Blended Call Center with Idling Times during the Call Service

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1 Blene Call Center with Iling Times uring the Call Service Benjamin Legros a Ouali Jouini b Ger Koole c a EM Normanie, Laboratoire Métis, 64 Rue u Ranelagh, 7506, Paris, France b CentraleSupélec, Université Paris-Saclay, Laboratoire Genie Inustriel, Grane Voie es Vignes, Chatenay-Malabry, France c VU University Amsteram, Department of Mathematics, De Boelelaan 08a, 08 HV Amsteram, The Netherlans benjamin.legros@centraliens.net ouali.jouini@centralesupelec.fr ger.koole@vu.nl IIE Transactions. To appear, 207 Abstract We consier a blene call center with calls arriving over time an an infinitely backlogge amount of outboun jobs. Inboun calls have a non-preemptive priority over outboun jobs. The inboun call service is characterize by three successive stages where the secon one is a break, i.e., there is no require interaction between the customer an the agent for a nonnegligible uration. This leas to a new opportunity to efficiently split the agent time between inboun calls an outboun jobs. We focus on the optimization of the outboun job routing to agents. The objective is to maximize the expecte throughput of outboun jobs subject to a constraint on the inboun call waiting time. We evelop a general framework with two parameters for the outboun job routing to agents. One parameter controls the routing between calls, an the other oes the control insie a call. We then erive structural results with regar to the optimization problem an numerically illustrate them. Various guielines to call center managers are provie. In particular, we prove for the optimal routing that at least one of the two outboun job routing parameters has an extreme value. Keywors: Call centers; blening; task overlapping; routing; performance evaluation; waiting times; optimization; queueing systems; Markov chains. Introuction Context an Motivation: New technology-riven innovations in call centers are multiplying the opportunities to make more efficient use of an agent as she can hanle ifferent types of workflow, incluing inboun calls, outboun calls, s an chats. However, several issues on the management of call center operations emerge also as a result of avance technology. In this paper, we consier a call center with two types of jobs, inboun an outboun jobs. We focus on

2 how to efficiently share the agent time between the two types of jobs in orer to improve the call center performance. Call center situations where inboun calls an outboun jobs (outboun calls or s) are combine is referre to as blening. The key istinction of problems with blening comes from the fact that outboun jobs are less urgent an can be inventorie to some extent, relative to incoming calls. Therefore, managers are likely to give a strict priority to inboun calls over outboun jobs. An important question here is what shoul be the best way of routing outboun jobs to agents, i.e., as a function of the system parameters an the service level constraints (on calls an outboun jobs), when shoul we ask the agent to treat outboun jobs between the call conversations (Bernett et al., 2002; Bhulai an Koole, 2003; Pang an Perry, 204; Legros et al., 205). The outboun job routing question is further important in the context of the call center applications we consier here. We encountere examples where a call conversation between an agent an a customer contains a natural break. We mean by this a time interval with no interaction between the agent an the customer. During the conversation, the agent asks the customer to o some necessary operations in her own (without the nee of the agent availability). After finishing those operations, the conversation between the two parties can start again. Insie an unerway conversation, the agent is then free to o another job if neee. For an efficient use of the agent time, one woul think about the routing of the less urgent jobs not only when the system is empty of calls, but also uring call conversations. In practice, such a situation often occurs. For example, an agent in an internet hotline call center asks the customer to reboot her moem or her computer which may take some time where no interactions can take place. It is also often the case that a call center agent of an electricity supplier company asks the customer for the serial number of her electricity meter box. This box is usually locate outsie of the house an is locke, so, the customer nees some non-negligible time to get the require information. Another example is that of commercial call centers with a financial transaction uring the call conversation. After some time from the start of the call conversation, the customer is aske to o an online payment on a website before coming back to the same agent in orer to finish the conversation. The online payment requires the customer to locate her creit car, next she enters the creit car numbers, next she goes through the automate safety check with her bank (using SMS for example), which may take some minutes. As an illustration, we provie real-life ata from a vehicle glass repair call center company where the service process consists of three phases an a break in the secon phase. Figure gives the empirical probability ensity functions of the three service phase urations for 5986 calls. We observe that these urations are ranom an fin a goo log-normal fit. The p values associate with the statistical χ 2 tests are all above the threshol of 5%. For phases, 2 an 3, they are 0.056, 0.56 an 0.0, respectively. In particular, we observe that the average break uration (phase 2) 2

3 is 2.39 min, an represents in average aroun 30% of the total service uration. It is natural for such a setting that the system manager thinks about using the opportunity to route outboun jobs (or back-office tasks in general) to an agent uring the break of an ongoing call conversation, an not only when no calls are waiting in the queue. The avantage is an efficient use of the agent time an therefore a better call center performance. l el l l l el l el el l el l l l W l l W e e (a) Service phase (b) Service phase 2 l l l el el l l W l (c) Service phase 3 Figure : Empirical probability ensity functions of phase urations Main Contributions: We consier a call center with an infinite amount of outboun jobs. Inboun calls arrive over time, an in the mile of an inboun call conversation, a break is require. Given this type of call centers, we are intereste in optimizing its functioning by controlling how the resource shoul be share between the two types of jobs. Calls are more important than outboun jobs in the sense that calls request a quasi-instantaneous answer (waiting time in the orer of some minutes), however outboun jobs are more flexible an coul be elaye for several hours. An appropriate functioning is therefore that an agent works on inboun calls as long as there is work to o for them. The agent can then work on outboun jobs when she becomes free from calls, i.e., after a service completion when no calls are waiting in the queue, or uring the call conversation break. We assume that calls have a strict non-preemptive priority over outboun jobs, which means that if a call is busy with an outboun job (that has starte after a service completion or uring the break), the agent will finish first the outboun job before turning to a new arrive call to the queue 3

