Note—A Mathematical Model for Evaluating Cross-Sales Policies in ...

MANUFACTURING & SERVICE OPERATIONS MANAGEMENT

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Vol. 9, No. 1, Winter 2007, pp. 1–8 issn 1523-4614
eissn 1526-5498
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doi 10.1287/msom.1060.0121 © 2007 INFORMS

Note A Mathematical Model for Evaluating Cross-Sales Policies in Telephone Service Centers Reynold E. Byers, Kut C. So

The Paul Merage School of Business, University of California, Irvine, Irvine, California 92697 {[email protected], [email protected]}

C

ross-selling in telephone service centers is seen by industry as a useful means to generate profits from an existing customer base. Introducing cross-sales to a service center, however, needs to be properly managed as it could degrade customer service quality. Real-time customer and system information can be used to set the appropriate control policy for cross-sales to enhance profitability without causing undesirable service degradation for calling customers. The objective of this paper is to illustrate the value of using real-time information in selecting optimal control policies for cross-sales in telephone service centers. Specifically, we develop an introductory mathematical model to incorporate the use of two types of information, system status (queuing congestion) and customer profile (likelihood of purchase) in determining the optimal control policy to maximize the expected operating profit of the system. We hope to stimulate further research in this area by providing a groundwork for further modeling. We show that using more information increases a call center’s operating profit. Improvements were dramatic for environments with intermediate utilization and high customer heterogeneity. Key words: call center management; service management; customer relationship management; cross-selling; threshold policies; optimal queue control History: Received: January 15, 2004; accepted: January 4, 2006.

1.

Introduction

cross-sales policies in call centers with many success stories. Some banking telephone centers, for example, have made significant contributions to the banks’ business, including hundreds of millions of dollars in new loans, deposits, and contribution to net income (see Stoneman 2003). Cross-sales need to be properly managed to ensure that potential revenue gains are not offset by service quality degradation. Cross-sales add work load to the customer sales representative, which can lead to capacity overload and long waiting times for newly arriving calls. Balancing such factors in managing cross-sales in a telephone service center represents a potentially rich research area. The objective of this paper is to propose a simple mathematical model for analyzing cross-sales policies and present some interesting results. We hope our work will stimulate further research in this area. Fundamental to setting an effective control policy is the use of available information that drives decision making. Representatives may not want to waste effort

Traditionally, telephone service centers have been used to simply respond to incoming service requests from existing customers. The cost of providing quality customer service was considered part of doing business. Recently, increasing competition and the need to grow profits have led companies to find new ways to generate revenue from their existing customer base; cross-sales is one of those ways. Cross-selling at telephone service centers involves selling some new service or product to existing customers who call with various customer service needs. Customer sales representatives bundle a sales pitch detailing available products or services during the process of satisfying customer service requests. For example, a banking telephone center representative may learn during a customer’s service call that the customer needs a loan or other bank product. The representative can then explain the bank’s available product in an attempt to sell the customer on that product. Managers in some industries, such as retail banking, have implemented 1

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on customers who would not make a purchase under any circumstance. Also, sales might need to be curtailed during high traffic periods. Customer relationship management (CRM) systems can provide customer purchase history data, and telephone center management software can provide real-time queueing information. We focus our model on these two readily available sources of information in implementing control policies. Specifically, we consider four different policies that employ customer and queueing information and compare their performance. These control policies address the basic question: When should a representative attempt to sell a product to a customer? Our results show that using both types of real-time information optimally always improves the performance of the system. More importantly, we show that using only queueing information generally outperforms using only customer profile information when the system is congested. Thus the queue length information is essential to managing waiting costs properly. On the other hand, using only customer profile information outperforms using only queueing information when there is high variability in the customer profiles. This implies that it is more important to differentiate cross-sales effort to a diverse customer base. Finally, our results show that using both types of information is best when the system is highly congested, the revenue from a sale is high, and customers are heterogeneous. We see several possible directions for extending this work. First, we present only the single-server model in this paper, so a full treatment will require a model of multiserver systems. We are able to extend our single-server model to the multiple-server case (see Byers and So 2004). Furthermore, in our analysis, we segment our customer base into only two classes of customers. A finer customer segmentation may lead to a richer selection of control policies to further increase the profitability of cross-selling opportunities. Also, we assume in our model that the firm is able to accurately estimate the probability of a successful cross-sale for each customer. An interesting extension would be to relax this assumption to the situation where a firm has imperfect information regarding these probabilities. Finally, as it is often the case that customer service representatives can obtain better information about the customer than managers can,

