Capacity Sharing and Cost Allocation among Independent Firms with Congestion

Transcription

1 Capaity Sharing and Cost Alloation among Independent Firms with Congestion Yimin Yu Department of Management Sienes City University of Hong Kong Kowloon, Hong Kong Saif Benjaafar Department of Industrial and Systems Engineering University of Minnesota Minneapolis, MN Yigal Gerhak Department of Industrial Engineering Tel Aviv University, Tel Aviv, Israel Abstrat We analyze the benefit of prodution/servie apaity sharing 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 possibly subjet to servie level onstraints on delay. If firms deide to operate a shared faility they must also deide on a sheme for sharing the apaity ost. We formulate the problem as a ooperative game and identify settings under whih apaity sharing is benefiial and there is a ost alloation that is in the ore under either the FCFS poliy or an optimal priority poliy. We show that apaity sharing may not be benefiial in settings where firms have heterogeneous work ontents and servie variabilities. In suh ases, we speify onditions under whih apaity sharing may still be benefiial for a subset of the firms. Key words: Capaity sharing, queueing systems, joint ventures, ost alloation, ooperative game theory

2 1 Introdution Capaity sharing refers to the fulfillment of demand that arises from multiple soures from a single faility instead of failities dediated to eah demand soure. In a system without apaity sharing, eah dediated faility fulfills its own demand relying solely on its apaity. It has long been known that apaity sharing an be 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 sharing an also be 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, servie, and inventory. Capaity sharing has been studied mostly in situations where a single firm, or a sub-division within a 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 share apaity and how muh apaity to aquire. In this paper, we onsider a system with n independent firms, or sub-divisions within a firm, 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 share apaity, they must also deide on how to alloate the ost of the shared faility. They must do so in a manner that benefits everyone and prevents any of the firms from defeting and perhaps sharing a faility with a subset of the firms or staying on their own. Hene, firms that ontribute more to the ost of the shared faility (beause of their higher usage of apaity or lower tolerane for apaity shortage) are expeted to pay a greater share of total ost. In this paper, 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 times. The apaity at eah faility is determined by the rate at whih ustomers an be proessed. Beause ustomers are proessed one 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. 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 or waiting time 1

3 that ustomers experiene. Alternatively, firms may hoose to olletively operate a shared faility. In that ase, in addition to determining the optimal amount of apaity (taking into aount the delay osts and servie levels of all the firms), the firms must also determine how the orresponding osts must be alloated. In both ases, of either shared or individual failities, the failities are modeled as single server queues, with apaity determined by the assoiated servie rate. Capaity sharing among independent firms in the presene of ongestion, and with heterogeneous delay osts and servie level requirements, arises in a variety of settings. For example, firms (or sub-divisions within a firm) an deide to share support servies, suh as repair and maintenane failities, instead of investing in failities of their own 1. In the ase of repairs, repair requests for eah firm arise randomly over time (beause of the randomness in breakdowns) with repair times that an be stohasti. Given the limited repair apaity, this an lead to ongestion. Eah firm may have its own delay ost (e.g., osts orresponding to the opportunity ost of equipment down time) or may have a speified servie level it guarantees to its own ustomers. Capaity in this ase is determined by the speed with whih repairs an be undertaken (e.g., the proessing rate of the main bottlenek proess). Other examples inlude ommon servies, suh as printing, testing, and rapid prototyping, whih sub-divisions within the same firm may hoose to share. In this ase too, the variability in the arrival of servie requests and proessing times an lead to ongestion. Capaity, in terms of the speed with whih servie requests an be proessed, must again take into aount the requirements of the different subdivisions, inluding their individual sensitivity to delay. The main ontributions of our paper are summarized below. We provide a framework for modeling apaity sharing 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. Under the M/M/1 setting with the FCFS poliy, we formulate apaity sharing as a ooperative game, in whih the partiipating firms optimize shared apaity taking into aount the harateristis of individual firms, their delay osts and their servie level requirements. We haraterize settings where the ore of the game is non-empty. That is, there exists a apaity ost alloation rule for whih all the firms are better off than under any other alternative sharing arrangement, inluding being on their own. 1 For example, several airlines share repair and maintenane failities; see reent announements by Air Frane and Lufthansa and Delta and Aero Mexio 2

