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Optimal Workforce Mix in Service Systems

with Two Types of Customers

Srinivas R. Chakravarthy • Saligrama R. Agnihothri

Department of Industrial and Manufacturing Engineering and Business, Kettering University

(Formerly GMI-EMI), Flint, Michigan 48504-4898, USA

School of Management, State University of New York at Binghamton, Binghamton, New York 13902-6000, USA

schakrav@kettering.edu • agni@binghamton.edu

W

who serve either type of customers. Cross-trained workers are more flexible and help reduce system

delay, but also contribute to increased service costs and reduced service efficiency. Our objective is to

provide insights into the choice of an optimal workforce mix of flexible and dedicated servers. We

assume Poisson arrivals and exponential service times, and use matrix-analytic methods to investigate

the impact of various system parameters such as the number of servers, server utilization, and server

efficiency on the choice of server mix. We develop guidelines for managers that would help them to

decide whether they should be either at one of the extremes, i.e., total flexibility or total specialization,

or some combination. If it is the latter, we offer an analytical tool to optimize the server mix.

e consider a service system with two types of customers. In such an environment, the servers can

either be specialists (or dedicated) who serve a specific customer type, or generalists (or flexible)

Key words: service system design; queuing models; flexibility; cross-training; matrix analytic method;

optimization

Submissions and Acceptance: Received March 2003; revisions received October 2003 and July 2004;

accepted September 2004.

1.

Complex high technology products, equipment, and

systems have become a critical part of the infrastruc-

ture of today’s businesses and homes. In recent years,

companies selling high-tech manufactured products

are realizing that a customer’s product purchase deci-

sion is not only influenced by the product’s value, but

also by the service support available after the sale of

the product. In addition, in high-tech product markets

the product profit margins are low while service profit

margins are high. Thus, for a product business, service

opportunities are greater than the initial product sale

value over the product life cycle, both in terms of

revenue and profits (Blumberg, 1991). Hence, provid-

ing good after-sales service support is an excellent

business strategy and can play an important role in

achieving a competitive advantage.

A major part of after-sales support includes instal-

lation, maintenance, and repair of equipment at cus-

tomers’ site, which is referred to as field service.

Equipment failures reduce productivity. Reducing

Introduction

machine downtime by providing prompt and success-

ful field service is very important for customer satis-

faction. One of the factors that impacts downtime is

the extent of worker cross-training employed. Since

the machine failures are random, the demand for skill

types needed to repair the machine is also random.

Making sure that a technician with the right skill is

readily available is not easy. Firms have the choice of

employing specialists (or dedicated) who are dedi-

cated to serve one type of customer, or generalists (or

cross-trained or flexible workers) who can serve more

than one type of customers. Cross-trained workers

represent flexible capacity, since they can be readily

assigned to serve any type of incoming customers.

Although cross-training increases server flexibility

and improves responsiveness, it also increases the ser-

vice costs and may reduce service efficiency (see, for

example, Pinker and Shumsky 2000). On the other

hand, using specialized servers increases the down-

time because of unbalanced server loads due to lack of

work sharing. The objective of this paper is to study

POMS

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Vol. 14, No. 2, Summer 2005, pp. 218–231

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the extent of cross training required to balance the cost

of workforce staffing and the cost of machine down-

time for service systems.

Most models in the literature evaluate the benefits

of flexibility without considering the cost of acquiring

this capability. In fact, whether it is the flexibility of

facilities (see, for example, Jordan and Graves 1995), of

machines (see, for example, Sheikhzadeh et al. 1998),

or of workers (see, for example, Brusco and Johns

1998), a very significant finding is that limited flexibility

yields most of the benefits of total flexibility. This finding

considers only the benefits of flexibility and ignores

the cost of flexibility. As pointed out by Karmarkar

and Kekre (1987), the extent of cross-training needed

to increase server flexibility should be evaluated by

considering the tradeoff between the cost of acquiring

this flexibility and the benefits obtained. In this paper,

our objective is to evaluate the extent of flexibility

needed, taking into account both the costs and benefits

of flexibility.

We consider a service system with two types of

customers. There are three types of servers, a specialist

who can serve only a particular customer type or a

flexible server who can serve both customer types.

Assuming Poisson arrivals and exponential service

times, and taking into account service cost and cus-

tomer downtime cost, we evaluate the impact of var-

ious system parameters such as the number of servers,

server utilization, and server efficiency on the optimal

server mix of flexible and dedicated servers. We will

investigate situations in which a manager should

choose between the two extreme choices of total flex-

ibility and total specialization, and those in which it is

important to choose a combination. If it is the latter,

we offer an analytical tool to optimize the server mix.

The model presented in this paper was inspired by

observing after-sales support operations of a leading

supplier of high-tech capital equipment in the elec-

tronics industry. This firm maintains sales, service,

product training, and parts distribution centers world-

wide. Currently, the firm’s installed base of 16,000

active machines is supported by a global organization

of more than 200 field engineers, who perform ma-

chine and system installations, as well as a variety of

warranty and contract services. The number of field

engineers in a territory varies between 3 and 25. Ma-

chine failures can be aggregated into two major failure

categories: electrical and mechanical. Within each cat-

egory, there are approximately a dozen specific tasks

that may be required to remedy any given failure. A

‘dedicated field engineer’ is one who is trained in one

of these categories over some set of machine types.

Since this is high-tech equipment, the training time

is long and varies between three months and a year.

As a result, the cost of cross-training a field engineer is

quite significant. Equipment is redesigned continu-

ously to incorporate the changes in technology. This

necessitates periodic retraining of field engineers.

Since labor cost is a major component of the service

cost, when training is expensive, firms must give at-

tention to maintain a right workforce mix to balance

the company profitability and customer satisfaction.