but may be reticent to cross-sell, it would be useful to construct a principal-agent model to address the issue of devising appropriate incentive mechanisms for service representatives in such situations. The rest of this paper is organized as follows. Section 2 discusses relevant literature. Section 3 describes the model formulation and presents the analysis for determining the optimal policy under each of the four classes of control policies. Section 4 discusses the results of the numerical experiments and summarizes the major insights.

2.

Related Literature

Our work builds directly on some recent research on modeling cross-selling in call centers. Two papers are especially relevant to our study here. First, Gunes and Aksin (2004) addressed customer segmentation and incentives in cross-selling and other value creation service activities. They segmented the customer base into high- and low-value customers and analyzed cross-sales attempts to one or both customer segments. Gunes and Aksin (2004) also modeled incentives for telephone center representatives to do sales by using a principal-agent modeling framework. Our work extends their queuing model by also using realtime queue length information to form the optimal control policies. However, we do not consider incentive issues in our paper. Second, Aksin and Harker (1999) modeled the effects of adding cross-selling to a telephone service center. They demonstrated that cross-sales can put a strain on the telephone center’s resources, resulting in service degradation. They suggest that appropriate managerial actions can ameliorate the congestive effects of cross-selling. Our paper analyzes the use of real-time information in determining appropriate control policies for managing cross-sales to avoid the negative effects they describe. There is considerable research interest in analysis of the design and operations of telephone call centers. Gans et al. (2003) provide a broad look at telephone centers. In their paper, they provide a tutorial into the basic operation of and management issues in telephone centers. They survey a broad range of academic literature and discuss future research opportunities. Koole and Mandelbaum (2002) also provide a survey of using queuing models to analyze call centers.

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Queuing control problems have also been well studied in the literature. Relevant to the models in our paper are a variety of queuing control models (e.g., see Dudin 1998 and Federgruen and So 1991). These models used real-time queue length information to optimally control the server operations in the underlying system.

3.

Model Formulation

We model a telephone call center as a queuing system, which uses a control policy for cross-sales. For simplicity in our exposition, we only present a singleserver model. An extension of our model to the multiple-server case can be found in Byers and So (2004). We first introduce the following notation. h: Unit holding cost per customer in the system R: Reward for each successful sale : Arrival rate to the system 1 : Service rate for bundling the regular service request with the cross-selling effort 2 : Service rate for providing the regular service request only Define 1 = /1 and 2 = /2 . To simplify our analysis, we assume that the service time distributions for providing the regular services and bundling the regular service with cross-selling are both exponential, with rates 1 and 2 , respectively, such that we can use a birth-and-death process to analyze the performance of the underlying service systems. Furthermore, we assume that 1 < 2 and 2 < 1 throughout this paper, meaning that cross-sales require more work and the system is stable when there is no cross-selling. All customers are assumed to be first come, first served. We also assume preemptive service, where the servers are allowed to switch between the two service modes at any time, while a particular customer is still being served. For instance, a server can start to cross-sell to a new incoming customer, however, as the server observes additional callers have arrived in the system, the server can decide to terminate his cross-selling effort to this customer and switch to provide only the regular service. The preemptive service assumption helps simplify our analysis. We have found this policy being used in some telephone centers; however, it could prove to be restrictive in other call center applications. Nevertheless, we