4 When the ore exists, 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. We onsider systems that operate under an optimal priority poliy. We show that there exists a apaity ost sharing sheme that is in ore. We aomplish this by showing that the orresponding ooperative game is submodular. Under the FCFS poliy, we also haraterize settings where the ore may not exist beause apaity sharing may not be benefiial. These settings inlude systems where servie levels are speified in terms of waiting time instead of total delay and ases where the firms are heterogeneous in their harateristis, inluding their work ontents, servie time variability, and delay osts. All these indiate that apaity sharing should be onsidered with aution in ontrast to the ommon belief that risk pooling is benefiial. For these ases, we haraterize onditions, under whih apaity sharing may still be benefiial for a subset of the firms. These onditions provide insights into the harateristis of firms that would benefit from forming sub-oalitions. We extend our results to a variety of queueing systems, inluding M/G/1 queues, and GI/G/1 queues. In doing so, we extend known results regarding the benefit of apaity pooling in eah system by endogenizing apaity instead of assuming it remains onstant with and without apaity sharing. We should note that there is a rih literature that models manufaturing and servie systems as queueing systems (see Setions 2 and 3 for further disussion). Surprisingly very little of this literature addresses the issue of ooperation and apaity sharing when there are independent firms. Therefore, we view our paper as a step toward a more omprehensive examination of the issue of ooperation in queueing systems, whether it arises in manufaturing, servies 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 Setion 3, we treat the ase with no apaity sharing. In Setion 4, we analyze the ase with apaity sharing. In Setion 5, we onsider apaity sharing when there are servie priorities. In Setion 6, we extend our analysis to systems with heterogeneous work ontents and systems with general arrival and servie proesses. In Setion 7, we offer onluding omments. 3

5 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. Tekin et al. (2009) use approximations to evaluate the benefit of partitioning servers in multiple pools instead of a single large one. Sheikhzadeh et al. (1998), Gurumurthi and Benjaafar (2004) and Jordan et al. (2005) study the haining of servers, where eah server an proess ustomers from two ustomer streams and eah ustomer an be routed to two servers. They show that in systems with homogeneous demand rates and servie time requirements, haining an ahieve most of the benefitsof total server pooling; see also Hopp et al. (2004), Iravani et al. (2004), Bassambo et al. (2008), Aksin et al. (2008), Wallae and Whitt (2005) and the referenes therein. These papers belong to the growing literature on queueing systems with server flexibility (or ross-training); see Jouini et al. (2008), 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 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 4

6 with identifying ost alloation shemes under whih all firms prefer a single shared faility to any other apaity sharing arrangement, inluding remaining on their own. The literature dealing with apaity sharing in the ontext of independent firms is limited. Gonzalez and Herrero (2004), and also Garia-Sanz et al. (2008), onsider a speial ase of the M/M/1 model we onsider. However in both ases, they do not optimize apaity (before or after pooling) and do not onsider the delay ost. 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. Anily and Haviv (2010) treat a related M/M/1 model where the issue is how to alloate delay ost to ensure that the alloation is in the ore. They show that a Shapley alloation based on theservie levels is in the ore. In this literature, the ommon approah is to useonavity as a basis for proving that the ore exists and that a Shapley alloation is in the ore; see Gonzalez and Herrero (2004), Garia-Sanz et al. (2008), and Anily and Haviv (2010). In ontrast to the above literature, a Shapley alloation may not be in the ore in our ase. This is in part due to our treatment of apaity as endogenous and to the requirement that eah firm absorbs its own delay ost and only apaity osts are alloated among the firms. We also treat queueing systems other than the M/M/1 queue, inluding queues with servie priorities, M/G/1 queues, and GI/G/1 queues. 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. In Operations Management, there is growing literature that applies ooperative game theory to joint ordering problems, partiularly in the ontext of eonomi order quantity models (see Anily and Haviv (2007), Dror and Hartman (2007) and the many referenes therein), eonomi lot sizing models (see for example van den Heuvel (2007) and Chen and Zhang (2009), among others), and news-vendor models (see Muller et al. (2002), Nagarajan and Sošić (2008), Kemahlioglu-Ziya (2004), Chen and Zhang (2007), and Hanany and Gerhak (2008) and the referenes therein). Finally, we should note that there is a rih literature on outsouring where multiple firms may be served by the same supplier, inluding for settings where the outsouring supplier is modeled as a queueing system; see for example, Cahon and Harker (2002), Allon and Federgruen (2006), Gans 5

