah server is aessible only to a subset of the demand streams. Examples inlude Gurumurthi and Benjaafar (2004), Jordan et al. (2005) and Jouini et al. (2008); see also Aksin et al. (2005) and Koole and Pot (2005) for reent reviews. The treatment in this paper is different from the above literature in three important aspets. First, we do not assume that there is a single deision maker that determines whether or not to pool. Instead, we onsider multiple firms that deide independently on either operating their own failities or sharing one with other firms (pooling here does not imply a merger however). Seond, we do not assume that servie apaity, either in terms of servie rate or number of servers, is exogenously given. We allow for this to be an outome of an optimization arried out by the firms either individually or jointly. Third, we are onerned with identifying ost alloation shemes under whih all firms prefer the pooled system to any other pooling arrangement, inluding remaining on their own. 4

6 The literature dealing with pooling in the ontext of independent firms is limited. Herrero (2004), and also Garia-Diaz et al. (2007), onsider a speial ase of the M/M/1 model we onsider in Setions 3 and 4. However in both ases, they do not optimize apaity (before or after pooling), and do not onsider delay osts. In our ase, the presene of delay osts signifiantly ompliates the proess of ost alloation sine we seek alloations that ould allow for eah firm to absorb its own ost of delay. We also onsider systems other than the M/M/1 queue, inluding systems with general servie times, systems with multiple servers, and systems where a lead firm makes the apaity investment deision. Outside the queueing literature, there is growing researh in Operations Management that deals with the pooling of other forms of apaity, partiularly inventory, among independent firms. Reent reviews an be found in Nagarajan and Sošić (2007), Kemahlioglu-Ziya (2004), and Chen and Zhang (2007). Our work is of ourse related to the vast literature on ooperative game theory and, more broadly, the eonomis of oalition formation and joint ventures; see Moulin (1995) for a general introdution to the topi. Some of this literature has foused on ooperation involving sequening and sheduling; see for example Moulin and Stong (2002), Maniquet (2003), and Katta and Sethuraman (2006). This literature sometimes refers to these problems as queueing problems. However, they typially involve a finite population of ustomers who simultaneously arrive to the system, and therefore are not onerned with steady state behavior and ongestion in the way that we are in this paper. 3 The Distributed System We onsider a system onsisting of a set N = {1,..., n} of n independent firms. Eah firm makes an independent deision about apaity in the form of a servie rate. Firm i, i N, faes an independent demand stream with ustomers arriving aording to a Poisson proess with rate λ i. Eah firm serves its ustomers one at a time on a first-ome, first-served (FCFS) basis. We assume servie times are independent and identially distributed random variables denoted by X i where X i is of the form Y/µ i and Y is a random variable that is exponentially distributed with a mean equal to 1. Hene, servie time is also exponentially distributed with mean E[X i ] = 1/µ i. The parameter µ i, (µ i > 0) is a saling parameter that orresponds to the servie rate or apaity. The random variable Y an be viewed as the work ontent assoiated with eah ustomer. We assume that work ontent is homogeneous aross firms. We relax this assumption in Setion 6.1. Given the exponential nature of both ustomer inter-arrival times and servie times, eah firm behaves like an M/M/1 queue. There is signifiant literature on the eonomis of queues in ompetitive settings 5

7 that primarily fouses on the M/M/1 queue (and where the servie rate is the deision variable); see Hassin and Haviv (2003) for a review of that literature. Our treatment of the M/M/1 queue an be viewed as a omplement to that literature in ooperative settings. We assume that servie rate an be varied ontinuously and that firms inur a apaity ost per unit of servie rate per unit time. This is justified in settings where apaity an be ontinuously saled over a suffiiently large interval (e.g., the speed of omputing failities, the bandwidth of ommuniation networks, or the throughput of prodution lines). It is also onsistent with treatments elsewhere in the literature (see for example Kalai et al. (1992), Mendelson and Whang (1990), Allon and Federgruen (2007, 2006), Cahon and Zhang (2007), and the vast literature reviewed therein). In Setion 6.2, we disuss the ase where the servie rate is fixed but the number of parallel servers an be varied. In that ase, apaity is disrete. The assumption of linear apaity ost implies that there are neither eonomies nor diseonomies of sale. This is an important ase that has been widely studied in the literature (see Allon and Federgruen (2006, 2007), Dewan and Mendelson (1990), and Stidham (1992), among others), leads to tratable analysis, and provides a useful benhmark for other ost strutures. In Setion 6.3, we disuss the ase where apaity ost is not linear. The objetive of eah firm is to minimize its apaity investment while limiting the amount of delay its ustomers experiene. Limiting ustomer delay an be ahieved by enforing a servie level onstraint or by assoiating a ost with the amount of delay ustomers experiene. A servie level onstraint may take several forms, inluding a onstraint on the probability of ustomer delay not exeeding a speified threshold, or a onstraint on expeted delay not exeeding a ertain maximum amount. Servie level onstraints are managerial deisions that typially reflet either a position in the marketplae that a firm would like to take or ontratual obligations that a firm has negotiated with its ustomers. Delay osts an reflet either diret or indiret osts. Diret osts are penalties inurred by the firm due to delays experiened by its ustomers (for example, payments to ustomers to ompensate for the total time they spend in the system) or indiret osts due to loss of ustomer goodwill. Hene, delay osts are not unlike bakorder osts, ommon in inventory settings (Zipkin 2000). Delay osts may also reflet the ost of work-in-proess aumulation when there is a physial produt released to the queue with the arrival of eah ustomer, as in many manufaturing appliations. The use of delay osts and servie levels are both ommon in the literature; see for example Dewan and Mendelson (1990, Mendelson and Whang (1990), Allon and Federgruen (2006, 2007) and the referenes therein. 6