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believe that the basic insights provided by our model remain valid. We assume that customer profile information is available from which the firm can a priori estimate the probability of a successful cross-sale for all its customers. For instance, such probability estimates can be obtained from previous customer purchase history information, which is readily available from many CRM software systems. Our model further assumes that the probability distribution of a successful crosssale for the existing customer base is uniformly distributed in q −  q + , where 0 ≤ q − ≤ q + ≤ 1. In other words, the probability that any random caller will buy the product is uniformly distributed in q −  q + . Thus, q is the average probability that a random customer will make a purchase, and is a measure of the variability of a successful cross-sale across the existing customer base. We next evaluate four different operating policies for using the customer profile information and/or the queue length information to maximize the expected operating profit in the system. 3.1.

The Full Policy: Using Both Customer Profile and Queue Length Information The full policy uses both types of real-time information under which each customer is classified into one of two possible types, high-value or low-value (indexed by the symbols h and l, respectively), based on their individual estimated probability of a successful sale. The full policy is specified by p nh  nl , where we classify the top p (0 ≤ p ≤ 1) proportion of customers having the highest probability of a successful crosssale as high-value customers, and the bottom 1 − p proportion as low-value customers. Cross-selling to a high-value (low-value) customer happens if and only if i ≤ nh (i ≤ nl ), where nh (nl ) denotes the queue length threshold for cross-selling to high-value (low-value) customers. We assume that 0 ≤ nl ≤ nh , implying that we are more willing to cross-sell to the high-value customers than to the low-value customers. Let S = i j i ≥ 0 j ∈ l h denote the state space of the underlying birth-and-death process, where i denotes the total number of customers in the system, and j represents the current customer type in service. Let i j denote the steady-state probability at state i j. Also, define h i and l i as the service rates for the high-value and low-value customers

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when there are i customers in the system, respectively. Under the full policy, h i = 1 for 0 ≤ i ≤ nh and h i = 2 for i > nh . Similarly, l i = 1 for 0 ≤ i ≤ nl and l i = 2 for i > nl . The steady-state probabilities  satisfy the following set of transition equations:

operating profit under the full policy is equal to nl −1   F p nl  nh  = Rq i + Rpq + 1 − p · nl +

For 1 ≤ i ≤ nl 

−h

i = 1 i 0 p + 2   2  + 1  + p1 − 2  nl

nl +1 l =

1 − p + 1    2  + 1  + p1 − 2  nl

(1)

For nl + 1 < i ≤ nh  i l =

1 − p2  2  + 1  + p1 − 2  i−1 h  − p + 1   2  + 1  + p1 − 2  i−1 l
 nh +1 = 2 nh  h + nh  l  +

For i ≥ 1

nh +i = 2 i−1 nh +1 

and nl  i=0

i +

nh  i=nl +1

i h + i l  +

nh +1

1 − 2

= 1



i=nl +1

ii +

i l + i h  nh  i=nl +1



ii h + i l 

nh 1 + 1 − 2 1 − 2 2

  (3)

3.2.

p + 2   2  + 1  + p1 − 2  i−1 h p2   2  + 1  + p1 − 2  i−1 l



nh −1

+ nh +1

For nl + 1 < i ≤ nh 

+

 nl  i=1

nl +1 h =

i h =

i=0



(2)

Note that under the full policy p nh  nl , “highvalue” customers correspond to the set of customers whose probability of a successful cross-sale exceeds q + − 2 p. Therefore the average probability of a successful cross-sale to any random high-value customer is equal to q + 1 − p . Thus the average reward per unit time because of successful crosssale to high-value customers is equal to Rpq + 1 − p . Similarly, the expected reward per unit time because of successful cross-sale to low-value customers is equal to R1 − pq − p . Thus the expected