In field services, downtime is a key measure of cus-

tomer satisfaction. One of the major issues facing this

firm is to study the extent of cross training required to

balance the cost of workforce staffing and the cost of

machine downtime. Although we address the prob-

lem from a field service system perspective, the results

obtained here could be applied to other manufactur-

ing and service systems as well.

2.

There is extensive literature on strategic use of flexi-

bility in manufacturing (see, for example, Sethi and

Sethi 1990; Singhal et al. 1987; Bordoloi et al. 1999; Jack

and Raturi 2003; Gaimon and Morton 2005). In the

service context, Roth and Menor (2003) provide the

service strategy triad that includes target market, ser-

vice concept, and service delivery system. Deciding on

the right mix of flexible and dedicated workers is part

of designing the service delivery system. They present

architecture for service delivery systems and explain

the content of a service operations management strat-

egy.

General issues in field services are discussed in Hill

(1992) and Agnihothri and Karmarkar (1992). Sim-

mons (2001) establishes a framework for field service

flexibility. There are very few papers in the literature

dealing with field service problems with multiple job

types requiring different worker skills. Begley et al.

(1983) study a problem faced by a computer com-

pany’s field service engineering department with

three types of technical service calls. They consider

only three servers with four cross-training configura-

tions and use machine downtime as the performance

measure. They did not consider server or downtime

costs as we do here, and study a specific situation

using a simulation model.

Several authors have developed queueing models to

study the impact of cross training on system perfor-

mance. For a system with m customer types and m

servers, Agnihothri (2001) developed queueing mod-

els to examine when it is economical to use all flexible

servers instead of all specialists. Green (1985), Stanford

and Grassmann (1993), and Shumsky (2004) studied

models with two customer types and two server types,

where type 1 servers can serve only type 1 customers

and the type 2 servers can serve both customer types.

Assuming exponential inter-arrival and service times,

they developed Markov models and used matrix-geo-

metric procedures to evaluate system performance

Literature Review

Chakravarthy and Agnihothri: Optimal Workforce Mix in Service Systems with Two Types of Customers

Production and Operations Management 14(2), pp. 218–231, © 2005 Production and Operations Management Society

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measures. Both Green (1985) and Stanford and Grass-

mann (1993) apply their model to solve problems with

eight and six servers, respectively, and point out that

only relatively small server systems can be solved due

to the complexity of the model. In addition, they were

interested in a specific application and did not use

their model to evaluate the impact of server flexibility

on system performance. However, Shumsky (2004)

develops an approximation procedure that could be

used to solve large problems and applies it to demon-

strate the benefits of server flexibility in a telephone

call center with 40 servers. He concludes that a rela-

tively small proportion of cross-trained servers

achieved most of the benefits of flexibility. A major

difference between their model and the model pre-

sented in this paper is that they have specialized

server for only one customer type, and as a result, the

other customer type is always served by a cross-

trained server. Hence, they assume that cross-training

is inevitable, and then find the minimum number of

cross-trained servers. We have specialized servers for

each customer type and our objective is to decide

whether to cross-train at all, and if so, how many

servers to cross-train. Furthermore, we exploited the

special structure of the coefficient matrices that appear

in the steady state equations to analyze systems with

large number of servers. One of the key aspects of the

computational methods using matrix-geometric ap-

proach is the exploitation of the special structure of the

generator of the Markov process governing the queue-

ing model under study.

In a related setting, Pinker and Shumsky (2000)

develop an integrated model to determine the optimal

server mix of specialized and cross-trained severs tak-

ing into account the quality of service provided by the

cross-trained servers. They integrate a stochastic ser-

vice system model (a queueing model) with models

for tenure and experience-based service quality. Al-

though their queueing model assumes Poisson arriv-

als and exponential service times with three server

types and two customer types as we do, they assume

a loss system without any waiting room space, and

use server utilization and throughput as performance

measures. The result of their queueing model taken in

isolation confirms that cross-trained servers provide

more throughput with fewer workers than specialized

servers. However, the integrated model reveals that

cross-trained workers may not gain sufficient experi-

ence to provide high-quality service to any one cus-

tomer, and what is gained in efficiency is lost in qual-

ity. The objective of their paper is to investigate the

efficiency-quality tradeoff of cross-trained workers by

evaluating the impact of learning rate, tenure process,

and system size on staffing configuration choice.

While we do not consider employee learning and

turnover issues in this paper, we do consider cross-

trained server inefficiency through increased mean

service time. The objective of our paper is to evaluate

the impact of server utilization, staffing and downtime

costs, system size, and cross-trained server ineffi-

ciency on cross-training decisions.

Many researchers investigated the benefits of

worker cross-training in the dual resource constrained

(DRC) manufacturing systems (see Hottenstein and

Bowman 1998, for a review). Recently, Hopp and Van

Oyen (2004) classify workforce agility architectures

and investigate strategic implications of developing an

agile workforce. These papers largely address manu-

facturing environments, with issues that are different

in nature and hence, we omit the review of the DRC

literature here.

We present the modeling assumptions in the next

section and provide some applications of the model in

Section 4. The final section summarizes the paper and

points out a few future research directions. The details

of the model development and analysis are presented

in the appendix.

3.

We develop a queuing model (Figure 1) with the

following assumptions.

• There are two customer types. The arrivals of

customer type i occur according to a Poisson pro-

cess with rate ?i, i ? 1, 2.

• There are three different types of servers, referred

to as 1, 2, and ‘12’. A type i server is a dedicated

server who can only serve customer type i, i ? 1,

2, and a type 12 server is a flexible (cross-trained)

server who can serve either type of customers.

The service times are exponentially distributed.

The service rate of a type i dedicated server is ?i,

The Modeling Assumptions

Figure 1Description of the Queuing Model Under Consideration.

