Pooling Strategies for Call Center Agent Cross-training

Eylem Tekin∗

Department of Statistics and Operations Research

University of North Carolina-Chapel Hill

Chapel Hill, NC 27599-3180

eylem@unc.edu

Wallace J. Hopp

Department of Industrial Engineering and Management Sciences

Northwestern University

Evanston, IL 60208-3119

hopp@northwestern.edu

Mark P. Van Oyen

Department of Information Systems & Operations Management

Loyola University Chicago

25 E. Pearson Street

Chicago, IL 60611-2196

mvanoye@luc.edu

November 1, 2004

∗Corresponding author: Phone: (919) 962-3834 Fax: (919) 962-0391

Abstract

We consider call center environments where agents serve distinct customer types, and

investigate the efficiency benefits achievable via cross-training. We do this by first considering

specialized agents grouped into N departments according to the customer type they serve.

Then we examine cross-training policies that pool a set of departments K (selecting k = |K|of N) into a single larger department in which every agent serves all of the pooled call

types. For the pooled department, we analyze both First-Come-First-Served (FCFS) and

Non-Preemptive Priority (NPP) service disciplines. By comparing the resulting queueing

models via standard queueing approximations and numerical analysis, we characterize the

impact of system parameters, such as department sizes, arrival rates and service times, on

the decision of what departments to pool.

Keywords: Multi-class queueing systems, call centers, pooling, cross-trained servers, average

waiting time, flexibility.

1 Introduction

The proliferation of call centers in the global economy has elevated interest in the efficient

design and management of these systems. A practice that is particularly well suited to

improving call (or contact) centers is agent cross-training. Cross-training can increase staffing

and scheduling flexibility to improve customer service through more choices for matching

agents to customer service requests and enhance efficiency by reducing the total number of

agents needed to handle a given call load. Effective use of cross-training requires two steps:

(1) design, that determines which agents are trained to handle which customer types, and (2)

control, in which customers are dynamically assigned to agents. In this paper, we concentrate

on the design step by considering situations where the control step is straightforward due to

the departmental structure of the system.

A call center has a departmental structure if the agents can be divided into groups

such that each customer type is unambiguously assigned to a single group (department) for

service. The customer type to group assignment can be one-to-one, or if agents are cross-

trained to serve more than one type, many-to-one. For example, in a multilingual call center,

departments could correspond to groups of agents that handle customers from a specific set

1

of languages (Brigandi et al. 1994, Aksin and Karaesmen 2002). If all French speaking agents

also speak English and vice-versa, then we could pool the French and English departments

and route both call types to the combined department. However, if some, but not all

French speakers speak English, then the system no longer has a department structure, since

French/English agents cannot be assigned to a unique department. When the cross-training

pattern follows a non-departmental structure, the control problem of assigning customers to

agents become more complex (see Gans et al. 2003 for discussions). However, in many call

centers, such as financial services support, where customer types correspond to the products

about which they are calling, grouping agents into merged departments is a common practice

(Evenson et al. 1999). In such environments, it is important to know which departments to

pool via cross-training of agents. This paper provides insight into this problem.

To understand call centers with departmental structures, we consider environments in

which agents are initially grouped into departments based on the customer type(s) they are

assigned. We assume that departments differ with respect to parameters such as arrival rate,

mean service time, variability in service time and number of agents. Then we examine the

impact of pooling, namely combining two or more departments into a larger department with

the agents in the pooled department cross-trained to handle all of the call types of those

departments. This involves two fundamental issues: (1) how many departments to pool and

(2) which departments to pool. The specific focus of this paper is to determine how the

various system parameters – arrival rates, mean service times, variability in service times

and the number of agents – effect these pooling decisions.

Pooling has been widely studied in the literature (for example, see Mandelbaum and

Reiman 1998), usually as a comparison between the two extreme scenarios of a dedicated

system versus a fully pooled system. In a dedicated system, each group of servers is

responsible for a specific customer type while in a fully pooled system all servers are merged

into a single group to serve all customer types. However, in many real life situations, and

particularly in call centers, cross-training all agents to handle all call types may not be

feasible due to cost and/or quality penalties arising from cross-training, excessive stress,

and/or scarcity of agents capable of handling all of the call types. Although dedicated and

fully pooled systems are considered in this paper, we emphasize partial pooling scenarios in

which only some of the departments are pooled, while others continue to function as dedicated

2

departments. Specifically, we consider a call center consisting of N dedicated departments,

and seek to pool k ≤ N departments into a larger department so as to minimize the average

waiting time of customers in queue. However, because we recognize that using average

waiting time as the objective can lead to policies that degrade service to some customer

types, we also derive the sufficient conditions for pooling to reduce average waiting of all

customer types.