The p-Policy: Using Customer Profile Information In the p-policy (0 ≤ p ≤ 1), the server will cross-sell to only the top p proportion of customers who have the highest probability of a successful cross-sale. Under a given p-policy, the service system can be modeled as an M/G/1 queue, where the arrival rate to the system is equal to  and the service time distribution is equal to an exponential distribution with parameter 1 with probability p and is equal to an exponential distribution with parameter 2 with probability 1 − p. It is straightforward to show that this service time distribution has a mean of p = p/1 + 1 − p/2 and variance of p2 = p2 − p/21 + 1 − p2 /22 − 2p1 − p/1 2 . Because 2 > 1 , it is easy to see that p is decreasing in p. We define all admissible p-policies to be those under which the corresponding M/G/1 queue is stable, i.e.,  < p . As discussed in §3.1, the expected reward per unit time because of a successful cross-sale under the p-policy is equal to Rpq + 1 − p . Using the Pollaczek-Khintchine formula (e.g., Gross and Harris 1998), the average system size under the p-policy is given by Lp = p +

2 2p + 2 p2 21 − p 

= p1 + 1 − p2 +

p21 + 1 − p22  1 − p1 + 1 − p2

Thus the expected operating profit under any admissible p-policy is equal to  F p = Rpq + 1 − p − h p1 + 1 − p2 +

 p21 + 1 − p22  1 − p1 + 1 − p2

(4)

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We prove two important properties about the above profit function. The proof of Proposition 1 can be found in the appendix.

We prove two important properties about the above profit function. The proof of Proposition 2 can be found in the appendix.

Proposition 1. (a) F p is strictly concave in p. (b) Let p∗ -policy denote the optimal p-policy. Then, the optimal p∗ is nondecreasing in the ratio R/h and is nondecreasing in q.

Proposition 2. (a) The optimal n∗ is bounded above by Rq1 − 2 / h1 − 2  , and (b) The function F n is unimodal in n for n ≥ 1.

Proposition 1(a) shows that the optimal p is unique, and we can use a simple bisection search for finding the optimal p that maximizes F p. Proposition 1(b) shows that the firm should cross-sell to a higher proportion of its customers as the ratio of the expected reward of a successful cross-sale to the holding cost rate increases and when the average probability of a successful cross-sale increases. 3.3.

The n-Policy: Using Queue Length Information In the n-policy, cross-selling attempts are made if and only if i ≤ n. Let i denote the service rate when there are i customers in the system, and i denote the steady-state probability of the system at state i. Then, the steady-state probabilities i satisfy the set of equations i = /ii−1 for i ≥ 0. Under the n-policy, i = 1 when 0 ≤ i ≤ n and i = 2 when i > n. It is straightforward to show that  1 i 0 1≤i≤n (5) i =  n  i−n  i > n 1 2 0 where



0 = 1 +

n  i=1

i1

+ n1

 k=1

k2

−1  −1 1 − n1 n1 = +  1 − 1 1 − 2

Using (5), we can deduce the average system size under the n-policy as 

 + n − 1n1  1 − nn−1 1 ii = 0 1 Ln = 1 − 1 2 i=1  n n − n − 12  + 1 1 − 2 2 Thus the expected operating profit under the n-policy is given by     +n−1n1 1−n1  1−nn−1 1 −h 1 F n = 0 Rq 1−1 1−1 2  n n − n − 12  (6) + 1 1 − 2 2

The results in Proposition 2 imply that we can readily compute the optimal n∗ , for n ≥ 1, by using a simple bisection search between 1 and the upper bound Rq1 − 2 /h1 − 2  . Then, we can compare F n∗  with F 0 to determine the optimal n-policy. 3.4.