4 or a call that has accomplishe the requeste operations an wants to start again the conversation to finish her service. The non-preemption priority rule is coherent with the operations in practice an also to the call center literature (Bhulai an Koole, 2003; Deslauriers et al., 2007). It is not appropriate to stop the service of a low priority customer. We focus on the research question: when shoul the agent treat outboun jobs? Between calls, or insie a call conversation, or in both situations? Given the nature of the job types, a call center manager in practice woul be intereste in maximizing the number of treate outboun jobs while respecting some service level objective on the call waiting time (Bhulai an Koole, 2003). For inboun calls, we are intereste in the steay-state performance measures in terms of the expecte waiting time, the probability that the waiting time is less than a given threshol, an the probability of elay. For outboun jobs, we are intereste in the steay-state performance in terms of the expecte throughput, i.e., the number of treate outboun jobs per unit of time. Despite its prevalence in practice, there are no papers in the call center literature aressing such a question. Most of the relate papers only focus on the outboun job routing between call conversations but not insie a call conversation. To answer this question, we evelop a general framework with two parameters for the outboun job routing to agents. One parameter controls the routing between calls, an the other oes the control insie a call conversation. Although this moeling is not optimal, its performance measures as shown later are close to the optimal ones an its routing policy is easier to implement than the complex optimal routing. For the tractability of the analysis, we first focus on the single server case. We then iscuss the extension of the results to the multi-server case an its applicability to a more complex setting with may inclue abanonment, general service time istribution an a time-epenent arrival process. For the single server moeling, we first evaluate the performance measures using a Markov chain analysis. Secon, we propose an optimization metho of the routing parameters for the problem of maximizing the outboun job expecte throughput uner a constraint on the service level of the call waiting time. As a function of the system parameters (the server utilization, the outboun job service time, the severity of the call service level constraint, etc.), we erive various guielines to managers. In particular, we prove for the optimal routing that at least one of the two outboun job routing parameters has an extreme value. As etaile later, an extreme value means that the agent shoul always o outboun jobs insie a call (or between calls) or not at all. In other cases, the parameters lea to ranomize policies. We also solve the optimization problem by proposing four particular cases corresponing to the extreme values of the probabilistic parameters. We analytically erive the conitions uner which one particular case woul be preferre to another one. The interest from the particular cases is that they are easy to unerstan by agents an managers. Several numerical experiments are use to illustrate the analysis. We then focus on the routing optimization problem for the multi-server case in a more general setting, using simulation an approximations evelope 4

5 uner the light an heavy-traffic regimes. We foun that most of the observations of the single server case are still vali (in particular the result stating that at least one control parameter has an extreme value). This justifies the applicability of our results to real-life call centers. Paper Organization: The rest of the paper is organize as follows. In Section 2 we review some of the relate literature. In Section 3, we escribe the blene call center moeling an the optimization problem. In Section 4, we consier a single server analysis an evelop a metho base on the analysis of Markov chains in orer to erive the performance measures of interest for inboun an outboun jobs. Next, we focus on optimizing the outboun job routing parameters. In Section 5, we exten the previous analysis to the multi-server case using simulation an also asymptotic approximations. In Section 6, we assess the applicability of our analysis to a more setup with abanonment, general service time istribution an time-epenent arrival process. Finally, Section 7 conclues the article. 2 Literature Review There are three relate streams of literature to this paper. The first one eals with blene call centers. The secon one is the Markov chain analysis for queueing systems with phase-type service time istributions. The thir one is relate to the cognitive analysis, or in other wors the ability for an agent to treat an switch between ifferent job types. The literature on blene call centers consists of eveloping performance evaluation an optimal blening policies. Deslauriers et al. (2007) evelop a Markov chain for the moeling of a Bell Canaa blene call center with inboun an outboun calls. The performance measures of interest are the rate of outboun calls an the waiting time of inboun calls. Through simulation experiments they prove the efficiency of their Markov chain moel to reflect reality. Brant an Brant (999) evelop an approximation metho to evaluate the performance of a call center moel with impatient inboun calls an infinitely patient outboun calls of lower priority than the inboun traffic. Bhulai an Koole (2003) consier a similar moel to the one analyze in this paper, except that the call service is one in a single stage without a possible break. The moel consists of inboun an outboun jobs where the inboun jobs have a non-preemptive priority over the outboun ones. For the special case of ientically istribute service times for the two jobs, they optimize the outboun jobs routing subject to a constraint on the expecte waiting time of inboun jobs. Gans an Zhou (2003) stuy a call center with two job types where one of the jobs is an infinitely backlogge queue. They evelop a routing policy consisting in the reservation of servers in orer to maximize the expecte throughput on the jobs of the infinitely backlogge queue. Armony an Maglaras (2004) an Legros et al. (206) analyze a similar moel with a call-back option for incoming customers. The customer behavior is capture through a probabilistic choice moel. 5

6 Other references inclue (Bernett et al., 2002; Keblis an Chen, 2006; Pichitlamken et al., 2003). The analysis in this paper is also relate to the analysis of queueing systems with phase-type service time istributions. We moel the call service time through three successive exponentially istribute stages, where the secon stage may also overlap with the service of one or several outboun jobs with an exponential time uration for each. The performance evaluation of such systems involves the steay-state analysis of Markov chains an is usually aresse using numerical methos. We refer the reaer to Kleinrock (975) for simple moels with Erlang service time istributions. For more complex systems, the reaer is referee to Bolotin (994); Brown et al. (2005); Guo an Zipkin (2008). Our approach to erive the performance measures is base on first eriving the stationary system state probabilities for two-imension an semi-infinite continuous time Markov chains. One may fin in the literature three methos for solving such moels. The first one is to truncate the state space, see for example Seelen (986) an Keilson et al. (987). The secon metho is calle spectral expansion (Daigle an Lucantoni, 99; Mitrani an Chakka, 995; Chouhury et al., 995). It is base on expressing the invariant vector of the process in terms of the eigenvalues an the eigenvectors of a matrix polynomial. The thir one is the matrixgeometric metho (Neuts, 995). The approach relies on etermining the minimal positive solution of a non-linear matrix equation. The invariant vector is then expresse in terms of powers of itself. In our analysis, we reuce the problem to solving cubic an quartic equations, for which we use the metho of Caran an Ferrari (Gouron, 994). Finally, we briefly mention some stuies on human multi-tasking, as it is the case for the agents in our setting. Glastones et al. (989) show that a simultaneous treatment of jobs is not efficient even with two easy jobs because of the possible interferences. In our moels, we are not consiering simultaneous tasks in the sense that an agent cannot talk to a customer an at the same time treats an outboun job. More interestingly, Charron an Koechlin (200) stuie the capacity of the frontal lobe to eal with ifferent tasks by alternation (as here for calls an outboun jobs). They evelop the notion of branching: capacity of the brain to remember information while oing something else. They show that the number of jobs one alternatively has to be limite to two to avoi loss of information. Dux et al. (2009) showe that training an experience can improve multi-tasking performance. The risk from alternating between two tasks is the loss of efficiency because of switching times. An important aspect to avoi inefficiency as pointe out by Dux et al. (2009) an Charron an Koechlin (200) is that the alternation shoul be at most between two tasks quite ifferent in nature (like inboun an outboun jobs). 6