7 and Zhou (2007), and Benjaafar et al. (2007). In general, the fous of this literature is different as it does not deal with ost alloation or oalition formation. 3 Systems without Capaity Sharing Consider a system onsisting of a set N = {1,...,n} of n firms. Firm i, i N, faes an independent demand stream with ustomers arriving aording to a Poisson proess with rate λ i (we treat more general arrival proesses in Setion 6). When firms operate independently, eah firm invests in a separate servie faility and hooses a ertain level of apaity in the form of a servie rate. We refer to this senario as the senario without apaity sharing. One the failities are built, eah firm serves its ustomers from its own faility 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 the work ontent is homogeneous aross firms. This assumption is justified if firms provide servie to similar ustomers (e.g., repairing similar equipments in the ase of a maintenane faility). Given the exponential nature of both ustomer inter-arrival times and servie times, eah firm behaves like an M/M/1 queue. There is a signifiant literature on the eonomis of queues in ompetitive settings 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 and see Cahon and Harker (2002), Cahon and Zhang (2007), Benjaafar et al. (2007), and Allon and Federgruen (2007), among many others, for example appliations. Our treatment of the M/M/1 queue is onsistent with assumptions made in that literature and an be viewed as omplementing it for 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. It is onsistent with treatments elsewhere in the literature (see for example Kalai et al. (1992), Mendelson and Whang(1990), Ha(2001), Allon and Federgruen (2007, 2008), Cahon and Zhang (2007), and the vast literature reviewed therein). This assumption an also be found in the signifiant literature on apaity planning, as noted reently by Bassambo et al. (2008). The assumption of linear apaity ost implies that there are neither eonomies nor 6

8 diseonomies of sale. This is an important ase that has been widely studied in the literature (see Allon and Federgruen (2007, 2008), Dewan and Mendelson (1990), Stidham (1992), Cahon and Harker (2002), and Bassambo et al. (2008) among others), leads to tratable analysis, and provides a useful benhmark for other ost strutures. We assume that the demand rate for eah firm is known. This of ourse does not mean that demand is deterministi. Inter-arrival times between onseutive ustomers are stohasti. Therefore, the number of ustomers that arrive over a given period of time is random. The assumption of known demand rate is onsistent with most of the existing literature on apaity planning in queueing systems (and indeed in most of the queueing literature); see for example Kleinrok (1976), Cahon and Harker (2002), Bassambo et al. (2008), and Allon and Federgruen (2007, 2008), among many others. The objetive of eah firm is to minimize its apaity investment while limiting the amount of delay, as measured by either total time in system, or waiting time in the queue, 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. 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. We assume that all firms hoose the same type of servie level onstraints sine they are in the same industry. 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 that 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). 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), Ha (1998, 2001), Allon and Federgruen (2007, 2008) and the referenes therein. In this paper, we onsider the ase where a unit 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) and the objetive is to minimize the long run expeted delay ost. Moreover, we onsider the ase where servie level is expressed in terms of a probability that delay in the system for eah ustomer, whih we define as the sum of waiting time in the queue and time in servie, does not exeed a 7