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i ? 1, 2, and the service rate of a flexible server is

?12. Since the flexible server serves two different

job types, we assume that average service time of

a flexible server is higher than that of a specialist.

That is, ?12? ?i; i ? 1, 2. There are sitype i

dedicated servers and r flexible servers and the

total number of servers in the system is s ? s1

? s2? r. Note that s1, s2, and r are integers

between 0 and s.

• The efficiency of the flexible server compared to

the type i dedicated server is defined as Ei?

?12/?i. Since ?12? ?i, 0 ?Ei?, i ? 1, 2. When all

the flexible servers have the same efficiency, we

denote it by E.

• An arriving type i customer waits in a line in front

of a type i server, if all type i servers are busy, and

a waiting room space is available. There is an

infinite waiting room for type 1 customers. How-

ever, the waiting room capacity (buffer) for type 2

customers is of finite size, K, so that at any time,

we can have a maximum of K type 2 customers

waiting in the queue. Any type 2 customer arriv-

ing to find the buffer full is considered to be lost.

Ideally, in practice, the buffers have finite capac-

ity and by requiring one of them (here it is the

type 1 buffer) to be of infinite size makes it rela-

tively easy to quantify certain results. However,

the algorithmic analysis proposed here could eas-

ily be modified to take into account the case when

both buffers are finite. The case when both buffers

are of infinite size can be handled numerically by

truncating on one of the two types of customers.

Instead, here we chose to restrict one of the types

(namely type 2) to have a finite buffer. This

should not be viewed as a limitation of the meth-

odology used to study the queueing model.

• The service discipline used is FCFS. We assume

that type 2 customers are more time-sensitive

with a higher rate of downtime cost and hence are

given a non-preemptive priority for service with

flexible servers. That is, when all dedicated serv-

ers are busy, a free flexible server will first offer

service to a type 2 customer if there is any; oth-

erwise, a type 1 customer will be served. This also

results in a reduced number of type 2 customers

lost due to finite buffer size.

• The system is in steady state.

With these assumptions, we develop a Markov

model with a three-dimensional state space. We then

establish many system performance measures of in-

terest using the steady state distribution of the Markov

chain. The details of the model development and anal-

ysis are presented in the Appendix. In the next section,

we will apply these steady state results to analyze the

impact of adding flexible servers on the system per-

formance measures.

4.

The objectives of this section are three-fold. First, we

will demonstrate how the analytical results could be

used to determine the optimum mix of dedicated and

flexible servers. Second, it is well known that cross-

trained workers represent flexible capacity. Obvi-

ously, the extent and the value of added flexible capac-

ity depend heavily on the system conditions. We

evaluate how much flexible capacity could be gained

by cross-training. In order to evaluate the value of the

flexible capacity, we develop a model that takes into

account both the costs and the benefits of cross-train-

ing and investigate when and how much flexible ca-

pacity is worthwhile to have. Third, we will show that

total flexibility is not always optimal when the cost of

adding flexibility is considered. We will develop a

table that would provide guidelines for managers to

decide whether they should be at either one of the

extremes, i.e., total flexibility or total specialization, or

some combination.

Note that we are interested in a field service system

where demand should be met without excessive de-

lay. Hence, among all the measures developed in the

appendix, we will extensively use the mean downtime

(Ws, see xiv in the Appendix) as the measure of system

performance. In order to better understand the results,

we initially assume that ?1? ?2. Later, we investigate

situations when ?1? ?2. Without loss of generality, we

will assume the service rates of dedicated servers to be

1, which corresponds to measuring time in the scale of

mean service times. Initially, we will also assume that

the flexible servers are 100% efficient, and hence, ?1

? ?2? ?12and E ? 1. We will relax this assumption

later. For all the examples considered here, we choose

the waiting room capacity K large enough to make

sure that the proportion of type 2 customers lost is

insignificant (of the order of 10?4). In addition, in our

cost model, we assign a very large cost for the loss of

a type 2 customer. We denote the fraction (proportion)

of flexible servers by fF? (r/s). Note that fF? 0, (1/s),

. . . , 1.

Applications of the Model

4.1.

In order to evaluate the impact of replacing dedicated

servers by flexible servers on the mean downtime, we

consider a system with 20 servers. We denote the

server utilization (see xv in the Appendix) correspond-

ing to a system with all dedicated servers by U0. We

first calculate the common arrival rate that results in a

U0of 0.9. We maintain this (common) arrival rate and

the total number of servers, but vary the proportion of

flexible servers in the system. Figure 2 displays the

mean downtime for type 1 and type 2 customers and

the aggregate mean downtime. Since flexible servers

give a nonpreemptive priority to type 2 customers

over type 1 customers, the mean downtime for type 1

Mean Downtime

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customers initially increases as flexible servers replace

the dedicated servers before it starts decreasing. As

expected, the mean downtime for type 2 customers

and mean aggregate downtime decrease as the frac-

tion of flexible servers increases. Furthermore, the rate

of decrease in the mean downtime also decreases as

the fraction of flexible servers increases, indicating

that the value of adding flexible servers decreases as

the fraction of flexible servers increases.

4.2.

How much flexible capacity could be gained if we

replace dedicated servers by flexible servers? Figure 3

shows the mean downtime as server utilization in-

Flexible Capacity

creases for various values of the fraction of flexible

servers when s ? 20. As we can see from Figure 3, a

given mean downtime can be achieved by using dif-

ferent fraction of flexible servers, each with a different

utilization rate. By increasing the fraction of flexible

servers, we can also increase the server utilization in

order to provide the same mean downtime. For exam-

ple, by increasing the fraction of flexible servers from

0.1 to 0.2, the server utilization increases from U1to U2

(see Figure 3), with a net capacity gain of (U2-U1). Note

that with the assumption that ?1? ?2? ?12? 1,

(U2-U1) corresponds to a net increase in the total ar-

rival rate that could be handled by the system while

Figure 2Mean Downtime as the Fraction of Flexible Servers Increases (s ? 20, U0? 0.9).