7 3 Problem Description an Moeling We consier a call center moeling with s ientical agents an two types of jobs: inboun calls an outboun jobs. The arrival process of inboun calls is assume to be Poisson with mean arrival rate λ. There is an infinite amount of outboun jobs ( s, back-office tasks, etc.) that are waiting to be treate in a eicate first come, first serve (FCFS) queue with infinite capacity. We consier call center applications where the communication between the agent an the customer inclues a break (the customer oes not nee the agent availability). We moel the service time of a call by three successive stages. The first stage is a conversation between the two parties. The secon stage is the break, i.e., no interactions between the two parties. The thir an final step is again a conversation between the two parties. The service completion occurs as soon as the thir stage finishes. We moel each stage uration as an exponentially istribute ranom variable, with rate µ i for stage i. The urations of the three stages are jointly inepenent. This Markovian assumption, which is common in moeling in service operations, is reasonable for systems with high service time variability where service times are typically small but there are occasionally long service times. An agent hanles an outboun job within one single step without interruption. The time uration of an outboun job treatment is ranom an assume to be exponentially istribute with rate µ 0. Moreover, we assume the service urations are not known by the system manager before realization. So, the routing ecisions cannot be base on such information. This is a common assumption for call centers. For other applications, like packet elivery on internet, the ranom variables may be realize before service. Examples of relate stuies inclue Zhang (995); Le Bouec (998); Gevros et al. (200). The queueing moel is epicte in Figure 2., λ µ µ 2 µ 3 µ 0 Figure 2: Queueing moel Bhulai an Koole (2003) an Gans an Zhou (2003) consier a similar moel with inboun customers who arrive accoring to a homogeneous Poisson process an an infinite amount of outboun jobs, exactly like in our moel. Their objective is to maximize the throughput of serve outbouns with a constraint on the waiting time of inbouns. They show, using a Markov ecision process approach, that a strict priority shoul be given for inboun calls. Using their results, we also assume the strict priority of inboun calls. More precisely, upon arrival, a call is immeiately 7

8 hanle by an available agent, if any. If not, the call waits for service in an infinite FCFS eicate queue. Inboun calls have a non-preemptive priority over outboun jobs. Non-preemption is a natural assumption for our application since outboun jobs coul be for example outboun calls. We are intereste in an efficient use of the agent time between inboun calls an outboun jobs. More concretely, we want to answer the question when shoul we treat outboun jobs for the following optimization problem { Maximize the expecte throughput of outboun jobs subject to a service level constraint on the call waiting time in the queue. () We numerically aress Problem () using a Markov Decision Process (MDP) approach. As shown in Section of the online appenix, the optimal policy is very complex. It epens on six parameters: the number of inboun calls waiting in the queue, the number of agents working on each stage of service, the number of outboun jobs being in service between calls an the number of outboun jobs being in service insie the break. This makes the optimal policy har to obtain in general an then har to implement in practice. We instea propose a simpler moel for the routing of outboun jobs to agents. It is referre to as probabilistic moel or Moel PM an is escribe below. Although this moel is not optimal, we numerically show its efficiency through a comparison with the optimal policy (see Section of the online appenix). It is moreover easy to implement an to unerstan by a system manager. Probabilistic Moel (Moel PM): We istinguish the two situations when an agent is available to hanle outboun jobs between two call conversations, or insie a call conversation. Between two calls: just after a call service completion (as soon as the thir stage finishes) an no waiting calls are in the queue, the agent treats one or more outboun jobs with probability p (inepenently of any other event), or oes not work on outboun jobs at all with probability p. In the latter case, the agent simply remains ile an waits for a new call arrival to hanle it. In the former case (with probability p), she selects a first outboun job to work on. After finishing the treatment of this outboun job, there are two cases: either a new call has alreay arrive an it is now waiting in the queue, or the queue of calls is still empty. If a call has arrive, the agent hanles that call. If not, she selects another outboun job, an so on. At some point in time, a new call woul arrive while the agent is working on an outboun job. The agent will then hanle the call as soon as she finishes the outboun job treatment. Insie a call: Just after the en of the first stage of an unerway call service (regarless whether there are other waiting calls in the queue or not), the agent treats one or more outboun jobs with probability q (inepenently of any other event), or oes not work on outboun jobs at all with probability q. In the latter case, the agent simply remains ile an waits for the currently serve 8

9 customer to finish her operations on her own (corresponing to the secon call service stage, i.e., the agent break). As soon as the customer finishes by herself her secon service stage, the agent starts the thir an last service stage. In the former case (with probability q), she selects a first outboun job to work on. After finishing the treatment of this outboun job, there are two cases: either the currently serve customer has alreay finishe her secon service stage, or not yet. If she oes, the agent starts the thir stage of the customer call service. If not, she selects another outboun job, an so on. At some point in time, the currently serve call woul finish her secon service while the agent is working on an outboun job. The agent will then hanle the call as soon as she finishes the the outboun job treatment. Moel PM is epicte in Figure 3. µ µ 2 µ 3 µ µ 2 µ 3 µ µ 2 µ 3 Agent works on s, with proba p Agent works on s, with proba q Figure 3: Moel PM The ecision to initiate an outboun job is only taken at an inboun service completion (control parameter p) or at the completion of the first phase of service of an inboun (control parameter q). After each outboun job service completion, a new outboun service is automatically initiate. One may think of moifying Moel PM by allowing to switch an stop serving outboun jobs. Actually, ue to the exponential assumptions for the inter-arrival, the secon phase of service an the outboun service times, Moel PM is equivalent to a moel where a new ecision may be taken upon each outboun service completion. The proof of the result is given in Section 2 of the online appenix. We further consier next four particular cases of Moel PM as shown in Table. Although these moels might appear to be too restrictive to solve Problem (), we show later their merit in Section when we focus on the optimization of p an q in Moel PM. Moreover, they have the avantage of being easy to implement in practice, easy to unerstan by managers, an easy to follow by agents. Note that in Moel, the expecte throughput of outboun jobs is zero. The interest from Moel is in the extreme case of a very high workloa of calls or a very restrictive constraint on the call waiting time. The objective for the system manager is to fin the optimal values for the control parameters p an q or to etermine among the particular cases of Moel PM which one woul better answer Problem (). The knowlege of the system parameters is essential to obtain implementable results. Section 3 of the online appenix is evote to the estimation of these parameters base on real ata. 9