9 speified threshold. This measure is onsistent with servie levels used elsewhere in the literature; see for example Allon and Federgruen (2007, 2008), among others. We also onsider servie levels expressed in terms of waiting time the queue alone. Servie level measured in terms of total time in system is appropriate in appliations suh as omputing, teleommuniation, and manufaturing where ustomers are onerned about the total fulfillment of their orders/servie requests. Servie level measured in terms of waiting time is appropriate in settings, suh as all enters and other servie systems, where ustomers are partiularly sensitive to time spent in the queue. 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 delay (waiting time in the queue + servie time) that a ustomer of firm i experienes and P(W i ) the probability that ustomer delay does not exeed where 0 (we will onsider later the ase where servie level is expressed in terms of waiting time). 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 P(W i ) = 1 e (µ i λ i ) α i, (2) λ i /µ i 1. 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. (3) 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 ). (4) 8

10 Surprisingly, the buffer apaity ln( 1 ) 1 α i is independent of the demand rate, a result of the fat that the delay distribution depends on λ i and µ i through µ i λ i only. This feature is also present if servie levels are speified in terms of expeted delay. If we let α i now denote the threshold on the maximum expeted delay, then the servie level onstraint is given by 1 µ i λ i α i. This leads to an optimal apaity given by µ i = λ i + 1 α i. Note that the buffer apaity is again independent of the demand rate. 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 that inreases in either the ratio h iλ i or the servie level α i. The expressions in equations (3) and (4) are not new. Similar expressions have been derived elsewhere; see for example Kleinrok (1976), Allon and Federegruen (2008) 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, (5) where Substituting µ i η i = max{ ln( 1 1 α i ) hi λ i, }. (6) in (1), we obtain the optimal expeted ost for firm i as z i = (λ i +η i )+ h iλ i η i. This leads to a total system ost of z 1,...,n = i N z i. In systems where h i λ i ln( 1 ) 1 α i for all i N, the optimal ost simplifies to z i = λ i +2 h i λ i. This leads to a total system ost, z 1,...,n, given by z 1,...,n = i N λ i+2 i N hi λ i. In the ase of idential firms, with λ i = λ and h i = h for all i N, the optimal total ost redues to z 1,...,n = nλ + 2n hλ, and the total apaity in the system to hλ i N µ i = n(λ + ). 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 hλ ln( 1 1 α ). 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. Next we onsider the ase where the servie level is speified in terms of waiting time in the queue. Let Q i, a random variable, denote the time a ustomer of firm i spends waiting in the queue before servie starts and let P(Q i q 0 ) be the probability that ustomer waiting time does not 9

11 exeed q 0 where q 0 0. The problem faed by firm i an then be restated as Minimize z i (µ i ) =µ i + h iλ i µ i λ i (7) subjet to P(Q i q 0 ) = 1 λ i µ i e (µ i λ i )q 0 α i, (8) λ i /µ i 1. In the absene of servie level onstraints, the optimal apaity level is given by (3). In systems with servie level onstraints but no delay osts, the optimal apaity level is given by the smallest µ i that satisfies inequality (8) or, equivalently, the optimal apaity is the solution to the following equation ln(λ i q 0 )+λ i q 0 ln(1 α i ) = ln(µ i q 0 )+µ i q 0. (9) Unfortunately, there is no expliit solution for the above equation. However, we are able to show the following important result. Lemma 3.1 Let µ i (α i,λ i ) be the solution to (9) and η i (α i,λ i ) = µ i (α i,λ i ) λ i, the amount of buffer apaity ( µ i (α i,λ i ) > λ i ). Then, given α i, η i (α i,λ i ) is nondereasing in λ i, with η i (α i,λ i )/λ i being noninreasing in λ i. (The proof of this and of all subsequent results an be found in the Appendix A). This lemma indiates that, in ontrast to the ase where servie level is delay-based, buffer apaity is inreasing in demand, although the rate of inrease is less than one. This result is due to the fat that the delay is exponentially distributed with the rate of the buffer apaity while the waiting is not exponentially distributed. As we will see in the next setion, this signifiantly affets the benefit derived from apaity sharing. 4 Systems with Capaity Sharing In this setion, we onsider the senario where the firms deide to form a oalition and invest in a single shared faility (a joint venture) from whih the demand of all the firms will then be satisfied. We assume that the rules governing the joint venture (as negotiated by members of the oalition) require that the hoie of apaity, in the form of a servie rate, for the shared faility takes into aount the demand levels of eah member of the oalition, their delay osts, and their servie level requirements. In partiular, we assume that the servie rate is hosen by the managers of the 10