Figure 3 Impact of Increasing Fraction of Flexible Servers on Mean Downtime When s ? 20.

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maintaining the same mean downtime. We can also

see from this figure that the net gain in capacity de-

creases as we increase the fraction of flexible servers.

To see this clearly, we display the net increase in

capacity for every 10% increase in flexible servers in

Figure 4, when s ? 20, U0? 0.7, and Ws? Ws(fF? 0).

Here, Ws(fF? 0) denotes the mean downtime for a

system with all dedicated servers.

Although at higher server utilization, the gain in

actual flexible capacity due to increased fraction of

flexible servers is small, the value of flexible capacity is

high. This is due to the fact that the aggregate mean

downtime increases rapidly as server utilization in-

creases. In order to understand the value of flexible

capacity, we now develop a model that takes into

account the costs associated with the delay and the

flexible servers.

4.3.

In this section, we first develop an expression for the

average total cost per unit time. We then find opti-

mum fraction of flexible servers by minimizing the

average total cost per unit time under varying system

conditions. We make the following additional as-

sumptions:

• Service cost per unit time (in salary, benefits and

amortized training cost) is c ? 0 for dedicated

servers of either type, and is c ? p for flexible

servers, where p is a cost premium. Since flexible

servers have multiple skills and hence cost more,

we assume that p is non-negative.

• For customers of type i, the downtime cost (or

delay cost) per unit time per customer is di, i ? 1,

Optimum Fraction of Flexible Servers

2. We assume that type 2 customers are more

time-sensitive and have a higher downtime cost

per unit time (0 ? d1? d2? ?). Hence, it is

reasonable to assume a non-preemptive priority

policy with flexible servers for type 2 customers.

• There is a cost of l ? 0 units for every type 2

customer lost due to finite buffer of size K. We

assume l to be very large so that choosing a small

value for K is discouraged.

Recall that mean downtime for type i customers is

denoted by Ws

function, then the average total cost per unit time is

given by (see, for example, Hillier et al. 2000)

i, i ?1, 2. If we assume a linear cost

Z1?s, s1, r? ? ?s1? s2?c ? r?c ? p? ? d1?1Ws

? d2?2

? sc ? rp ? d1?1Ws

? l?2?s2?K

1

eWs

2? l?2?s2?K

1? d2?2

eWs

2

(3)

Since ?2?s2?Kis the number of type 2 customers lost

per unit time (see v in the appendix), Equation (3)

gives the sum of the average service cost, the average

customer downtime cost, and the average cost of lost

customers per unit time. For a given set of parameters

(?1, ?2, ?1, ?2, K, c, p, d1, d2, l), we can find the optimal

number of servers by solving Mins,s1,rZ1(s, s1, r). How-

ever, to find the optimum fraction of flexible servers

for a given s, we consider the following objective

function

Z1?s, s1, r? ? sc ? d2Z2?s, s1, r?

(4)

where

Figure 4Amount of Increase in the Flexible Capacity for Every 10% Increase in the Fraction of Flexible Servers (s ? 20, U0? 0.7).

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Z2?s, s1, r? ? R1r ? R2?1Ws

1? ?2

eWs

2? R3?2?s2?K

(5)

R1? (p/d2), R2? (d1/d2), and R3? (l/d2). Here, 0 ? R2

? 1 since 0 ? d1? d2? ?. For a given s, since sc and

d2are constant, if s*1and r* minimize Z1(s, s1, r) in

Equation (4), then s*1and r* also minimize Z2(s, s1, r) in

Equation (5). Thus, for a given s, the objective is to find

Z*2(s) ? Mins1,rZ2(s, s1, r). We denote the optimal

fraction of flexible servers by f*Fand it is equal to (r*/s).

In the following examples, we assume that d1? d2

? d, so that R1? (p/d), R2? 1, and R3? l/d. Since p

is the cost premium rate for a flexible server and d is

the downtime cost per customer per unit time (either

type 1 or type 2), the ratio (p/d) measures the cost rate

of flexibility in terms of (or relative to) cost rate of

downtime. We call R1? (p/d) as “relative cost of flex-

ibility”. In the service system we studied, costs of

equipment are very high, resulting in a higher down-

time cost per unit time than the premium rate for a

flexible server. Hence, in general, relative cost of flexi-

bility varies between zero and one. In all the following

examples, for a given s and U0, we first calculate the

common arrival rate that results in a utilization of U0.

We maintain this arrival rate but vary the proportion

of flexible servers in the system. Note that for a given

K and s, when ?1? ?2? ?12? 1, the aggregate

utilization factor remains constant when we vary the

fraction of flexible servers.

4.3.1.

ure 5 depicts how the optimum fraction of flexible

servers changes as the total number of servers in the

system (s) and relative cost of flexibility (R1) change. As

we will see later, the need for flexibility is greater

when the server utilization is high. Hence, we assume

here U0? 0.9, R2? 1, R3? 100, and E ? 1. Obviously,

when R1? 0, the premium rate for flexible servers is

zero, and hence it is always optimal to have all flexible

servers no matter how many servers are in the system.

In the case when s is small, it is interesting to note that,

even with a significant increase in the relative cost of

flexibility R1,total flexibility is still optimal. Even for

large s (s ? 30), when R1increases from 0 to 0.1, the

optimum fraction of flexible servers drops from 1 to

only 0.55. Only for large s, the need for flexibility

rapidly decreases as R1increases. Hence, using very

limited flexibility, say 10%, is optimal only for large

values of s and R1(s ? 30, and R1? 0.6) and not

optimal when either s or R1is small. Furthermore, the

need for flexibility increases as the total number of

servers in the system decreases and the need for flex-

ibility decreases as the relative cost of flexibility in-

creases.