10 Moel Moel Moel 2 Moel 3 Moel 4 Table : Particular cases of Moel PM Description p = q = 0, no treatment of outboun jobs p = an q = 0, systematic treatment of outboun jobs only between two calls p = 0 an q =, systematic treatment of outboun jobs only uring the break p = q =, systematic treatment of outboun jobs between two calls an uring the break 4 Single Server Analysis In this section, we provie an exact metho to characterize the call waiting time in the queue an the outboun job expecte throughput for Moel PM an its extreme cases for a single server moel. We also evelop various structural results for the optimization problem. The tractable analysis in this section enhances our unerstaning of the system behavior. This woul not be possible to o irectly for the multi-server case since an exact analysis is very complex. However, we exten the analysis to multi-server case in Section 5 using light an heavy-traffic approximations. Our approach consists of using a Markov chain moel to escribe the system states an compute their steay-state probabilities. The computation of some of the steay-state probabilities involves the resolution of cubic (thir egree) or quartic (fourth egree) equations for which we use the Caran-Ferrari metho. 4. Performance Evaluation Let us efine the ranom process {(x(t), y(t)), t 0} where x(t) an y(t) enote the state of the agent an the number of waiting calls in the queue at a given time t 0, respectively. We have y(t) {0,, 2,...}, for t 0. The possible values of x(t) (corresponing to the possible states of the agent), for t 0, are - Agent working on the first stage of a call service enote by x(t) = A, - Ile agent that is waiting for the call to finish her secon stage of service enote by x(t) = B, - Agent working on an outboun job while an unerway call has alreay finishe her secon stage of service an is waiting for the agent to start her thir stage of service enote by x(t) = B, - Agent working on the thir stage of a call service enote by x(t) = C, - Agent working on an outboun job between two call conversations enote by x(t) = M, - Agent ile between two call conversations enote by x(t) = 0. Since call inter-arrival times, call service times in each stage, an outboun job service times are exponentially istribute, {(x(t), y(t)), t 0} is a Markov chain (Figure 4). For ease of exposition, we enote by P 0 the probability to be in state (0, 0), an for n 0 we enote by a n, b n, b n, c n an m n the probabilities to be in state (A, n), (B, n), (B, n), (C, n) an (M, n), respectively. We also efine ρ i = λ µ i, for i {0,, 2, 3}. In Proposition, we give the 0

11 C, 0 C, C, n C, n+ ( q ) µ 2 µ 0 B, 0 B, B, n B, n+ pµ 3 ( p ) µ 3 qµ 2 B, 0 µ 3 B, B, n B, n+ µ 0, 0 λ A, 0 A, A, n A, n+ µ 0 M, 0 λ M, M, 2 M,n+ M,n+2 Figure 4: Markov chain for Moel PM probability of elay of a call (probability of waiting) enote by P D an the expecte throughput of outboun jobs enote by T. Note that the stability conition of Moel PM is λ < Proposition For Moel PM, we have P D = p + pρ 0 ( ρ ρ 2 qρ 0 ρ 3 ), T = µ 0 ( + ρ0 + pρ 0 p ( ρ ρ 2 qρ 0 ρ 3 ) + q(ρ 2 + ρ 0 ) Proof. From the Markov chain of Moel PM, we have ). q µ 0 µ µ 2 µ 3 c 0 = ρ 3 (P 0 + m 0 ), c n = ρ 3 (a n + b n + b n + c n + m n ), for n. Then ( c n = ρ 3 {P 0 + m 0 + an + b n + b ) } n + c n + m n. (2) n=0 n=0 n= Since all system state probabilities sum up to, i.e., P 0 + n=0 (a n + b n + b n + c n + m n ) =,

12 Equation (2) becomes c n = ρ 3. (3) n=0 For the state (M, 0), we have pµ 3 c 0 = λm 0, or equivalently c 0 = ρ 3 m 0 p. Therefore c 0 = ρ 3 P 0 p. We then may write P 0 = p p m 0. (4) Combining Equation (3) an the relations µ 2 n=0 b n = µ 3 n=0 c n = µ 0 n=0 b n+( q)µ 2 n=0 b n = µ n=0 a n, we obtain b n = ρ 2, n=0 b n = qρ 0, a n = ρ. (5) n=0 n=0 For state (M, n), n, we have m n = ( ρ 0 +ρ 0 ) n m 0. Therefore m i = m 0 ( + ρ 0 ). Using now Equation (5) together with the normalization conition implies m 0 = Equation (4) then becomes P 0 = i=0 p + pρ 0 ( ρ ρ 2 qρ 0 ρ 3 )). p +pρ 0 ( ρ ρ 2 ρ 3 qρ 0 ). A new call enters service immeiately upon arrival, if an only if the system is in state (0, 0). Since the call arrival process is Poison, we use the PASTA property to state that the steay-state probabilities seen by a new call arrival coincie with those seen at an arbitrary instant. P D = P 0, which leas to the expression of P D. As for the outboun job expecte throughput, it is given by T = µ 0 (q i=0 b i + i=0 b i + ( ) i=0 m i), which may be also written as T = µ +ρ0 0 +pρ 0 p ( ρ ρ 2 ρ 3 qρ 0 ) + q(ρ 2 + ρ 0 ). This finishes the proof of the proposition. Thus Let us now efine W, a ranom variable, as the steay-state call waiting time in the queue, an P(W t) as its cumulative istribution function (cf) for t 0. Conitioning on a state seen by a new call arrival an averaging over all possibilities, we state using PASTA that P(W t) = P n=0 (P(W t (A, n)) a n + P(W t (B, n)) b n + P(W t (B, n)) b n) + P(W t (C, n)) c n + P(W t (M, n)) m n ). (6) For n 0, the quantities P(W t (A, n)), P(W t (B, n)), P(W t (B, n)), P(W t (C, n)) 2

13 an P(W t (M, n)) are the cf of the conitional call waiting times in the queue, given that a new arriving call fins the system in states (A, n), (B, n), (B, n), (C, n) an (M, n), respectively. In the Markov chain of Moel PM, these conitional ranom variables correspon to first passage times to state (0, 0) starting from the system state upon a new call arrival. They are convolutions of inepenent exponential ranom variables with arbitrarily rates, not necessarily all equal or all istinct. Using the expressions of the cf of an hypoexponential istribution (Amari an Misra (997), Legros an Jouini (205)), we can explicitly erive the expressions of P(W t (A, n)), P(W t (B, n)), P(W t (B, n)), P(W t (C, n)) an P(W t (M, n)), for n 0, as shown in Section 4 of the online appenix. It remains now to compute the probabilities a n, b n, b n, c n an m n in n, for n 0. One can compute them using the well-known matrix geometric solution approach (Neuts, 995). However, the numerical computation is not exact, because the minimal non-negative solution to the matrix quaratic equation is compute with a given error. In what follows, we instea use the Caran- Ferrari metho to solve the moment generating function an fin the roots, which leas to an exact numerical computation. From the Markov chain of Moel PM, we can write the following iterative equations λx n = GX n, (7) for n, where X n = a n b n b n c n m n, for n 0 is the vector of probabilities to be compute an µ λ λ λ λ µ λ + µ G = 0 qµ 2 λ + µ ( q)µ 2 µ 0 λ + µ λ + µ 0 The first step to solve Equation (7) is to fin the eigenvalues of the matrix λg. These are solutions of the equation et( λ G yi) = 0 with y as variable. One obvious eigenvalue is + ρ 0 (see the last line of G), an the remaining ones are those of a 4 4 matrix (erive from λ G by removing the last line an the last column) an they are solutions of the following quartic equation σ 4 y 4 (3σ 4 + σ 3 )y 3 + (3σ 4 + 2σ 3 + σ 2 )y 2 (σ 4 + σ 3 + σ 2 + σ )y + + ρ 0 ( q) = 0, (8) 3