12 joint venture so that it minimizes the total ost for the oalition (the sum of expeted delay osts experiened by ustomers of all the firms and the ost of apaity) and satisfies all servie level onstraints. We assume that all members of the oalition are truthful in their reporting of their demand rates, delay osts, and servie levels. We assume throughout that, although independent, the firms are not ompetitors so that their demands are exogenously determined and are not affeted by deisions made by any of the firms. The assumption of full information applies to settings where the information is publi and an be independently verified by all the firms. For example, delay penalties and servie level guarantees ould be publily advertised by the firms themselves as part of their marketing strategy. In some ases, delay penalties and servie levels may also adhere to well-known industry standards. In settings where delay osts are diretly inurred by the shared faility (e.g., the shared faility is responsible for handling delay penalty payments to the ustomers), firms would also need to provide the shared faility with the orret delay osts. Similarly, servie levels must be known to the shared faility if ontratual agreements with the ustomers regarding servie levels are handled diretly by the shared faility. Demand rates are in most ases verifiable sine demand would eventually be satisfied from the shared faility. Firms an be indued to dislose their true demand rates by imposing high penalties if the originally reported rates are higher than the realized rates (measured over a suffiiently long period of time) one the faility is in operation. The assumption of full information is of ourse appliable to the ase where the firms are all subdivisions of a single large firm. We refer to the servie rate in the shared faility from whih the demand of all firms is satisfied as µ N (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 shared 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. Hene, the system with the shared faility behaves again as an M/M/1 queue. 4.1 Capaity Optimization First, we onsider the ase where the servie level is speified in terms of delay (see Setion 4.3 for a disussion of waiting time-based servie levels). We assume that the terms of the joint venture 11

13 between the partiipating firms in the oalition require that the shared faility invests in apaity so as to minimize the total ost to the oalition while satisfying the servie level onstraint of eah firm. The total ost to the oalition onsists of the sum of apaity ost and expeted delay ost (experiened by ustomers of all the firms over the long run). Satisfying the servie level onstraints of all the firms requires satisfying the highest of these servie level onstraints. If we let z N (µ N ) denote total system ost and let W N, a random variable, refer to ustomer delay, then the apaity optimization problem an be stated as follows: Minimize z N (µ N ) =µ N + i N h iλ i µ N i N λ i subjet to P(W N ) = 1 e (µ N i N λ i) α N, λ i /µ N 1, where α N = max(α 1,...,α n ). Then, the optimal apaity is given by i N (10) µ N = i N λ i +η N, where η N = max{ ln( 1 1 α N ) i N, h iλ i }. Similar to the system without apaity sharing, the optimal apaity 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. The results are similar if the servie level is speified in terms of a threshold on expeted delay. In that ase the optimal buffer apaity is given by η N = max{ 1 i N α N, h iλ i }. The following theorem shows that by investing in a shared faility, the firms are able to redue total ost in the system while investing in less apaity. Theorem 4.1 z N z 1,...,n and µ N n i=1 µ i, where z N is the optimal ost in the shared faility. The potential magnitude of the savings from apaity sharing an be more easily seen in a system hλ with idential firmswhereα i = α, h i = h, and λ i = λ for all i N. Considerthease where ln( 1 1 α ). This leads to µ N = nλ+ nhλ, z N = nλ+2 nhλ, and E(WN ) = nhλ 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 ase of no apaity sharing). In 12

14 nhλ the ase where ln( 1 1 α ), we have µ 1 ln( 1 α N = nλ+ ), zn 1 ln( 1 α = (nλ+ ) )+ nhλ ln( 1 ), and 1 α E(WN ) = ln( 1 ). Here, the magnitude of savings on apaity is even larger with buffer apaity 1 α redued by a fator of n, but expeted delay remains unhanged from the ase without apaity sharing. 4.2 Cost Sharing We have so far showed that apaity sharing 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. We assume that eah firm inurs its own delay ost and pays a fration of apaity ost. A firm would prefer the shared faility if the sum of its share of apaity ost and its long run expeted delay ost is lower than the ost it would inur without apaity sharing. 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 share apaity with only a subset of the firms. This ould lead firms to form groupings around multiple smaller shared failities. A single shared faility would be preferred by all firms only if there exists a ost alloation under whih the firms are better off than under any other apaity sharing arrangement, inluding operating individual failities. Hene, it is desirable that the ost alloation for the shared 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 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 expeted 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 expeted ost in the grand oalition that is inurred by 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 z i and to be stable for a oalition J if i J φ i z J. 13