4.3.2.Total Cost Curve. In order to see how the

total cost curve changes as R1increases, we present the

curves for the average total cost per unit time in Figure

Varying the Total Number of Servers. Fig-

Figure 5Impact of Total Number of Servers on Optimum Fraction of Flexible Servers as R1Increases (U0? 0.9, R2? 1, R3? 100, and E ? 1).

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6, when s ? 20 and U0? 0.9. As we can see from the

figure, the total cost corresponding to R1? 0 is some-

what flat in the range of high values of fraction of

flexible servers. Hence, even for a small increase in the

relative cost of flexibility (from 0 to 0.3), the optimum

fraction of flexible servers drops from 1 to 0.4.

4.3.3.Vary Server Utilization. The impact of

server utilization on the optimal fraction of flexible

servers is depicted in Figure 7. When the server utili-

zation is high, the optimal fraction of flexible servers

does not drop rapidly from 1, as R1increases. Hence,

using a very limited fraction of flexible servers, say

10%, when utilization is high and R1is reasonably

high is not optimal. When the server utilization is

large (U ? 0.9), using very limited flexibility is appro-

priate only for large value of R1(R1? 0.5). Overall, the

need for flexibility increases as server utilization in-

creases and the need for flexibility decreases as the

relative cost of flexibility increases.

We now summarize the results obtained in Sections

4.3.1 and 4.3.3 to provide some approximate guide-

lines for managers to choose appropriate server mix.

The basic questions we address here are: (1) “What are

the conditions under which a manager can choose

between the two extreme choices, full flexibility and

very limited or no flexibility?”, and (2) “When should a

manager choose an optimal server mix?” Accordingly,

we group the options available in the server mix

choice into three categories: (a) A full or nearly full

flexibility option corresponds to the situation when 0.9

? f*F? 1, (b) A limited or no flexibility option corre-

sponds to the situation when 0 ? f*F? 0.2, and (c) An

optimized server mix option corresponds to the situation

when 0.2 ? f*F? 0.9. Table 1 summarizes these guide-

lines. As we can see from the table, optimizing server

mix is important in the following three situations:

(i) {0.1 ? R1? 0.5; 0.6 ? U ? 0.9; s ? 6}

(ii) {0.1 ? R1? 0.5; U ? 0.9; s ? 6}, and

(iii) {0.5 ? R1? 1; U ? 0.9; s ? 15}.

In all other situations, a manager can choose one of

the two extreme choices listed in (a) and (b) that is

appropriate.

Figure 6 Curve Representing Average Total Cost Per Unit Time Corre-

sponding to Selected Values of R1(s ? 20, U0? 0.9, R2? 1,

R3? 100, and E ? 1).

Figure 7 Impact of Server Utilization on Optimum Fraction of Flexible Servers for Selected Values of U0(s ? 20, R2? 1, R3? 100, and E ? 1).

Chakravarthy and Agnihothri: Optimal Workforce Mix in Service Systems with Two Types of Customers

Production and Operations Management 14(2), pp. 218–231, © 2005 Production and Operations Management Society

225

Page 9

4.3.4.

previous examples, we assumed that the flexible serv-

ers are as fast as specialized servers. However, cross-

training in multiple skills or working with multiple

customer types could lead to a loss of efficiency as

compared to a server who is dedicated to one cus-

tomer type. Intuitively, if the efficiency is reduced sig-

nificantly, it is not advisable to conduct extensive cross-

training.Notethatefficiencyofflexibleserversisdefined

as E ? ?12/?. Unlike before, reducing server efficiency

increases the server utilization due to an increase in

the mean service time. The system becomes unstable

when the flexible server efficiency is too low. We set ?

? 1, U0? 0.8 when E ? 1, and varied E (by varying

?12) from 0.80 to 1. The results are shown in Figure 8.

As we can see from Figure 8, the impact of the

efficiency of cross-trained servers on the optimal

server mix decisions is quite significant. First, we note

Inefficient Cross-Trained Servers. In all the

that when the efficiency of cross-trained workers is

less than 100%, full flexibility is never optimal. Sec-

ond, even when the relative cost of flexibility is zero (R1

? 0), a small drop in efficiency to 95% results in the

reduction of optimal fraction of flexible servers from

100% to 30%. At an efficiency of 80%, only 10% of the

servers should be flexible. As we can see from this

example, the benefits of having flexible servers vanish

rapidly as the flexible servers’ efficiency decreases.

Thus, when flexible server efficiency is less than 100%,

it is always desirable to use very limited or no flexi-

bility. Third, from Table 1, we see that for low server

utilization, limited or no flexibility is always optimal

when E ? 1. This result, together with extensive nu-

merical analysis, reveals that when the cross-trained

servers’ efficiency drops below 80%, using all dedi-

cated servers is always optimal (irrespective of the

system size and the server utilization factor).

Table 1 Approximate Guidelines for Server Mix Decisions When Flexible Servers are 100% Efficient

Server utilization

Total Number of Servers (System Size)

s ? 6 (Small)6 ? s ? 15 (Medium)s ? 15 (Large)

0 ? U ? 0.6

(Low)

0.6 ? U ? 0.9

(Medium)

Limited or no flexibilityLimited or no flexibilityLimited or no flexibility

0 ? R1? 0.1: Nearly/Full flexibility

0.1 ? R1? 0.5: Optimize server mix

R1? 0.5: Limited or no flexibility

0 ? R1? 1: Nearly/Full flexibility

R1? 1: Limited or no flexibility

0 ? R1? 0.1: Nearly/Full flexibility

R1? 0.1: Limited or no flexibility

0 ? R1? 0.1: Nearly/Full flexibility

R1? 0.1: Limited or no flexibility

0.9 ? U ? 1

(High)

0 ? R1? 0.1: Nearly/Full flexibility

0.1 ? R1? 1: Optimize server mix

R1? 1: Limited or no flexibility

0 ? R1? 0.1: Nearly/Full flexibility

0.1 ? R1? 0.5: Optimize server mix

R1? 0.5: Limited or no flexibility

Figure 8Impact of Flexible Servers’ Efficiency on Optimum Fraction of Flexible Servers for Selected Values of E (s ? 20, U0? 0.8, R2? 1, and R3? 100).