14 with y as variable, σ = ρ 0 + ρ + ρ 2 + ρ 3, σ 2 = ρ 0 ρ + ρ 0 ρ 2 + ρ 0 ρ 3 + ρ ρ 2 + ρ ρ 3 + ρ 2 ρ 3, σ 3 = ρ 0 ρ ρ 2 + ρ 0 ρ ρ 3 + ρ 0 ρ 2 ρ 3 + ρ ρ 2 ρ 3, an σ 4 = ρ 0 ρ ρ 2 ρ 3. Since the constant term + ρ 0 ( q) in Equation (8) is strictly positive, zero cannot be a solution of that equation. Then, λg is invertible. Therefore the eigenvalues of λg are solutions of ( + ρ 0 ( q))x 4 (σ 4 + σ 3 + σ 2 + σ )x 3 + (3σ 4 + 2σ 3 + σ 2 )x 2 (3σ 4 + σ 3 )x + σ 4 = 0, (9) where x = y. We solve the quartic Equation (9) using the Caran-Ferrari metho. In Section 5 of the online appenix, we escribe the etails of this metho. The explicit expressions of the probability components of the vector X n, for n 0, can be erive, however they are too cumbersome for Moel PM. We go further in proviing their expressions for the extreme cases of Moel PM in Section 6 of the online appenix, an also using a light-traffic approximation in Section 7 of the online appenix. In all cases, an exact numerical metho is straightforwar an easy to implement. Numerical illustrations are shown later in Section 4.2. Let us now compute the expecte call waiting time in Moel PM, enote by E(W ). Consier first a moel similar to Moel PM except that outboun jobs can only be treate insie a call conversation. We enote this moel by Moel PM, an its call expecte waiting time by E(W ). With a little thought, one can see that the expecte call waiting time in Moel PM is that of Moel PM plus p µ 0. The reason is mainly relate to the memoryless property of outboun job service times. This result is proven in Lemma. Lemma The expecte waiting time in PM is elaye by p µ 0 compare to that in PM, for p [0, ]. Proof. See Section 8 of the online appenix. Note that the proof of Lemma also hols for all inepenent an ientically generally istribute call service times. Let us now compute the expecte waiting time in Moel PM, E(W ). We use the Pollaczeck-Kinchin result for an M/G/ queue. From Pollaczeck (930), we have E(W ) = ρ2 (+c 2 v) 2λ( ρ), where c v is the coefficient of variation of the service istribution (stanar eviation over expecte value) an ρ is the server utilization (expecte arrival rate over expecte service rate). Because of the possibility to o outboun jobs between calls, the ranom variable measuring the service time uration, say S, can be written as S = S + S 2 + US 0 + S 3, where S i, a ranom variable, follows an exponential istribution with rate µ i, for i = 0,..., 3, an U follows a Bernoulli istribution with parameter q. We enote by E(Z) an V (Z) the expecte value (first moment) an the variance of a given ranom variable Z, respectively. The first moment of S is given by E(S) = µ + µ 2 + q µ 0 + µ 3, 4

15 an its variance can be written as (using the inepenence between S i an S j for i j {0,..., 3}) V (S) = V (S ) + V (S 2 ) + V (US 0 ) + V (S 3 ) = µ 2 + µ q q2 µ µ 2. 3 Then c 2 v(s) = + + 2q q2 + µ 2 µ 2 2 µ 2 0 µ 2 3 ( µ + µ2 + q µ 0 + µ 3 ) 2. After some algebra, we obtain which leas to E(W ) = (ρ + ρ 2 + ρ 3 ) 2 + ρ 2 + ρ2 2 + ρ qρ 0(ρ 0 + ρ + ρ 2 + ρ 3 ), 2λ( (ρ + ρ 2 + qρ 0 + ρ 3 )) E(W ) = p + (ρ + ρ 2 + ρ 3 ) 2 + ρ 2 + ρ2 2 + ρ qρ 0(ρ 0 + ρ + ρ 2 + ρ 3 ). (0) µ 0 2λ( (ρ + ρ 2 + qρ 0 + ρ 3 )) This closes the performance measure analysis of Moel PM. Note that for the waiting time of inboun calls, we ignore the waiting time they may have before starting stage 3. We expect that the waiting experience is more comfortable before stage 3 than before starting service. The customer sensitivity to uncertain waiting shoul ecrease after being connecte to an agent, because she woul consier such waiting time as a part of her service time. For instance, one coul expect that abanonments woul be higher while waiting before stage than before stage 3. For the rest of the paper, waiting before stage 3 is ignore. We however provie in Section 9 of the online appenix the etails to characterize the istribution of the whole waiting time incluing that before stage 3. For the four extreme cases of Moel PM (Moels,...,4), we erive the expressions of the outboun job expecte throughput, the call probability of elay, an the call expecte waiting time, we simply apply the previous analysis an state the results as shown in Table Comparison Analysis an Insights We start in Section 4.2. by a comparison analysis between the extreme cases Moels,...,4. The comparison is base on the optimization problem (). We erive various structural results an properties for this comparison. In particular, we investigate the impact of the mean arrival rate intensity of calls on the comparison between Moels,...,4. One coul think of a call center manager that ajusts the job routing schema as a function of the call arriving workloa over the ay. In Section we focus on the general case Moel PM. We prove that the optimization 5