The Static Policy: Using No Real-Time Information The static policy uses no real-time information; the server will either always cross-sell or never cross-sell. If 1 ≥ 1, then the average system size goes to infinity and the servers should never cross-sell. Thus, assume that 1 < 1. If the servers always cross-sell, the underlying service system is an M/M/1 queue with 1 , and the expected operating profit is given by Fcross-sell = Rq −

h1  1 − 1 

(7)

where the first term represents the expected revenue per unit time (arrival rate times expected reward because of a successful cross-sale), while the second term represents the expected customer holding cost in the underlying M/M/1 queue. On the other hand, if the servers never cross-sell, the underlying service system is an M/M/1 queue with 2 , and expected operating profit is given by Fno-cross-sell = −

h2  1 − 2 

(8)

Thus the expected operating profit is the maximum of (7) and (8). We summarize the result in the next proposition. The proof of Proposition 3 is straightforward, and we omit the details here. Proposition 3. The optimal static policy is to always cross-sell if and only if Rq 1 − 2 2 − 1  > =  h 1 − 1 1 − 2  2 − 1 − 

(9)

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4.

Numerical Results

We conducted extensive numerical experiments across a broad range of parameter values as given in Table 1. The first two parameters are fixed in all cases, while the last five parameters vary within the given ranges of values in our experiments. We next present the most interesting results regarding the effect of congestion and the effect of customer profile variability on the system. We remark that these basic insights also apply for the multiple-server case (see Byers and So 2004). 4.1. Effect of Congestion Figure 1 shows the expected profit generated by the optimal policy among each class of control policies under a wide range of arrival rates when the reward for a successful cross-sale is low and customer profile variability is high. Note that at very low and very high utilizations, information has little or no value as all policies perform the same. Further, note that information-based control policies show their greatest advantage for intermediate utilization levels. Table 1

Parameter Values in Numerical Analysis

Parameter q: Average likelihood of purchase h: Holding cost rate for customers 1 : Service rate for sales 2 : Service rate for service R: Expected reward for a sale : Arrival rate

: Variability in customer profile

Value or Range 0.5 1 1 11 12 2 1 2     50 02 03     19 005 01     05

Figure 1

Effect of Congestion at High Customer Profile Variability

Impact of congestion on profit (R = 50, = 0.5) Full

p

65%

75%

n

Static

20

10

Profit

The result in Proposition 3 shows that it is optimal to cross-sell only when the ratio of the expected reward weighted by the average success probability of a cross-sale to the holding cost rate exceeds some cutoff level as given in (9). Otherwise, the optimal policy is to never cross-sell. Also, observe from (9) that the cutoff level is increasing in , which implies that the firm will be less inclined to cross-sell as the traffic to the system increases. Remarks. Note that the static policy is a special case of either the n or p policies, which in turn are special cases of the full policy. Therefore, optimally using both types of real-time information always improves the performance of the system.

0 25%

35%

45%

55%

85%

95%

–10

–20

Utilization

Figure 1 illustrates some important observations. The optimal n-policy outperforms the p-policy as the system congestion increases. Additionally, the optimal full policy outperforms all policies over the same region. Thus, as the system congestion increases, queue length information becomes highly valuable in managing the cross-sales decisions. However, differentiation based on customer profile information is still valuable and can generate substantial incremental profit even in congested systems. We observe similar behavior under different variations of R and in our experiments. Our results lead to the following key insights. The p-policy performs best when there is a wide variance in customer profiles. Thus, while the value of the n-policy is in managing congestion, the value of the p-policy is in differentiating a heterogeneous customer base. Furthermore, an increase in the reward for a cross-sale leads to more cross-selling, greater congestion, and more emphasis on customer profile information. As a result, the use of real-time information in the full policy is of greater benefit when cross-sales rewards are high. 4.2. Effect of Customer Profile Variability Figure 2 summarizes the results that illustrate the effect of customer profile variability. First, note that both the static policies and n-policies are independent of the customer profile variability parameter . On the other hand, the expected profit under either the optimal p-policy or the optimal full-policy steadily increases as increases. Therefore, our results again support the earlier contention that the value of customer