15 An alloation is said to be a member of the ore if it satisfies the following inequalities: i J φ i zj, J N, (11) φ i = zn. (12) 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, (13) 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 (14) and, otherwise (if ln( 1 ) 1 α N i N > h iλ i ), η N γ i = i N,i imax h iλ i h i λ i i N,i imax h iλ i i N,i imax h iλ i if i = i max, and if i i max, (15) with again i max {i : α i = max(α 1,...,α n )}. Remark 3. If the set {i : α i = max(α 1,...,α n )} has multiple indies, then we an arbitrarily hoose any index in this set to be i max. We an also let the firms with the highest servie 14

16 i N,i imax level share the portion of the apaity ost η N h iλ i h i λ i i N,i imax h iλ i i N,i imax h iλ i. in addition to the portion Under the above alloation rule, eah firm (1) inurs its own delay ost, h iλ 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 ) 1 α N i N h iλ i, this fration is proportional to the firms demandweighted delay osts. If ln( 1 ) 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. 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 for example Gans and Zhou (2003, 2007) and Aksin et al. (2008). Theorem 4.2 The ost alloation rule φ = (φ 1,...,φ n ) as speified in (13)-(15) is in the ore. That is, under this ost alloation, no subset of the firms in N has an inentive to seede from the grand oalition. Remark 4. In general, a simple proportional ost alloation poliy may not be in the ore. For example, onsider the ase with pure servie level onstraints and with one of the firms requiring a muh higher servie level than the rest (the extreme ase being only one of the firms requiring a servie level). Then, learly, all but one of the firms prefer not to join the grand oalition. Similarly, we an show that other ommon alloation shemes, suh as the Shapley value, may not be in the ore. In general, our ooperative game is not a onave game. 4.3 The Case of Waiting Time-based Servie Levels In this setion, we onsider the ase where the servie level is speified in terms of waiting time. Let Q N, a random variable, refer to ustomer waiting time in the shared faility. Then, the apaity 15

17 optimization problem an be stated as where α N = max(α 1,...,α n ). i N Minimize z N (µ N ) = µ N + h iλ i µ N i N λ i i N subjet to P(Q N q 0 ) = 1 λ i e (µ N i N λ i)q 0 α N, (16) µ N λ i /µ N 1, i N Similar to the ase without apaity sharing, in the absene of servie level onstraints, the optimal apaity level is given by i N λ i + i N h iλ i. In systems with servie level onstraints but no delay osts, the optimal apaity level is given by the smallest µ N that satisfies the servie level onstraint or, equivalently, the optimal apaity is the solution to the following equation ln(λ N q 0 )+λ N q 0 ln(1 α N ) = ln(µ N q 0 )+µ N q 0, (17) where λ N = n i=1 λ i. Let µ N (α N,λ N ) be the solution to the above equation and η N (α N,λ N ) = µ N (α N,λ N ) λ N, the assoiated amount of buffer apaity (note that µ N > λ N ). In ontrast to the ase where servie level is speified in terms of delay, the amount of buffer apaity is not invariant to λ N and is indeed inreasing in λ N. This leads to the following result. Theorem 4.3 Capaity sharing may not be benefiial and it is possible for z N > z 1,...,n and µ N > n i=1 µ i. The above result an be proven using a ounter-example. Consider the ase where h i = 0 for all i and α 1 > 0 but α i = 0 for all i 1. For the ase without apaity sharing, we have η i (0,λ i ) = 0 for i 1. However, η 1 (α 1,λ 1 ) > 0 and is given by the solution to ln(1 α 1 ) = ln(1+ z λ 1 )+zq 0. For the system with a single shared faility, we have η N (α 1,λ N ), where η N (α 1,λ N ) is the solution to ln(1 α 1 ) = ln(1+ z λ N )+zq 0. Then, it is lear that η N (α 1,λ N ) > η 1 (α 1,λ 1 ) and, onsequently, z N = η N (α 1,λ N ) + λ N > i N z i = η 1 (α 1,λ 1 )+λ N. This is due to fat that buffer apaity is inreasing in the demand rate, with the demand rates of different firms having marginal effets on the buffer apaity. 16