16 Table 2: Expressions of T, E(W ) an P D for Moels,...,4 Moel Moel 2 T 0 µ 0 ( ρ ρ 2 ρ 3 ) E(W ) (ρ +ρ 2 +ρ 3 ) 2 (+ ρ2 +ρ2 2 +ρ2 3 (ρ +ρ 2 +ρ 3 ) 2 ) 2λ( ρ ρ 2 ρ 3 ) µ 0 + E(W ) P D ρ + ρ 2 + ρ 3 Moel 3 Moel 4 T µ 0 (ρ 0 + ρ 2 ) µ 0 ( ρ ρ 3 ) E(W ) (ρ +ρ 2 +ρ 3 +ρ 0 ) 2 (+ ρ2 +ρ2 2 +ρ2 3 +ρ2 0 (ρ +ρ 2 +ρ 3 +ρ 0 ) 2 ) 2λ( ρ ρ 2 ρ 3 ρ 0 ) µ 0 + E(W 3 ) P D ρ 0 + ρ + ρ 2 + ρ 3 of the parameters of Moel PM lea to extreme situations in the sense of a systematic outboun job treatment of outboun jobs either between calls or insie a call conversation, which gives an interest in practice for Moels,..., Comparison between the Extreme Cases We first compare between Moels,...,4 base on their performance in terms of the outboun job expecte throughput, enote by T,..., T 4, respectively. It is obvious that Moel 4 is the best an Moel is the worst (no outboun jobs at all). Let us now compare between Moels 2 an 3. From Table 2 we have T 2 = µ 0 ( ρ ρ 2 ρ 3 ) an T 3 = µ 0 (ρ 0 + ρ 2 ). Thus T 3 > T 2 is equivalent to λ > µ +, µ 2 where µ = µ 0 + µ + µ 2 + µ 3. Since the stability conition for Moel 3 is λ < µ, Moel 3 is better than Moel 2 if µ + < λ < µ. () µ 2 Denoting the left term in Inequality () by R, the conition uner which T 3 > T 2 is then R = µ 0 + µ + µ < λ < µ. (2) µ 2 From Inequality (2), we first see that treating outboun jobs only insie a call conversation (Moel 3) becomes better than treating them only between calls (Moel 2) is likely the case for high arrival workloas (in such a case, ile perio urations are reuce). We also see that R µ 2 > 0 for µ 2 > 0, R µ 0 > 0 for µ 0 > 0, R µ > 0 for µ > 0, an R µ 3 > 0 for µ 3 > 0. This means that 6

17 ecreasing the expecte uration of the call service secon stage (/µ 2 ) relative to the expecte urations of the other call service stages or the outboun job service uration (/µ, /µ 3 an /µ 0 ) increases the range of arrival workloas where it is preferre to use Moel 2 instea of Moel 3. In other wors, there is no sufficient time to treat outboun jobs insie the call conversation. Comparison with a Constraint on E(W ) As a function of the mean call arrival rate, we want to answer the question when shoul we treat outboun jobs (which moel among Moels to 4 shoul a manger choose?) for the following problem { Maximize T subject to E(W ) w, (3) where w is the service level for the expecte waiting time, w > 0. Let W i, a ranom variable, enote the expecte call waiting time in Moel i, i =,..., 4. It is clear that for some perios of a working ay with a very high call arrival rate λ, the manager is likely to choose Moel (no outboun jobs), an for other perios with a very low λ, she is likely to choose Moel 4 (outboun jobs between calls an insie a call). However for intermeiate values of λ, the optimal choice is not clear. This is what we analytically analyze in what follows. Uner the conition of stability of Moel i, E(W i ) is continuous an strictly increasing in λ (see Table ), for i =,..., 4. The constraint E(W i ) w is then equivalent to λ λ i, for i =,..., 4, where λ = λ 2 = λ 3 = λ 4 = 2w ( 3 ) ( 3 ) 2, (4) 2w µ i + µ i + 3 µ 2 i= i= i= i ) 2 (w µ0 ) 2 (w ( ) ( 3 3 ) 2, µ0 µ i + µ i + 3 i= i= µ 2 i= i 2w ( 3 ) ( 3 ) 2, 2w µ i + µ i + 3 µ 2 i=0 i=0 i=0 i ) 2 (w µ0 ) 2 (w ( ) ( 3 3 ) 2 µ0 µ i + µ i + 3 i=0 i=0 µ 2 i=0 i. For a given λ an uner the conition of stability of Moel i (i =,..., 4), the choice of Moel i happens if λ λ i an T i = max j {,...,4}, λ λj (T j ). When λ λ 4, the choice is obviously for Moel 4. When λ λ an λ > λ i for i = 2, 3, 4 the only possibility is Moel. Proposition 2 provies 7

18 the conitions uner which an optimal choice of Moel 2 or Moel 3 may happen. Proposition 2 The following hols:. For λ < λ 2 > µ µ 2 2. For λ < if µ 0 µ µ 2 µ 3, there exist some values of λ for which Moel 2 is optimal if an only if µ 3, there exist some values of λ for which Moel 3 is optimal if an only R λ 3 or λ 2 < λ We have λ 2 < λ 3 if an only if µ 0 < w < w, where w = 4 2 µ ( + + ) 4 µ 3 µ 2 µ µ µ 2 i= i i,j=;i j µ i µ j ( + + ). µ 3 µ 2 µ Proof. See Section 0 of the online appenix. Using Equation (4), the conition in the first statement of Proposition 2 may be rewritten as w > µ 0 or w < µ 0 ( 3 ) µ i= i i= µ 2 i ( 3 ). 2 µ i= i (5) The secon inequality in Relation (5) implies w < µ 0. Since at least the expecte waiting time in Moel 2 is strictly higher than µ 0 (any new call has at least to wait for the resiual time of an outboun job treatment), this secon inequality is impossible. Roughly speaking, the conition for the optimality of Moel 2 (for some values of λ) hols when the service level on the call waiting is higher than the expecte outboun job service time. In what follows, we numerically illustrate the analysis above. For various system parameters, Figure 5 gives the optimal moel choice as a function of the mean arrival rate of calls, λ. An intuitive reasoning of a manager woul choose the orering Moel 4 (outboun jobs between calls an insie a call), then 2 (outboun jobs only between calls), then 3 (outboun jobs only insie a call), then (no outboun jobs) as λ increases. 8