8 In this paper, we limit our analysis to the ase where servie level is expressed in terms of a probability that total delay in the system (time in the queue + time in servie) for eah ustomer in steady state does not exeed a speified threshold. We also limit ourselves to the ase where a delay ost h i is inurred for eah unit of time a ustomer spends in the system (time either in the queue or in servie in steady state). Let z i (µ i ) denote the expeted total ost inurred by firm i given a servie rate µ i (for stability, we assume that λ i /µ i < 1). Let W i, a random variable, denote the total time a ustomer of firm i spends in the system (ustomer delay) and P (W i ) the probability that ustomer delay does not exeed where 0. The problem faed by firm i an then be stated as follows Minimize z i (µ i ) = µ i + h iλ i µ i λ i (1) subjet to and P (W i ) = 1 e (µ i λ i ) α i, (2) λ i /µ i 1. (3) The objetive funtion in the above optimization problem onsists of two terms: a apaity ost term and a delay ost term, where the deision variable is the apaity level of firm i as determined by the servie rate µ i. The formulation aptures two important speial ases: (1) the ase where α i = 0 for all i N and (2) the ase where h i = 0 for all i N. The first orresponds to a pure ost-based formulation with no onstraints on servie levels, while the seond orresponds to a servie level-based formulation with no delay osts. In the absene of servie level onstraints, the optimal apaity level µ i an be obtained from the first order ondition of optimality, sine z i is onvex in µ i, as µ i = λ i + hi λ i. (4) In systems with servie level onstraints but no delay osts, the optimal apaity level is given by the smallest µ i that satisfies inequality (2). This leads to the following optimal apaity level µ i = λ i + ln( 1 1 α i ). (5) In both ases, the optimal apaity is the sum of two omponents. The first orresponds to the demand rate, λ i (sine all demand must be satisfied) while the seond orresponds to buffer apaity 7

9 that inreases in either the ratio h iλ i or the servie level α i. The expressions in equations (4) and (5) are not new. Similar expressions have been derived elsewhere; see for example Kleinrok (1976), Allon and Federegruen (2006) and Hassin and Haviv (2003). In the general ase, with both delay osts and servie level onstraints, the optimal apaity level is given by µ i = λ i + η i, (6) where Substituting µ i η i = max{ ln( 1 1 α i ) hi λ i, }. (7) in (1), we obtain the optimal expeted ost for firm i as z i = (λ i + η i ) + h iλ i η i. (8) This leads to a total system ost of z 1,...,n = i N z i. In systems where h i λ i i N, the optimal ost simplifies to ln( 1 ) 1 α i for all z i = λ i + 2 h i λ i. (9) This leads to a total system ost, z1,...,n, given by z 1,...,n = i N λ i + 2 i N hi λ i. (10) In the ase of idential firms, with λ i = λ and h i = h for all i N, the optimal total ost in (10) redues to and the total apaity in the system to z 1,...,n = nλ + 2n hλ, (11) ( µ i = n λ + i N ) hλ. (12) As we an see, both the optimal ost and the optimal buffer apaity in the system inrease linearly in the number of firms n. Similar observations an be made for systems in whih h i λ i ln( 1 ) 1 α i. That is, in this ase too, both the optimal ost and the optimal buffer apaity in the system inrease linearly in n when the firms have idential ost, servie level, and demand parameters. 8