Chakravarthy and Agnihothri: Optimal Workforce Mix in Service Systems with Two Types of Customers

Production and Operations Management 14(2), pp. 218–231, © 2005 Production and Operations Management Society

226

Page 10

4.3.5.

So far, we have assumed the arrival rates of the two

customer types are equal. We conducted numerical

analysis with varying the proportion of type 1 custom-

ers between 0 and 1. Since the arrivals are Poisson, the

uncertainty about the type of an arriving customer is

highest when ?1? ?2. Hence, the value of the flexibil-

ity is greatest when ?1? ?2. This also could be seen

from Table 2. To measure the impact of the variation

in the inter-arrival time on cross-training, the arrival

process should be assumed non-Poisson, and this will

be a topic for future research.

In all the analysis done so far, we held constant the

total number of servers and varied the proportion of

flexible servers. We conducted extensive numerical

analysis by relaxing this assumption to find an opti-

mal number of servers and optimum fraction of flex-

ible servers simultaneously. Since the results are not

quite revealing, we decided not to report them here.

Unequal Arrival Rates of Customer Types.

5.

In this paper, we considered a service system with two

types of customers and multiple servers who are either

specialists or cross-trained. We quantified flexible capac-

ity and showed that capacity could be gained by replac-

ing dedicated servers by flexible servers. However, the

net gain in capacity decreases as we increase the fraction

of flexible servers. Taking into account the service cost

and customer downtime cost, we showed that total flex-

ibility is not always the best choice. We developed

guidelines for managers to choose appropriate server

mix. In particular, we developed a table that could be

used to identify the conditions under which a manager

can choose between full flexibility and limited flexibility.

We showed that a manager needs to optimize the server

mix under the following system conditions: (i) Relative

cost of flexibility is between 0.1 and 0.5, server utilization

Summary and Future Research

is medium, and the system size is small; (ii) Relative cost

of flexibility is between 0.1 and 0.5, server utilization is

high, and the system size is medium or large; and (iii)

Relative cost of flexibility is between 0.5 and 1, server

utilization is high, and the system size is large. Finally,

we showed that the benefits of having flexible servers

vanish rapidly as the flexible servers’ efficiency de-

creases. When flexible servers’ efficiency drops below

80%, using only specialists is always optimal irrespective

of the system size and the server utilization.

Although there are several research opportunities in

this area, we identify three that we consider to be

important. First, the assumptions of exponential inter-

arrival times and service times could be relaxed to

incorporate general distributions. Second, we assume

in this paper that the initial diagnosis is perfect. How-

ever, the initial diagnosis could be imperfect in prac-

tice resulting in dispatching a wrong type of specialist

to repair the equipment. Since the travel times are

quite significant, the mismatch between a server type

and a customer type could lead to server redeploy-

ments and hence lengthy downtime. Therefore, mod-

els considering server mismatch could be quite useful.

Finally, a useful extension is to consider more than

two types of customers. Partial cross-training would

then be a choice, and as a result, number of possible

server types increases rapidly and developing exact

models become quite challenging. Using simulation,

Agnihothri and Mishra (2004) study a model with

three job types and with server-job mismatch. They

consider and compare a host of system configurations

including the concept of chaining.

Acknowledgments

The thoughtful comments and suggestions of three anony-

mous reviewers are very much appreciated.

Table 2Comparison of the Optimum Fraction of Flexible Servers When ?1? ?2, with the Case when ?1? ?2for Varying U0, s, and R1.

Here, p1Represents the Proportion of Type 1 Customers

R1

s ? 6s ? 10s ? 20

p1? 0.1

p1? 0.5

p1? 0.1

p1? 0.5

p1? 0.1

p1? 0.5

U0? 0.6

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

1

1

1

1

1

0

0

0

1

0.7

0.3

0.3

1

1

1

1

0.1

0.1

0.1

0.1

1

0.1

0.1

0.1

1

1

0.1

0.1

0.7

0

0

0

0.9

0.3

0.2

0

1

1

0.6

0.4

0.8

0

0

0

1

0.1

0.1

0

1

0.3

0.2

0.1

1

0

0

0

1

0.1

0

0

1

0.4

0.1

0.1

U0? 0.8

U0? 0.9

Chakravarthy and Agnihothri: Optimal Workforce Mix in Service Systems with Two Types of Customers

Production and Operations Management 14(2), pp. 218–231, © 2005 Production and Operations Management Society

227

Page 11

Appendix A.1.

Analysis

In this appendix, we develop and analyze the model for the

system defined in section 3. The underlying system can be

formulated as a continuous-time Markov process. For i ? 0,

1, . . . , r-1, define the set of states i*? {(i, j, k): i flexible

servers busy, j type 1 servers busy (with no customers in

queue), and k type 2 servers busy (with no customers in

queue); j ? 0, 1, . . . , s1; k ? 0, 1, . . . , s2}. For j ? 0, 1, . . .

define the set of states j ? {(j, k): all r flexible servers are

busy, there are j type 1 customers (either with type 1 servers

or waiting in the queue), and k type 2 customers (either with

type 2 servers or waiting in the queue); k ? 0, 1, . . . , s2? K}.

Then the state space of the Markov process governing the

system is ? ? {0*, 1*, . . . , (r-1)*, 0, 1, . . . }.