18 Hene, surprisingly, apaity sharing may not be even benefiial when servie level onstraints are imposed on waiting time in the queue even under the M/M/1 setting. This means that in ontrast to the ommon sense that risk pooling is benefiial, apaity sharing should be taken with aution when the servie level onstraints are imposed on waiting time in the queue. Notie that without servie level onstraints, the ooperative game an be shown as a submoduar game (whih is due to that in this ase the optimal ost z N = i N λ i +2 i N h iλ i, ) and the ore exists. As we an see that the existene of the ore for systems when the servie level onstraints are imposed on delay is due to the the buffer apaity for the servie level onstraints is independent of the total demand rate, i.e., when the servie level onstraints are ative, the optimal buffer apaity level is determined by the firm with the highest servie level only. In this ase, apaity sharing always lowers the buffer apaity. However, when the servie level onstraints are imposed on waiting time, in general the ore may not exist. This is due to that apaity sharing may lead to higher buffer apaity as we have shown in the above example sine the optimal buffer apaity level may depend on both the servie level onstraints and the total demand rate. In partiular, apaity sharing for a firm with high servie level requirement but low demand rate and a firm with low servie level requirement but high demand rate ould be detrimental. Although apaity sharing is not always benefiial in general, it is in the ase where the firms have idential servie level onstraint and α i = α for all i. Theorem 4.4 If α i = α, then z N z 1,...,n and µ N n i=1 µ i. Consider the following ost alloation sheme. φ i = h iλ i η N +λ i +γ i, (18) where γ i = h i λ i i N i N h h iλ i iλ i if η N (α,λ N ) i N h iλ i (19) and, otherwise (if η N (α,λ N ) > i N h iλ i ), γ i = λ i λ N η N (α,λ N ). (20) As stated in the following theorem, above alloation sheme is in the ore. Theorem 4.5 If α i = α for all i, then the ost alloation rule φ = (φ 1,...,φ n ) as speified in (18)-(20) is in the ore. That is, under this ost alloation, no subset of the firms in N has an 17

19 inentive to seede from the grand oalition. In summary, if the servie level onstraints are imposed on total delay, then apaity sharing is always benefiial and we an identify a ost alloation sheme that is in the ore. However, if the servie level onstraints are imposed on waiting times, then apaity sharing may not be benefiial beause buffer apaity is no longer invariant to total demand. To our knowledge, this is a new result in the literature on apaity pooling in queueing systems. The result also suggests that apaity sharing should be used with aution and there might be settings where apaity sharing involving only a subset of the firms may be preferable to the grand oalition. 5 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 when there are multiple ustomer lasses with different delay osts or different servie level requirements. For example, for a system without servie level requirements but with different delay osts for different ustomers lasses, the so-alled µ rule is known to be optimal; see, for instane, Jaiswal (1968) and Klimov (1974). 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 a higher servie priority would be given to ustomers with higher delay osts). In the presene of servie level requirements, the optimal poliy is more ompliated and must aount for the interation between delay osts and servie levels, as well as other parameters. In this setion, we extend our analysis to settings where an optimal priority poliy is used and investigate whether or not, under an optimal poliy, there is a ost alloation that is in the ore. The analysis of systems with priorities is notoriously diffiult and, to our knowledge, there are no results in the literature regarding the nature of the optimal priority poliy (in the presene of both delay osts and servie level requirements) and no known losed form expressions for performane evaluation for systems that operate under suh a poliy. Also, to our knowledge, there are no results regarding the existene of the ore for queueing systems that operate under a priority poliy, optimal or otherwise. Other than assuming that a priority poliy is used, the assumptions of the model we onsider are the same as those of our original model desribed in Setions 3 and 4. In order to analyze the assoiated ooperative game we resort to an indiret approah, the soalledahievableregionmethod. Withoutlossofgenerality, weassumethath 1 h 2 h n > 0. We assume that the shared system operates under the optimal preemptive priority poliy (the ase 18