10 4 The Pooled System In the pooled system, the n firms in the set N share a single faility with servie rate µ N whih the demand of all firms is satisfied (from heretofore, we shall index parameters assoiated with a set of firms with the name of that set while parameters assoiated with individual firms with the name of the firm). Beause the superposition of independent Poisson proesses is also a Poisson proess, the demand proess at the shared faility is Poisson with rate i N λ i. Similarly, beause the work ontent for eah ustomer regardless of its firm is exponentially distributed, the proessing time at the pooled faility is a random variable X N = Y/µ N with the exponential distribution and mean 1/µ N. We assume that ustomers regardless of their firm affiliation are served in a FCFS fashion (we onsider alternative priority shemes in Setion 7). Hene, the pooled system behaves again as an M/M/1 queue. We assume that a entral deision making entity determines the amount of apaity to aquire so as to minimize the sum of expeted delay ost experiened by ustomers of all firms and the ost of apaity subjet to the servie level onstraints assoiated with eah firm. This implies that firms share with this entral deision making entity information about delay osts, servie level requirements, and the distribution of demand and work ontent. This is a reasonable assumption in settings where the firms are all subsidiaries of a single large firm. It is also plausible that even independent firms would share information about demand and proessing times as they would eventually be observed by the pooled faility. Obviously firms would want to share information about servie levels sine the pooled faility must guarantee that they are satisfied. In settings where delay osts are diretly inurred by the pooled faility (e.g., the pooled faility is responsible for paying delay penalties diretly to the ustomers), firms would also need to provide the pooled faility with the orret delay osts. However, in settings where delay osts are unobservable by the pooled faility (e.g., delay penalties are handled independently by eah firm or the penalties orrespond to a private assessment of loss of ustomer good will), firms may have an inentive not to dislose the orret osts. In suh settings, a formulation that inludes servie level requirements but not delay osts would be more appropriate. Let z N (µ N ) denote total system ost and let W N, a random variable, refer to ustomer delay. Then the problem faed by the entral deision maker for the pooled system an be stated as follows: Minimize z N (µ N ) = µ N + i N h iλ i µ N i N λ i from (13) 9

11 subjet to and P (W N ) = 1 e (µ N P i N λ i) α N, (14) λ i /µ N 1, (15) i N where α N = max(α 1,..., α n ). Then, the optimal apaity is given by µ N = i N λ i + η N, (16) where η N = max{ ln( 1 1 α N ) i N, h iλ i } (17) Similar to the distributed system, the optimal apaity in the pooled system onsists of two omponents. The first orresponds to the total demand rate, while the seond to buffer apaity whih, in this ase, inreases in either the sum of the ratios h iλ i or the maximum servie level α N. Theorem 4.1 z N z 1,...,n and µ N n i=1 µ i. The proof of Theorem 4.1 and all subsequent results an be found in the Appendix. Pooling leads to lower total ost for the system and to lower investments in apaity. The potential magnitude of the savings an be more easily seen in a system with idential firms where α i = α, h i = h, and λ i = λ for all i N. First onsider the ase where. This leads to µ N = nλ+ nhλ, z N = nλ + 2 nhλ, and E(W N ) = nhλ nhλ ln( 1 1 α ) from whih we an observe that both buffer apaity and expeted delay, and onsequently delay ost, are redued by a fator of a square root of n (relative to those observed in the distributed system). For the ase where nhλ ln( 1 1 α ), we have µ N = nλ + ln( 1 1 α ), zn = (nλ + ln( 1 1 α ) ) + nhλ 1 ln( ), and E(W N ) = 1 ln( ). Here, the magnitude 1 α 1 α of savings on apaity is even larger with buffer apaity redued by a fator of n, but expeted delay remains unhanged from the distributed system. Although apaity investments vary in the two ases, it is important to observe that, in both ases, expeted delay in the pooled system is no worse than the expeted delay experiened in the distributed system. We onlude this setion by noting that, while it is onsistent with intuition that pooling would be benefiial when delay osts and servie levels of all the firms are idential, it is not at all obvious that it would be so when different firms have different delay osts or different servie levels (note that the pooled system imposes the highest of the individual firms servie levels). In fat, in other settings, suh as those where apaity is in the form of inventory, pooling an be detrimental when 10