In the following the notation e and I will stand, respectively,

for a unit column vector with 1’s and an identity matrix of

appropriate dimensions. We denote the transpose of vector e

by e?. The notation R; will stand for Kronecker product of two

matrices. Thus if A is a matrix of order m ? n, and B is a matrix

of order of order p ? q, then ARB will denote a matrix of order

mp ? nq whose (i, j)thblock matrix is given by aijB. For more

details on Kronecker product, we refer to Bellman (1960).

Based on the above formulation, the infinitesimal generator of

the Markov process governing the system is given by

Q ??

Markov Model and Steady State

B˜0

?12I

0

···

0

0

0

0

0

0

···

B˜1

0

B˜1

0

0

B˜1

···

0

0

0

0

0

0

···

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

0

0

0

···

B˜0? ?12I

2?12I

···

0

0

0

0

0

0

···

B˜0? 2?12I

···

0

0

0

0

0

0

···

?r ? 1??12I

0

0

0

0

0

···

0

0

0

···

0

0

0

···

0

0

0

···

0

0

0

···

B˜0? ?r ? 1??12I

e1

e2

···

es1?1

?C1

0

···

0

0

0

···

· · ·

es1?B2

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

e1?B2

C0

?1I

···

0

0

···

e2?B2

?1I

C0? ?1I

···

0

0

···

e3?B2

0

?1I

···

0

0

···

??C1

??C1

?

0

0

0

···

0

0

0

···

0

0

0

···

0

0

0

···

0

0

0

···

0

0

0

···

0

· · ·

· · ·

· · ·

···

· · ·

· · ·

· · ·

···

· · ·

· · ·

es1?1??B2? ?1I?

0

0

···

C0? s1?1I

A1

···

0

0

···

s1?1I

0

···

?1I

A0

···

?1I

···

···?

where, letting a ? ? ? r?12, b ? ? ? r?12? s1?1,

B˜0??

B0

?1I B0? ?1I ?1I · · ·

···

00

?1I

0· · ·

· · · s1?1I B0? s1?1I?,

0

0

···

0

0

···

···

···

· · ·

0

B˜1??

B1

0

···

0

0

B1

···

0

· · · 0

· · · 0

· · ·

0

0

···

···

· · · 0 ?1I ? B1?,

B˜2??

B2

0

···

0

0

B2

···

0

· · · 0

· · · 0

· · ·

0

0

···

···

· · · 0 B2? B3?,

B0??

??

?2

···

0

?2

0

?2

···

0

??? ? ?2?

···

0

· · ·

· · ·

· · ·

· · ·

· · ·

0

0

···

0

0

···

?2

??? ? ?s2? 1??2?

s2?2

??? ? s2?2??

,

B1??

0

0

···

0

0

0

···

0

· · ·

· · ·

· · ·

· · ·

0

0

···

?2?

,

B2? ?B1

0?,

B3? ??1I

0?,

C1??

r?12I

0 ?,

C0??

?a

?2

0

···

0

0

···

0

0

?2

0

?2

0

0

?2

···

0

0

···

0

0

??a ? ?2?

2?2

···

0

0

···

0

0

??a ? 2?2?

···

0

0

···

0

0

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

0

0

0

···

0

0

0

···

?2

0

0

0

···

0

?2

···

0

0

??a ? s2?2?

r?12? s2?2

···

0

0

??a ? s2?2?

···

0

0

Chakravarthy and Agnihothri: Optimal Workforce Mix in Service Systems with Two Types of Customers

Production and Operations Management 14(2), pp. 218–231, © 2005 Production and Operations Management Society

228

Page 12

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

0

0

0

···

0

0

···

0

0

0

···

0

0

···

?2

??a ? s2?2?

r?12? s2?2

??a ? ?2? s2?2??

,

A0??

?b

?2

0

···

0

0

···

0

0

?2

0

?2

0

0

?2

···

0

0

···

0

0

??b ? ?2?

2?2

···

0

0

···

0

0

??b ? 2?2?

···

0

0

···

0

0

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

0

0

0

···

0

0

0

···

?2

0

0

0

···

0

?2

···

0

0

??b ? s2?2?

r?12? s2?2

···

0

0

??b ? s2?2?

···

0

0

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

0

0

0

···

0

0

···

0

0

0

···

0

0

···

?2

??b ? s2?2?

r?12? s2?2

??b ? ?2? s2?2??

,

A1??

?r?12? s1?1?I

0

0

s1?1I?.

Let A ? A0? A1? ?1I and let ? be the steady state

probability vector of A. That is, ?A ? 0, ?e ? 1. Then

the condition that the Markov process with generator

Q is ergodic is given by

?A1e ? ?1. (1)

It is easy to verify that (1) can be written explicitly in

terms of the input parameters as follows:

?1? s1?1? r?12

??

?

j?0

s21

j!?

?2?

?2

?2?

j

?

j?0

s21

j!?

?2

?2?

j

?

1

s2!?

?2

s2?

k?1

K?

?2

r?12? s2?2?

k?

.

Hence, the steady-state probability vector of Q exists

and is of matrix-geometric type [See Neuts (1981)]. Let

? ? (?0, . . . , ?r-1, x0, x1, . . . ) be the steady-state

probability vector of Q. That is, ?idenotes the steady

state probability that the system is in state i*; i ? 0, 1,

. . . , (r-1); and xjdenotes the steady state probability

that system is in state j; j ? 0, 1, . . . . Note that ? Q ? 0

and ?e ? 1. The steady-state equations are given by

?0B˜0? ?12?1? 0

?i?1B˜1? ?i?B˜0? i?12I? ? ?i ? 1??12?i?1? 0,

1 ? i ? r ? 2,

?r?2B˜1? ?r?1?B˜0? ?r ? 1??12I?

??

j?0

s1

xj?ej?1

?