19 DDDD DDD t t (a) µ 0 = 2, µ = µ 3 =, µ 2 = 3, w = (b) µ i = 2 for i = 0,..., 3, w = e e DD D t D e e e e DD DD t D (c) µ 0 =, µ = µ 3 = 5, µ 2 = 0.8, w = 5 () µ 0 = 0.8, µ = µ 3 = 0, µ 2 = 0.5, w = 5 Figure 5: Comparison between Moels to 4 with a constraint on E(W ) The orering Moel 2 then Moel 3 is not always appropriate, an some situations may require to consier some counterintuitive orering. For instance, Moel 3 is better than Moel 2 for small values of λ if R λ 4 an λ 3 < λ 2, see Figure 5(c). In other wors, this happens when the constraint on E(W ) is not too restrictive an when the expecte secon stage service uration is long. Another more surprising orering, as λ increases, is Moel 2, then Moel 3, then again Moel 2 (see Figure 5()) which happens for system parameters such that λ 4 < R < λ 3 < λ 2. Comparison with a Constraint on P (W < A W T ) In the constraint of Problem (), we want that the probability that a call waits less than a given threshol, efine as A W T is at least a given service level, efine as SL, i.e., P(W < A W T ) SL. Note that a special case of this constraint is that on P D, the call probability of waiting. The expressions involve in the analysis of P (W < A W T ) are quite complicate to allow an analytical comparison between the moels as we have one for a constraint on E(W ). We have then conucte a numerical comparison analysis (not totally illustrate here). The main qualitative conclusions are similar to those for the case of a constraint on E(W ). As λ increases, it is not always true as one woul intuitively expect that a manager shoul choose first Moel 2 an then at some point of λ she shifts to Moel 3 (Figure 6(a)). The optimal choice may change with the system parameters 9

20 an we may have the orering Moel 3 then Moel 2 (Figure 6(b)). DDD D Wtm e e DDD D Wtm (a) µ i = 2 for i = 0,..., 3 an α = 80% (b) µ 0 =, µ = 5 = µ 3, µ 2 = 0.8 an α = 20% Figure 6: Comparison between Moels to 4 with the constraint on P(W < )) α Optimization of Moel PM In this section we focus on the general case, Moel PM. We are intereste in the optimization of the parameters p an q in Moel PM for Problem (). Concretely, we want to fin the optimal routing parameters of Moel PM that allows the manager to maximize the outboun job expecte throughput while respecting a call service level constraint. Recall that the stability conition of Moel PM is λ < q µ 0 µ µ 2 µ 3 The expression of the outboun job expecte throughput T for Moel PM is given in Proposition. It is straightforwar to prove that for p, q [0, ] the maximum of T (best situation) is reache for p = q =. The proof is then omitte. Also, the expecte call waiting time of Moel PM (given in Equation (0)) is maximize (worst) for p = q =. Therefore in orer to solve Problem (), one woul be intereste analyzing the sensitivity of T with respect to p an q. In Lemma 2 we prove a sensitivity result for T. The result will be use later in our analysis. Lemma 2 We have T p > T q if an only if 0 < ρ 0 < ρ 0, where ρ 0 = (p q ρ ρ 2 ρ 3 ) 2 4(p 2 p q)( ρ ρ 2 ρ 3 ) q + p (ρ + ρ 2 + ρ 3 ) 2(q p 2. + p) Proof. We want to solve the following inequality in ρ 0 : T p = µ ( ρ ρ 2 qρ 0 ρ 3 )( + ρ 0 ) 0 ( + pρ 0 ) 2 > T q = µ ρ 0 ( p) 0. + pρ 0 This is equivalent to ( ρ ρ 2 ρ 3 qρ 0 ) ( + ρ 0 ) ρ 0 ( p) ( + pρ 0 ) > 0, or also (p 2 p q)ρ (p q ρ ρ 2 ρ 3 )ρ 0 + ρ ρ 2 ρ 3 > 0. (6) The iscriminant for this inequality is = (p q ρ ρ 2 ρ 3 ) 2 4(p 2 p q)( ρ ρ 2 ρ 3 ) > 0. 20

21 Equation (6) has then the two following solutions: (p q ρ ρ 2 ρ 3 ) 2 4(p 2 p q)( ρ ρ 2 ρ 3 ) q + p (ρ + ρ 2 + ρ 3 ) 2(q p 2 + p) an (p q ρ ρ 2 ρ 3 ) 2 4(p 2 p q)( ρ ρ 2 ρ 3 ) + q p + ρ + ρ 2 + ρ 3 2(q p 2. + p) Since the first solution is positive (enote by ρ 0 ) an the secon one is negative, ρ 0 [0; ρ 0 ), which finishes the proof of the lemma. In what follows we aress the question: starting from p = q =, in which irection shoul we ecrease T? Shoul we ecrease p or q first? ( (ρ ) For p = q =, we have ρ 0 = 2 + ρ 2 + ρ 3 ) 2 + 4( ρ ρ 2 ρ 3 ) (ρ + ρ 2 + ρ 3 ). Let us now prove (for p = q = ) that ρ 0 > ρ 0. From the one han, proving ρ 0 > ρ 0 is equivalent to proving (ρ + ρ 2 + ρ 3 ) 2 + 4( ρ ρ 2 ρ 3 ) > 2ρ 0 + (ρ + ρ 2 + ρ 3 ) or equivalently ρ ρ 0(ρ + ρ 2 + ρ 3 ) ( (ρ + ρ 2 + ρ 3 )) < 0 or also (ρ 0 + )(ρ 0 ( (ρ + ρ 2 + ρ 3 )) < 0. From the other han, we have ρ 0 + > 0. Also, the stability conition of Moel 4 (Moel PM with p = q = ) is ρ 0 + ρ + ρ 2 + ρ 3 <. Then ρ 0 < (ρ + ρ 2 + ρ 3 ). As a conclusion the inequality ρ 0 > ρ 0 is true, for p = q =. T p > T q Using Lemma 2, this means that starting from p = q =, we have > 0. As a consequence, when we nee to moify the values of p an q in orer to ecrease the expecte call waiting time (the constraint in Problem ()), the maximum of T is guarantee by first ecreasing q (the outboun job expecte throughput is less sensitive to the variation of q than that of p). The question now is: which irection to use next? In other wors when p = an some value of q such that 0 < q <, is it possible that it is better to ecrease p instea of q? The answer is no an the proof is as follows. For p =, let us try to fin a value of q for ( q ρ ρ which we have ρ 0 ρ 0. This is equivalent to 2 ρ 3 ) 2 +4q( ρ ρ 2 ρ 3 ) q+ ρ ρ 2 ρ 3 2q ρ 0. Thus, q 2 ρ qρ 0 ( ρ ρ 2 ρ 3 )( + ρ 0 ) > 0. This trinomial in q has two real solutions; q = + 4ρ (ρ +ρ 2 +ρ 3 )(ρ 0 +) 2ρ 0 an q 2 = + 4ρ (ρ +ρ 2 +ρ 3 )(ρ 0 +). Obviously q < 0. We also have q 2 > because: proving q 2 > 0 is equivalent to proving ρ (ρ + ρ 2 + ρ 3 )ρ 0 + > 0. The iscriminant of this latter trinomial in ρ 0 is negative an it is equal to (ρ + ρ 2 + ρ 3 ) 2 4. So q 2 > for any ρ 0 > 0. Therefore it is impossible to fin a value of q between 0 an for which 0 < T p < T q. In conclusion starting from p = q =, when we nee to change the values of p an q, the best irection to maximize T is to first ecrease q until q = 0 an only then start to ecrease p from p =. Consier now Problem () with a constraint on E(W ). 2ρ 0 From the one han, the constraint 2