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Figure 2

Effect of Customer Profile Variability

Impact of
on profit (µ2 = 2) 2.0 Full

p-policy

n-policy

Static

1.5

Profit

1.0 0.5 0 0.05 0.10

0.15

0.20

0.25

0.30 0.35

0.40

0.45

0.50

– 0.5 –1.0




profile information comes from segmenting a heterogenous customer base. Appendix

Proof of Proposition 1. (a) Differentiating F p in (4) w.r.t. p and after simplification, we obtain F p = Rq + 1 − 2p  −h1 − 2  1 + +

1 + 2 1 − p1 + 1 − p2

p21 + 1 − p22 1 − p1 + 1 − p2 2

and

 (10)



1 + 2 1 − p1 + 1 − p2 2  p21 + 1 − p22 +  (11) 1 − p1 + 1 − p2 3

F

p = −R2 − 2h1 − 2 2

Because the p-policy is admissible, we have  < p , which is equivalent to 0 < p1 + 1 − p2 < 1. Thus, F

p < 0 for all admissible p, which implies that F p is strictly concave for all admissible p. (b) First, observe that q + 1 − 2p = q −  + 21 − p ≥ 0 since q −  ≥ 0. It then follows easily from (10) that F p/h is nondecreasing in R/h. Suppose that the p1 -policy is optimal for the system with expected cross-sale reward R1 , and holding cost rate h1 and the p2 -policy is optimal for the system with expected cross-sale reward R2 and holding cost rate h2 , and that R2 /h2 ≥ R1 /h1 . Consider that 0 < p1 < 1. In this case, (a) implies that F p1  = 0 in (10) when R = R1 and h = h1 . Furthermore, we must have F p1 /h ≥ 0 when R = R2 and h = h2 . Because F

p < 0 for all admissible p, this follows again from (a) that p2 ≥ p1 as p2 is optimal for the system when R = R2 and h = h2 . If p1 = 1, then F 1/h ≥ 0 when R = R1 and h = h1 , which implies that F 1/h ≥ 0 when R = R2 and h = h2 . Therefore, p2 = 1. If p1 = 0, then it is trivial that p2 ≥ p1 . Therefore, we have shown that p2 ≥ p1 in all cases, which proves

that the optimal p∗ is nondecreasing in R/h. Similarly, we can show that the optimal p∗ is nondecreasing in q.
Proof of Proposition 2. (a) We use a Markov decision process (see Ross 1983) to analyze the optimal control problem of an M/M/1 queue with two service modes, where the decision is to choose either service rates 1 and 2 based on the queue length of the system. The cost structure consists of a linear holding cost rate and a expected reward qR if service rate 1 (corresponding to cross-selling) is chosen. The decision epochs are at customer arrivals and service completions. Let i
i = 0 1 2    denote the state space of this optimal decision problem, where i denotes the number of customers in the system at decision epochs. It has been shown (e.g., Tijms 1976) that a (stationary) threshold policy is optimal where service rate 2 is chosen at state i only when i > n for some fixed n. Specifically, let the n-policy be the stationary policy that chooses rates 1 when i ≤ n and chooses rate 2 when i > n. For any fixed n ≥ 0, because 2 = /2 < 1, state 0 is a positive recurrent state under the n-policy. We shall apply the same analysis technique used in Federgruen and So (1991) to establish the desired result. We briefly outline the analysis here and refer the interested reader to Federgruen and So (1991) for the details. Under any n-policy, define Mn i and Tn i to be the total expected profit and the total expected time incurred before reaching the positive recurrent state 0 when the system starts at state i, respectively. Also, define Vn i = Mn i − F nTn i, with Vn 0 = 0. Then, it follows that Vn i F n satisfy the following set of optimality equations: −F n + Vn 1  −hi F n  Vn i = − + V i + 1  + 1  + 1  + 1 n 1 + qR + Vn i − 1 1 ≤ i ≤ n  + 1 Vn 0 =

Vn i =

(12)