20 without preemption an be similarly analyzed). We assume that eah firm is subjet to a servie level onstraint on its expeted delay (Unfortunately, other forms of servie levels are substantially more diffiult to analyze). The ahievable region method onsiders the lass of mixed preemptive priority poliies. Note that for two strit preemptive priority poliies P 1 and P 2, if at the beginning of eah busy period, we use poliy P 1 with probability 1 β and poliy P 2 with probability β, the resulting poliy is a mixed preemptive priority poliy. Based on Coffman and Mitrani (1980), for oalition J given apaity µ (for µ > λ J,λ J = i J λ i), the feasible region for the vetor of the expeted delay in (E[W i ],i J) under any mixed preemptive priority poliy an be desribed by the following polyhedron: i V λ i E[W i ] λ V µ λ V,for all V J, (21) E[W i ] 0,for all i J. Suppose that the servie level onstraint for eah firm i is given by E[W i ] w i, i.e., firm i requires its expeted delay should not be more than w i. We an readily show that the problem of jointly deiding on the optimal apaity and the optimal priority order for oalition J is speified by the solution to the following optimization problem. z J = subjet to i V min E[W i ],i J;µ i J h i λ i E[W i ]+µ (22) λ i E[W i ] λ V µ λ V,for all V J, 0 E[W i ] w i,i J, µ > λ J. Let E[W P i,j ],i J and µp J be the optimal solution to (22). Noting that the above problem is a onvex optimization problem, it an be reformulated into an equivalent problem using the Lagrangian method (see Bertsekas 1999). Let θ i be the Lagrange multiplier for the servie level onstraint E[W i ] w i, for all i J. Then, the optimal solution to the following problem is also 19

21 optimal for the original problem in (22): ẑ J = max θ i 0,i J subjet to i V min E[W i ],i J,µ i J(h i +θ i )λ i E[W i ]+µ i J λ i E[W i ] λ V µ λ V,for all V J, E[W i ] 0,for all i J, µ > λ J. θ i w i (23) Based on Proposition of Bertsekas (1999), we an show that zj = ẑ J, i.e., there is no duality gap. Note that the Lagrange multiplier θ i an be viewed as the delay ost indued by the orresponding servie level onstraint for firm i. Hene, the presene of servie level onstraints might affet the optimal priority order. Let θ i,j oalition J. The Lagrangian problem implies the following result. be the optimal Lagrangian multiplier for firm i in Lemma 5.1 The optimal priority poliy for oalition J is a strit preemptive priority poliy and the priority order is dereasing in h i +θ i,j, with the firm with the highest value of h i+θ i,j having the highest priority. As we an see, in the presene of servie level onstraints, the µ rule may not be optimal and the optimal priority order is determined by the values h i +θi,j,i J. To the best of our knowledge, this is the first result in the literature to haraterize the optimal priority poliy in a system with both delays osts and servie level onstraints. Note that, although we do not have an expliit expression for the parameters θi,j, the θ i,js an be easily omputed sine the dual problem is a onave optimization problem (see Proposition of Bertsekas 1999). Next, we show that the optimal ost in a shared system is submodular in the set of firms involved. Theorem 5.2 z J T +z J T z J +z T game. for all J,T N, i.e., the ooperative game is a submodular The above result is important beause it is well known that a ooperative game that is submodular admits an alloation of total ost among that is in the ore. In partiular, the Shapley value is one suh alloation (see Shapley, 1971 for a more detailed disussion on submodular games). Under the Shapley value, firm i is alloated a fration of total ost speified by φ i = J N\{i} J!(n J 1)! [zj {i} n! z J],i = 1,,n. 20