12 delay-related osts, suh as bakorder osts, are suffiiently different and no appropriate priorities are instituted (see for example de Veriourt et al. (2002), Benjaafar et al. (2005) and Ben-Zvi and Gerhak (2005)). In these settings, pooling a firm with a low demand rate but a high delay ost with a firm with a high demand rate but low delay ost an lead to overall higher osts. In Setion 6.2, we show that this an also arise in our setting when apaity is in the form of a disrete number of servers instead of a ontinuous servie rate. Finally, let us note that our results show that pooled apaity is always lower than the total apaity in the distributed system. This is not the ase with other forms of apaity. For example, it has been observed that inventory pooling, while always benefiial, an inrease apaity (Gerhak and Mossman 1992). 5 Cost Sharing In the previous setion, we showed that pooling is system-optimal. However, whether or not it is also optimal for individual firms depends on how the ost of the shared faility is alloated among the firms. A firm benefits from pooling if its share of the ost is lower than the ost it would inur from operating its own independent faility. If both apaity osts and delay osts (if any) are absorbed by the pooled faility, eah firm is harged a usage fee (per unit time) that orresponds to a fration of total ost. If, on the other hand, only apaity ost is inurred by the pooled faility with delay osts inurred diretly by the individual firms, the usage fee would orrespond only to a fration of apaity ost. The net effet in both ases, is that eah firm would inur a fration of total system ost. Moreover, in many settings, the hoie is not just between a single faility shared among all firms or failities operated individually by eah firm. There may instead be a range of faility sharing options. For example, a firm may find it more advantageous to pool apaity with only a subset of the firms. This ould lead firms to form groupings around multiple smaller shared failities. A single pooled faility would be preferred by all firms only if there exists a vetor of usage fees under whih the firms are better off than under any other apaity sharing arrangement, inluding operating individual failities. Hene, ideally, the ost alloation in the pooled system would be designed so that it deters firms from breaking away and engaging in other faility sharing arrangements. The problem of determining whether or not there exists a ost alloation sheme under whih firms prefer to share a single faility to any other faility sharing onfiguration an be formulated as a ooperative game among the independent firms in the set N. Consistent with standard terminology from ooperative game theory, let us refer to the subset of firms J N as oalition J and to the 11

13 set N, the largest oalition, as the grand oalition. A ooperative game is then defined by a harateristi funtion whih speifies the value assoiated with eah oalition J. In our ontext, this orresponds to the total ost assoiated with a subset of firms J sharing a single faility. We refer to this ost as z J, where z J z J (µ J ). A vetor φ = (φ 1,, φ n ) is alled an alloation rule if φ i orresponds to the portion of total ost in the grand oalition that is assigned to firm i. If n i=1 φ i = zn, then the alloation rule is said to be effiient. An alloation rule is said to be individually rational if φ i zi and to be stable for a oalition J if i J φ i zj. An alloation is said to be a member of the ore if it satisfies the following inequalities: and i J φ i zj, J N, (18) φ i = zn. (19) i N When an alloation rule is in the ore, no subset of players would want to seede from the grand oalition and form smaller oalitions, inluding being on their own. Hene the existene of an alloation rule that is in the ore (the ore is non-empty) is suffiient in our ontext to show that it is optimal for all the firms to share a single faility. This single faility is a superior arrangement to any other arrangement that may involve a set of partially pooled failities shared among multiple subsets of the firms. In addition to the requirement of being in the ore, it is desirable for an alloation rule to be pereived as fair. In general, a fair alloation is one that assigns a higher portion of total ost to firms whose membership in the oalition ontribute more to total ost. In partiular, everything else being equal, firms with higher demand rates, higher delay osts, or higher servie levels should pay a greater portion of total ost. In what follows, we show that a relatively simple alloation rule has both the properties of being in the ore and satisfying the above intuitive notions about fairness (for a more extensive disussion of fairness in ost alloation rules see Moulin 1995). Consider the following ost alloation rule: φ i = h iλ i η N + λ i + γ i, (20) where γ i = h i λ i i N h iλ i η N if 1 ln( 1 α N ) i N h iλ i (21) 12