?C1? ? 0 (2)

?r?1?e1?B2? ? x0C0? ?1x1? 0

?r?1?ei?1?B2? ? ?1xi?1? xi?C0? i?1I?

? ?i ? 1??1xi?1? 0,1 ? i ? s1? 1,

?r?1?es1?1??B2? ?1I?? ? ?1xs1?1? xs1?C0? s1?1I?

? xs1RA1? 0,

xi? xs1Ri?s1,

i ? s1,

with the normalizing equation: ¥i?0

? xs1(I ? R)?1e ? 1, and R is the unique solution to the

matrix-quadratic equation:

r?1?ie ? ¥i?0

s1?1xie

R2A1? RA0? ?1I ? 0.

The coefficient matrices appearing in (2) are very

sparse and are exploited when using Gauss-Seidel

iterative method to solve the equations.

Appendix A.2.

We will list some important performance measures along

with their formulas. These measures are used to bring out

the qualitative behavior of the queueing model under study.

Some obvious results are stated for completeness and details

of the derivations are omitted wherever it is obvious.

i. P(system is idle)??000.

System Performance Measures

Chakravarthy and Agnihothri: Optimal Workforce Mix in Service Systems with Two Types of Customers

Production and Operations Management 14(2), pp. 218–231, © 2005 Production and Operations Management Society

229

Page 13

ii. P(j type 1 servers are busy)

??

?

i?0

?

i?0

r?1

?ije ? xje,

0 ? j ? s1? 1,

r?1

?is1e ? xs1?I ? R??1e,

j ? s1.

iii. P(k type 2 servers are busy)

??

?

i?0

?

i?0

r?1?

j?0

s1

?ijk??

?ijs2??

j?0

?

xjk,

0 ? k ? s2? 1,

r?1?

j?0

s1

j?0

??

l?ss

s2?K

xjl,

k ? s2.

iv. P?i flexible servers are busy?

??

?ie,

?

j?0

0 ? i ? r ? 1,

?

xje,

i ? r.

The calculation of the last two performance measures

is made easy on noting

? ??

j?0

?

xj? xs1?I ? R??1? ?

j?0

s1?1

xj.

v. The throughput ? of the system is given by ?

? ?1? ?2

arrival rate of type 2 customers, and ?s2?Kis the prob-

ability that the buffer for type 2 customers is full.

vi. The probability mass function for queue length

of type 1 customers is given by

e, where ?2

e? ?2(1 ? ?s2?K) is the effective

fj

1? xs1Rje,

j ? 0.

vii. The probability mass function for queue length

of type 2 customers is given by

fk

2??

j?0

?

xj,s2?k,0 ? k ? K.

viii. The means and standard deviations of the

queue lengths of type 1 and type 2 customers, and

number of busy type 1, type 2, and flexible servers, can

be computed using their respective probability func-

tions. We denote these in the sequel by

Nq

i? E?number of type i customers in the queue?,

i ? 1, 2;

E?BSi? ? E?number of busy type i servers?, i ? 1, 2;

E?BFS? ? E?number of busy flexible servers?.

ix. The mean waiting times in the queue of a type 1

and type 2 customers are obtained using Little’s law.

x. The waiting time distribution of a type 2 cus-

tomer in the system is of phase type with representa-

tion (?, T) of order K?2 where the components of ?

are given by

?1? P(at least one dedicated type 2 server is avail-

able at an arrival of an admitted type 2 customer)

???

i?0

j?0

k?0

i?0

?2? P (at least one flexible server is available with all

dedicated type 2 servers busy at an arrival of an

admitted type 2 customer)

???

i?0

j?0

?i? P [there are (i ? 3) type 2 customers in the queue

at an arrival of an admitted type 2 customer]

? ?xs1?I ? R??1es2?i?/?1 ? ?s2?K?,

and the matrix T is of the form

T ??

00

r?1?

s1?

s2?1

?ijk??

??

j?0

s2?1

xij???1 ? ?s2?K?,

r?1?

s1

?ijs2??(1? ?s2?K),

3 ? i ? K ? 2,

??2

0

s2?2

0

0

···

0

00

0

0

0

??12

r?12

0

0

···

0

??s2?2? r?12?

?s2?2? r?12?

0

···

0

0

??s2?2? r?12?

?s2?2? r?12?

···

0

0

??s2?2? r?12??

0

0

0

0

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

· · ·

0

0

0

0

0

···

0

0

0

0

0

···

0

??s2?2? r?12?

···

0

0

??s2?2? r?12?

0

.

The mean waiting time in the system for a type 2

customer is calculated as

Ws

2? ???T??1e

??1

?2?

?2

?12?

?r ? s2?

r?12? s2?2?

j?2

K?2

?j

?

1

r?12? s2?2?

j?3

K?2

?j ? 2??j.

xi. The mean number of type 2 customers in the

system is calculated using Little’s law as

Ns

2? ?2

eWs

2.

Chakravarthy and Agnihothri: Optimal Workforce Mix in Service Systems with Two Types of Customers

Production and Operations Management 14(2), pp. 218–231, © 2005 Production and Operations Management Society

230

Page 14

xii. The mean number of type 1 customers in the

system is calculated as

Ns

1? Nq

1? Nq

2? E?BS1? ? E?BS2? ? E?BFS? ? Ns

2.

xiii. The mean waiting time in the system (mean

downtime) for a type 1 customer is calculated as

Ws

1? Ns

1/?1.

xiv. The mean waiting time in the system for a

typical customer is calculated as

Ws??

?1? ?2

?1

e?Ws

1??

?2

e

?1? ?2

e?Ws

2.

xv. The server utilization factor (expected number

of busy servers/total number of servers) is calculated

as

U ? ?E?BS1? ? E?BS2? ? E?BFS??/?s1? s2? r?.

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