22 E(W ) w implies p + (ρ + ρ 2 + ρ 3 ) 2 + ρ 2 + ρ2 2 + ρ qρ 0(ρ 0 + ρ + ρ 2 + ρ 3 ) w, µ 0 2λ( (ρ + ρ 2 + qρ 0 + ρ 3 )) for p, q [0, ], or equivalently q 2λ( ρ ρ 2 ρ 3 )(w p/µ 0 ) (ρ + ρ 2 + ρ 3 ) 2 ρ 2 ρ2 2 ρ2 3 2ρ 0 (ρ 0 + ρ + ρ 2 + ρ 3 + λ(w, (7) p/µ 0 )) for p, q [0, ]. On the other han, the conition in Lemma 2, 0 < ρ 0 < ρ 0, is equivalent to q < (ρ + ρ 2 + ρ 3 )( + ρ 0 ) ρ 0 ( + ρ 0 ) + ρ 0 + ρ 0 p + ρ 0 + ρ 0 p 2, (8) for p, q [0, ]. Let us enote the right han sies of Inequalities (7) an (8) by the functions in p [0, ] f(p) an g(p), respectively. Illustrations of these functions are given in Figure 7. l l el el el el el el el el l l l l l l l l l l l l l l el el l (a) λ = 0.5, µ 0 = µ = µ 2 = µ 3 = 2, w = 0 l l l l el el l (b) λ = 0.22, µ 0 = µ = µ 2 = µ 3 = 2, w = l el el el el l l l l l l l l l el el l (c) λ = 0., µ 0 = 0., µ = µ 2 = µ 3 = 2, w = 5 Figure 7: Behavior of f(p) an g(p) In what follows we prove an interesting result on the optimal values of p an q. Consier for example Figure 7(a) an assume that the agent is in a situation such that (p, q) belongs to Zone or 2. Then the constraint on E(W ) is respecte, but T can be improve. Using Lemma 2, we shoul increase p first (q first) for Zone (Zone 2). From Figure 7(a), we also see that we shoul 22

23 ecrease p first (q first) for Zone 3 (Zone 4). It is clear that the optimal couple (p, q) will be on the curve of f. Moreover, we prove in Theorem that the optimal point is such that p {0, } or q {0, }. Theorem For p, q [0, ], the optimal values of p an q of the optimization problem { Maximize T subject to E(W ) w, (9) are always extreme values (0 or ) for at least p or q. Proof. We want to maximize the outboun job expecte throughput T (p, q) while respecting a constraint on the expecte call waiting time (E(W )(p, q) w ). We istinguish the following cases. Case : λw < ) (ρ +ρ 2 +ρ 3 ) (+ 2 ρ2 +ρ2 2 +ρ2 3 (ρ +ρ 2 +ρ 3 ) 2 2( ρ ρ 2 ρ 3 ) time cannot be met an Problem () has no solution. Case 2: λw ρ 0 + ) (ρ +ρ 2 +ρ 3 +ρ 0 ) (+ 2 ρ2 +ρ2 2 +ρ2 3 +ρ2 0 (ρ +ρ 2 +ρ 3 +ρ 0 ) 2 2( ρ ρ 2 ρ 3 ρ 0 ). In this case the constraint on the expecte waiting. Since T (p, q) increasing in both p an in q an the constraint on E(W ) is met for p = q =, the optimal values for the control parameters are p = q = (Moel 4). ) (ρ +ρ 2 +ρ 3 ) (+ 2 ρ2 +ρ2 2 +ρ2 3 (ρ +ρ 2 +ρ 3 ) 2 ) (ρ +ρ 2 +ρ 3 +ρ 0 ) (+ 2 ρ2 +ρ2 2 +ρ2 3 +ρ2 0 (ρ +ρ 2 +ρ 3 +ρ 0 ) 2 2( ρ ρ 2 ρ 3 ρ 0 ) Case 3: 2( ρ ρ 2 ρ 3 ) < λw < ρ 0 +. In this case, since T is increasing in p an in q, the constraint on E(W ) shoul be saturate. We use the metho of Lagrange multipliers to fin the optimal point (p, q). Let us enote the Lagrange multiplier by α (α is real). Then α an the extremum (p, q) of our optimization problem are solutions of the set of the 3 equations D(T + α(w w )) = 0, where D is the ifferential applicator in α, p an q. These 3 equations are (T + α(w w )) ( ρ ρ 2 qρ 0 ρ 3 )( + ρ 0 ) = µ 0 p ( + pρ 0 ) 2 + α = 0, µ 0 (20) (T + α(w w )) ( p)ρ 0 = µ 0 q + pρ 0 (2) + α ρ 0 (2(ρ 0 + ρ + ρ 2 + ρ 3 )( (ρ + ρ 2 + ρ 3 )) + (ρ + ρ 2 + ρ 3 ) 2 + ρ 2 + ρ2 2 + ρ2 3 ) 2λ ( (ρ + ρ 2 + ρ 3 + qρ 0 )) 2 = 0, (T + α(w w )) α From Equation (20), we obtain α = µ 2 0 Equation (2) leas to = p µ 0 + (ρ + ρ 2 + ρ 3 ) 2 + ρ 2 + ρ2 2 + ρ qρ 0(ρ 0 + ρ + ρ 2 + ρ 3 ) 2λ( (ρ + ρ 2 + qρ 0 + ρ 3 )) w = 0. (22) (+ρ 0 )( (ρ +ρ 2 +ρ 3 +qρ 0 )) (+pρ 0. Substituting this expression in ) 2 ( p)ρ 0 ( + ρ 0)(2(ρ + ρ 2 + ρ 3 + ρ 0 )( (ρ + ρ 2 + ρ 3 )) + (ρ + ρ 2 + ρ 3 ) 2 + ρ 2 + ρ2 2 + ρ2 3 ) + pρ 0 2( + pρ 0 ) 2 = 0, ( (ρ + ρ 2 + ρ 3 + qρ 0 )) 23