14 and, otherwise (if ln( 1 ) P 1 α N i N > h iλ i ), η N γ i = P i N,i imax h iλ i P P h i λ i i N,i imax h iλ i i N,i i h iλ i if i = i max, and if i i max, (22) with i max {i : α i = max(α 1,..., α n )}. Under the above alloation rule, eah firm is harged a fee onsisting of two parts: (1) its own delay ost, h i λ i η N and (2) a portion of total apaity ost, λ i + γ i. The portion of total apaity ost has itself two parts: (a) an amount proportional to the firm s demand rate that an be diretly attributed to eah firm (this amount orresponds to the minimum ost needed to satisfy demand from this firm) and (b) a portion of the ost of buffer apaity. This portion is non-dereasing in the demand rate, delay ost, and servie level of eah firm. If ln( 1 ) P 1 α N i N h iλ i, this fration is proportional to the firms demand-weighted delay osts. If ln( 1 ) P 1 α N i N > h iλ i (the ase where the servie level onstraint is more restritive), firm i max determines the servie level requirement for the entire system. Therefore, it is treated differently to ensure that it is alloated a portion of the ost that is suffiiently high so that other firms do not break away from the oalition. Theorem 5.1 The ost alloation rule defined by the vetor φ = (φ 1,..., φ n ) is in the ore. That is, under this ost alloation, the grand oalition is optimal for all firms in N. The above alloation rule is valid regardless of whether or not delay osts are absorbed by the pooled faility. If they are, then φ i is indeed the fee that is harged to firm i. If, on the other hand, delay osts are inurred diretly by the individual firms, the usage fee that is harged to firm i orresponds only to the portion of the apaity ost given by (λ i + γ i η N ). In systems without delay osts, firms are harged only the portion of apaity ost. It is perhaps surprising, given the heterogenous delay osts and servie level requirements among the different firms, that the alloated ost an always be broken down in this fashion, with firms always being harged, diretly or indiretly, their own delay osts. This alloation appears to be onsistent with those observed in pratie, where ombinations of volume based and apaity/servie level based fees are ommon; see Gans and Zhou (2003, 2005) and Aksin et al. (2004). The alloation rule speified in (20)-(22) is not unique. It is possible to onstrut other ost alloations that are also in the ore. As we show in the following theorem, a well known ost alloation, the Shapley value is in the ore for two important speial ases. Theorem 5.2 In systems with either only delay osts or only servie level onstraints, the apaity 13

15 sharing game is a onave game, i.e., z S + z T z S T + z S T, S, T N. Therefore, the Shapley value is in the ore. The Shapley value is an alloation sheme that has been shown to possess several desirable fairness properties and has been widely studied in other ontexts (see for example Moulin (1995)). However, the Shapley value does not neessarily separate the alloated ost the way our proposed alloation does, whih may be a required or desirable in ertain appliations. 6 Is Capaity Pooling Always Benefiial? We have so far studied a setting where pooling is always benefiial to eah firm and there is a fair ost alloation sheme under whih all firms prefer to partiipate in a single pooled faility. In this setion, we highlight situations where apaity pooling may not be desirable and firms may prefer to either operate independent failities or to form smaller oalitions. In partiular, we illustrate the effet of three fators that impat the desirability of pooling: (1) work ontent heterogeneity among the firms, (2) apaity disreteness, and (3) onvexity of apaity ost. For the sake of brevity, we fous on systems with only delay osts (i.e., without servie level onstraints). The treatment an be easily extended to the ase where delay osts and servie level onstraints are both in effet. 6.1 The Effet of Heterogeneous Work Content In the model desribed in Setions 3 and 4, we assumed that work ontent is homogenous aross ustomers from different firms, so that proessing times are identially distributed regardless of firm affiliation. In this setion, we onsider a system where the work ontent assoiated with ustomers vary from firm to firm. Customer proessing time is still a random variable of the form Y i /µ, for firm i, but Y i is now exponentially distributed with mean ω i (reall that, in the original model, ω i = 1 for all i N ). Customers are still served on a FCFS basis. In a pooled systems, with n firms sharing a single faility, servie times are no longer exponentially and identially distributed. Instead the distribution of servie time is a mixture of exponential distributions (i.e., the distribution is hyperexponential) with mean E(S) = i N p iω i /µ N and seond moment E(S 2 ) = 2 i N p iωi 2/µ2 N, where p i = λ i / i N λ i. The resulting squared oeffiient of variation C 2 s = E(S2 ) E(S) 2 E(S) 2 is stritly greater than 1 (Cs 2 = 1 is the ase where firms have idential work ontents) and inreases with the differenes in the amount of work ontent of different firms. 14

16 In a distributed system, the problem faed by firm i an be stated as minimize z i (µ i ) = µ i + h i λ i µ i /ω i λ i (23) subjet to λ i ω i /µ i 1, (24) and in the pooled system as minimize z N (µ N ) = i N h i λ i { (1 + C2 s )E(S) 2(1 ρ) + ω i µ N } + µ N (25) subjet to ρ = i N λ iω i µ N 1. (26) The optimization problem in both ases is onvex and therefore an be easily solved numerially. However, a losed form analytial solution is diffiult to obtain in the ase of the pooled system beause the pooled faility no longer behaves like an M/M/1 queue and, instead, behaves like a M/G/1 queue. Note that expeted delay now depends on both the mean and the variane of servie time via the oeffiient of variation C s. It is diffiult to haraterize analytially the optimal apaity in a shared faility. Therefore, it is also diffiult to haraterize analytially onditions under whih the ore may or may not be empty. In order to get some insights, and isolate the effet of work ontent variability, let us onsider the ase where h i = h and λ i = λ for all i N. In this ase, the expression for Cs 2 simplifies to Cs 2 = 2C2 ω + 1 where C ω is the oeffiient of variation of the means of the individual work ontents. Total ost in the pooled system an now be expressed as z N (µ N ) = µ N + h{ (1 + C2 ω)ρ 2 (1 ρ) + ρ}. It is easy to see that, everything else remaining the same, total ost for a given servie rate µ N is inreasing in C ω and, onsequently, the optimal apaity level is also inreasing in C ω. This suggests that for suffiiently high C ω, the pooled system ould be less desirable than a system with independent failities. This observation is onfirmed by the numerial results shown in Figures 1 where the perentage ost differene between the pooled and distributed system, (zn z 1,...,n )/z 1,...,n, is shown for different values of C ω for an example system with two firms (to obtain different values of C ω, we vary ω 1 and ω 2 while keeping ω 1 + ω 2 = 20). There is a threshold on C ω above whih 15