−hi F n  − + V i + 1  + 2  + 2  + 2 n 2 + V i − 1 i > n  + 2 n

Suppose that the n-policy is optimal. Then, the optimality equation stipulates that, choosing service rate 2 (no crosssell) is a better action than choosing service rate 1 at any state i > n. (Otherwise, there exists an improvement policy that chooses service rate 1 at some state i > n.) Therefore, the optimality of the n-policy implies that, for all i > n, F n  1 −hi − + V i + 1 + qR + Vn i − 1  + 1  + 1  + 1 n  + 1 ≤

F n  2 −hi − + V i + 1 + V i − 1  + 2  + 2  + 2 n  + 2 n

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n-policy, contradicting the assumption that F n ≥ F n + 1. Therefore

which, after simplification, is equivalent to 2 − 1 −hi − F n +  + 2 1 Rq + 2 − 1 Vn i + 1 − Vn i − 1 ≤ 0

(13)

From the definition of Vn ,

≤

Vn i +1−Vn i = Mn i +1−Mn i −F nTn i +1−Tn i  Observe that, for any i ≥ 0, state i + 1 must first reach state i before reaching state 0 under the n-policy. Therefore the quantity Vn i + 1 − Vn i is equal to the total expected profit incurred until reaching state i when starting at state i + 1 minus F n times the total expected time to reach state i when starting at state i + 1 under the n-policy. Furthermore, for any i ≥ n, the average queue length in the system under the n-policy during the time from state i + 1 to state i is simply equal to the average queue length during the one busy period of an M/M/1 queue with service rate 2 plus i additional customers in the system. For an M/M/1 queue with service rate 2 , the expected length of one busy period is equal to 1/2 −  and the average queue length during one busy period is equal to 2 /2 − . Thus Vn i + 1 − Vn i − 1 = Vn i + 1 − Vn i + Vn i − Vn i − 1      1 2 − F n = −h i + 2 −  2 −       1 2 − F n  + −h i − 1 + 2 −  2 − 

F n  2 −hn + 1 − + V n + 2 + V n  + 2  + 2  + 2 n  + 2 n

which, after simplification as before, is equivalent to the inequality that Rq1 − 2  2 F n − ≤ n + 1 − h1 − 2  1 − 2 h Obviously, it follows from the above inequality that, for all i ≥ 2, Rq1 − 2  2 F n − ≤ n + i − h1 − 2  1 − 2 h which implies that, for any fixed k ≥ 2, the n + k-policy, where service rate 1 is chosen at state i = n + 1     n + k, while keeping all other decisions the same as in the n-policy cannot be an improvement policy over the n-policy. Therefore F n ≥ F n + k for any k ≥ 2. In other words, any local maximum for F n is also the global maximum, and so F n is unimodal.


References

Substituting into (13), we obtain 2 − 1 −hi − F n +  + 2 1 Rq     22 2 − 1  −h 2i − 1 + − 2F n ≤ 0 + 2 −  2 −  which, after simplification, is equal to F n 2 Rq1 − 2  − − ≤ i h1 − 2  1 − 2 h

−hn + 1 F n  1 − + V n + 2 + qR + Vn n  + 1  + 1  + 1 n  + 1

(14)

This above analysis shows that if the n-policy is optimal, Equation (14) must hold for all i ≥ n. Also, F n = maxk≥0 F k, which implies that F n ≥ F 0 = −h2 /1−2 . Therefore  2 F n Rq1 − 2  − − n≤ h1 − 2  1 − 2 h   Rq1 − 2  F 0 2 Rq1 − 2  = − −  ≤ h1 − 2  1 − 2 h h1 − 2  (b) To show F n is unimodal, assume that F n ≥ F n+1. This implies that, under the n-policy, choosing service rate 1 (cross-sell) cannot be a better action than choosing service rate 2 (no cross-sell) at state n + 1, because otherwise, the n + 1-policy is an improvement policy over the

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