17 the distributed system beomes preferable. The effet of inreasing C ω is partiularly signifiant when delay ost is high (as shown in Figure 1) or when the demand rates or apaity ost are low (as observed in additional numerial results not shown here for brevity). In the following proposition, we provide a suffiient ondition for pooling to be benefiial. Theorem 6.1 Pooling is always benefiial if i N h iλ i i N h iλ i ω i i N λ iω i i N λ iωi 2. (27) An important ase that not only satisfies (27) but also leads to the existene of the ore is desribed in the following orollary. Corollary 6.2 If h i /ω i = h j /ω j for all i, j, then the orresponding game is onave and the ore exists. The above ase orresponds to settings where delay osts are proportional to work ontent, so that firms that bring more work to the system per ustomer are also those that are more sensitive to delay. This prevents situations where a firm with a small work ontent but high delay ost leads the pooled faility to invest in a large amount of apaity to mitigate the delay ost of that firm. In that ase, the other firms may do better by exluding the high delay ost/low work ontent firm from the oalition. Note that the ondition in the orollary is independent of the demand rates. This points to the dominant role that differenes in work ontent and delay osts play in determining whether or not pooling is benefiial. Note that the fat that total pooling may not be superior to multiple independent failities does not mean that all forms of pooling are not desirable. In Figure 2, we ompare the performane of three faility pooling senarios in a system with 4 firms. Senario 1 orresponds to the grand oalition, senario 2 to four independent failities, and senario 3 to two pooled failities: one shared by firms 1 and 2 and the other by firms 3 and 4. The firms are idential with respet to all parameters, exept for their mean work ontents whih are given by ω 1 = 2.5, ω 2 = 2.5, ω 3 = 100, and ω 4 = 450. As we an see, senario 3 is superior to both senarios with the differene inreasing as delay osts inrease. That is, firms with similar work ontent find it benefiial to form smaller shared failities. 6.2 The Effet of Disrete Capaity We have so far assumed that apaity an be varied ontinuously in the form of a servie rate. This allowed us to model the servie faility, regardless of the number of partiipating firms and their 16

18 orresponding demand rates, as a single server queueing system. In this setion, we onsider settings where apaity an be aquired only in disrete quantities with eah unit of apaity orresponding to a separate server (e.g., servers are human operators, omputer servers, or teleommuniation lines). The servie rate of eah server is fixed but the number of servers is variable. Hene, failities an now be viewed as multi-server queues with a fixed servie rate per server. Consistent with our treatment in our original model of Setions 3 and 4, let us assume that firm i, for all i N, faes a Poisson demand with rate λ i, the servie rate of eah standard server is µ, servie times are exponentially distributed (with mean 1/µ), ustomers are served on a FCFS basis, and the amortized ost per server is k. Consider first the distributed system. Let m i denote the number of servers in firm i s faility. Then, eah faility an be modeled as an M/M/m i queue. The problem faed by firm i in determining the optimal number of servers an be stated as follows: ( 1 minimize z i (m i ) = h i λ i µ + P (m ) i, λ i ) + km i, (28) m i µ λ i subjet to and ρ i = λ i < 1, (29) m i µ m i 1, m i : integer, (30) where P (m i, λ i ) = mi 1 n=0 (m i ρ i ) m i m i!(1 ρ i ) (m i ρ i ) n n! + (m iρ i ) m i m i!(1 ρ i ). (31) Using the fat that P (m i ) m i µ λ i is onvex in m i (see for example Dyer and Proll (1977)), we an show that z i (m i ) is also onvex. Therefore, the optimal number of servers is given by the smallest integer m i that satisfies the inequality: We denote the optimal apaity level by m i. z i (m i + 1) z i (m i ) > 0. Although it is diffiult to obtain a losed form expression for m i and the orresponding optimal ost zi z i (m i ), both are easy to ompute numerially. For the pooled system, the analysis is similar. Expeted ost, given a hoie of number servers m N, is given by: z N (m N ) = i N ( 1 h i λ i µ + P (m N, i N λ ) i) + km N, (32) m N µ λ i 17

19 where and ρ N = P (m N, i N λ i ) = (m N ρ p ) m N m N!(1 ρ N ) mn 1 n=0 (m N ρ N ) n n! + (m N ρ i ) m N m p!(1 ρ N ) P i N λ i m N µ. The optimal apaity m N and the optimal ost z N similar approah as in the ase of the distributed system., (33) an be obtained using a In ontrast to the ase where apaity is in the form of a servie rate that an be varied ontinuously, pooling is not always benefiial here. We show this by ounter example. In Figure 3, we provide numerial results for a system with two firms, 1 and 2, with the following harateristis: µ = 1, λ 1 = ɛ, ɛ > 0, h 1 = 1000 ɛ, λ 2 = 1 and h 2 = 10. The figure shows the impat of varying ɛ on the optimal ost for the pooled and distributed systems. As we an see, for suffiiently small values of ɛ, the total ost in the distributed system is lower than the ost in the pooled system. As ɛ gets small, it is possible to invest in fewer servers for firm 1 in the distributed system, although the delay ost is large. This is possible beause ustomers of firm 1 either experiene no wait or wait very little for servie. This is not the ase in the pooled system. The wait for ustomers of firm 1 an be signifiant even though ɛ is small (e.g., when servers are busy with ustomers from firm 2). This leads to higher delay ost sine h is inreasing in ɛ. In turn, this ould prevent the redution in the optimal number of servers seen in the distributed system. In the example shown in Figure 3, reduing ɛ from 2.5 to 0.5 redues the number of servers of firm 1 from 10 to 6 (the number of servers of firm 2 is unaffeted). However, in the pooled system the number of server remains unhanged at 23. In general, we have observed that pooling beomes less benefiial as variability in delay osts among ustomers of different firms inreases. Interestingly, this effet is absent in the model of setions 2-4 where apaity is in the form of a servie rate. In that ase, the benefit of pooling are muh more signifiant. In partiular, the redution in servie times that aompanies pooling is always suffiient to offset the negative impat of any inreased variability in delay osts. While it is possible for total or partial apaity pooling to be benefiial (e.g., when variability in delay ost among firms sharing the same faility is relatively moderate), it is diffiult to haraterize onditions under whih this ours. Therefore, it is also diffiult to identify analytially onditions under whih the grand oalition is optimal. An exeption is when the firms have idential delay osts, with h i = h for all i N. In this ase, we an show that pooling is system-optimal, with zn = z N (m N ) z 1,,n = i N z i(m i ); see the Appendix for a proof. Unfortunately, even in this ase, it is diffiult to show that the ore is not empty in general. However, it is possible to do so if we restrit ourselves to the limit regime where λ N and rely on approximations that are 18

20 asymptotially exat. This regime has gained a lot of attention in the literature reently; see Halfin and Whitt (1981), Reiman (1984), and Borst et al. (2004). It is important in pratie beause it is useful in modeling large sale servie systems suh as large all enters, say with hundreds of operators. We briefly sketh this approah and show how our alloation rule is in fat in the ore in this ase. Following the approah in Borst et al. (2004), the optimal number of servers an be approximated by the ontinuous parameter ˆm N where ˆm N = λ N /µ + x λ N /µ, x hp (x) = arg min{kx + }, (34) x>0 x 1 P (x) := 1+ x, and H( ) is the hazard rate funtion of the standard normal distribution. The H( x) value of ˆm N is asymptotially optimal with lim λ N ˆm N m = 1. Expeted delay an now be approximated (the approximation is also asymptotially exat) N as E(Ŵ N ) = P (x) ˆm N µ λ N + 1 µ, and expeted ost as ẑ N ( ˆm N ) = k ˆm N + hλ N E(Ŵ N ). Consider now the alloation rule speified by ˆφ i = hλe(ŵ N ) + kλ i/µ + (λ i /λ N )kx λ N /µ, for i N. This is a similar alloation rule to the one speified in equations (20)-(22). Eah firm is harged a fee onsisting of three parts: (1) its own delay ost, (2) the ost of apaity for whih it is diretly responsible, kλ i /µ, and (3) a portion of the ost of buffer apaity (λ i /λ N )kx λ N /µ whih is proportional to its demand rate (reall that h i = h for all i N ). Theorem 6.3 The alloation rule speified by ˆφ i ẑ N. for i N is in the ore under the ost funtion The above result suggests that the effet of disrete apaity beomes less signifiant when demand is suffiiently large and onsequently the number of servers is also large. The result also provides evidene of the robustness of the alloation funtion originally speified for systems with single servers. 19

21 Table 1: Optimal Coalitions for an Example System (h = 20, λ = 1, (µ) = 0.5µ 2 ) Number Optimal total ost (apaity) Optimal Optimal total ost (apaity) of firms under grand oalition oalitions under optimal oalitions (5.33) {1} 7.46 (5.33) (11.57) {5} (11.57) (12.84) {6} (12.84) (14.06) {7} (14.06) (15.24) {8} (15.24) (16.41) {9} (16.41) (17.55) {5,5} ({11.57,11.57}) (26.28) {9,9} ({16.41,16.41}) (27.34) {6,6,7} ({12.84,12.84,14.06}) (33.62) {8,8,9} ({15.24,15.24,16.41}) (34.66) {6,6,7,7} ({12.84,12.84,14.06,14.06}) 6.3 The Effet of Diseonomies of Sale In some settings, apaity ost may not be linear (in servie rate or number of failities) but instead may feature either eonomies or diseonomies of sale. The first ase may orrespond to situations where operating a faility involves fixed osts that are largely independent of the size of the faility. In this ase, the benefits of pooling, if any, would not be diminished and we expet the grand oalition to ontinue to be optimal. The seond ase may orrespond to situations where inreasing apaity gets inreasingly expensive (e.g., due to limits on tehnology or to inreasing managerial omplexity). Here, pooling may not always be benefiial. It is easy to onstrut examples where either the distributed system or a system onsisting of multiple small oalitions is optimal. In Table 1, we illustrate the number of optimal oalitions for an example system with n idential firms and apaity ost (µ) = 0.5µ 2 as we vary n (the notation {6, 6, 7} indiates that for n = 19, the optimal onfiguration onsists of 3 suboalitions, two with 6 firms eah and one with 7); the values in parentheses denote the orresponding optimal apaities). 7 Systems with Servie Priorities We have so far assumed that ustomers in a shared faility, regardless of their firm affiliation, are served on a FCFS basis. This poliy is simple to implement and evaluate and has the appearane of fairness. However, it is not system-optimal sine it does not minimize the overall expeted delay. A poliy that does is one where ustomers with higher delay osts are given higher priority. For an 20

22 M/M/1 queue with multiple ustomer lasses, where eah lass may have a different delay ost, the so-alled µ rule minimizes expeted delay in the system (see for example, Jaiswal (1968), Klimov (1974), Harrison (1975)). Under the µ rule, ustomers are assigned priorities based on the produt of their delay osts and their servie rates. In our setting, this means that higher servie priority is given to ustomers with higher delay osts. In this setion, we study the impat of using suh a poliy on the desirability of pooling and whether or not the grand oalition ontinues to be optimal. Other than the priority poliy, the assumptions of the model we onsider are the same as those of our original model desribed in Setions 3 and 4. Beause the analysis of systems with priority is onsiderably more diffiult than the FCFS ase, we limit our analysis to systems with no servie level onstraints. Without loss of generality, we let h 1 h 2 h n. Under the priority poliy, eah time the servie faility beomes available, the ustomer with the highest delay ost, among those waiting, is seleted for servie; a FCFS poliy is used to hoose among ustomers with the same delay ost. We assume that preemption is allowed so that the servie of a ustomer, regardless of its lass, an be interrupted for the servie of a higher priority ustomer (the analysis an be easily extended to systems where preemption is not allowed and the results are qualitatively the same). Under a priority sheme of this type, total expeted ost in a pooled faility with N firms and servie rate µ P N is given by n n zn P (µ P k=1 λ i µ N ) = h i λ i P N + 1 S i S i 1 µ P N S + µ P N, (35) i i=1 where S i = 1 i k=1 λ i, i 1 and S µ P 0 = 1. N Although it is not possible to provide an expliit expression for the optimal ost and the optimal apaity level, it is possible to show that both the optimal ost z P N under the priority poliy are lower than those under the FCFS poliy. and the optimal apaity µp N Theorem 7.1 z P N z N and µp N µ N, where z N and µ N and the optimal apaity under the FCFS poliy. refer respetively to the optimal ost Showing that z P N z N is trivial sine the expeted delay ost under the priority poliy is always lower than under the FCFS poliy for any hoie of µ N. In order to show that µ P N µ N, it is suffiient to show that (1) zn P is onvex in µp N and (2) zp N (µp N ) µ P µ P N N =µ P N 0. A proof that both onditions hold is provided in the Appendix. Although pooling under the priority poliy is more ost effiient for the system and requires less apaity, it does not guarantee that all firms would experiene a lower expeted delay. 21