The remainder of this paper is organized as follows: Section 2 presents a literature

review. Section 3 develops models of the pooling strategies we consider. Section 4 evaluates

the relative effectiveness of these strategies and draws general conclusions about the benefits

of pooling. Section 5 concludes the paper.

2 Literature Review

The literature on call centers has been growing rapidly in recent years. Gans at al. (2003)

provided the most recent survey of this literature, as well as an excellent overview of the

system mechanics of call centers. Most analytic call center literature has focused on queueing

models that are based on the assumption that the customers and agents are homogenous

(see Koole and Mandelbaum 2002 for a thorough review and Whitt 2002 for future research

directions).

A large body of the queuing literature on call centers focuses on dynamic routing of

calls to available agents with the right skills. Perry and Nilsson (1992) considered a system

in which two classes of calls are served by a single pool of agents (V-design) and determined

both the number of agents and the assignment policy needed to yield specified expected

waiting times. Bhulai and Koole (2000) also modeled a variant of this system. Stanford

and Grassman (2000) considered a call center with two call types and two agent groups

where one group can handle both call types and the other can handle only one of them (N-

design), and used fixed priority policies and matrix-geometric methods for performance and

staffing analysis. Shumsky (2000) considered the same problem and proposed an approximate

analysis. Borst and Seri (2000) proposed a dynamic routing rule that prioritizes the call type

that is farthest behind the schedule, and determined bounds for the number of fully cross-

trained agents to offer a given service level. Koole and Talim (2000) modeled a multi-skill

call center as a network of queues and approximated each queue as an M/M/r loss system

3

to minimize the number of unanswered calls. Gans and Zhou (2003) studied a call routing

problem with two call types where one call always has priority over the other one and used

a Markov decision process model to achieve service level constraints.

Another stream of queueing based call center research focuses on heavy traffic analysis

and obtains asymptotic results. Harrison and Lopez (1999) considered the optimal dynamic

assignment of n non-identical servers working in parallel to m types of customers to minimize

waiting costs. They showed that if the server skills overlap in an appropriate manner, then in

the limiting Brownian control problem all servers merge into a pool of single service capacity.

Bell and Williams (2001) proved the asymptotic optimality of threshold controls for the N-

design discussed above. In contrast, Van Mieghem (1995) proved asymptotic optimality of

a simple generalized cµ rule for a V-design with waiting costs that are convex increasing.

Harrison and Zeevi (2004) studied staffing large call centers using stochastic fluid models.

Armony and Maglaras (2004a, 2004b) considered multi-class, multi-server call centers with

a call-back option, and proposed asymptotically optimal routing and staffing policies.

The literature most closely related to this paper has focused on choosing appropriate

skill sets for servers usually by comparing dedicated and fully pooled systems. The basic

pooling models on which this work is based on is described in Kleinrock (1976 pp.272-290).

Smith and Whitt (1981) and Benjaafar (1995) showed that a pooled system is better than

a dedicated one if arrivals and service times have the same distribution. Buzacott (1996)

considered a serial system with n stages where each customer is served by a distinct server

at each stage, and transformed this system into a parallel system with n servers in which

each server can perform the operations required for all n stages. He showed that the pooled

system where all servers can perform all tasks is superior to the unpooled alternative, and

the higher the task variability the greater the advantage. Mandelbaum and Reiman (1998)

considered pooling of a Jackson network into an M/PH/1 queue. They compared the pooled

and the unpooled system in terms of the steady state mean sojourn times and showed that

depending on the system parameters, pooling can be either good or bad. Andradottir and

Argon studied the effects of pooling several adjacent stations in tandem lines where servers

are cross-trained and two or more servers can operate on the same job. They showed that

benefits of pooling can be substantial and the bottleneck station should be among the pooled

stations to obtain the greatest improvement. Our research differs from this literature in

4

that we consider parallel service environments (e.g., call centers), and by comparing the

performance of many (partially) pooled systems we provide guidelines for choosing the best

alternative.

Finally, there are also papers that propose alternatives to pooling. Sheikhzadeh et

al. (1998) and Jordan et al. (2004) studied chaining of servers where each customer can be

routed to one of two adjacent servers and each server can process customers from two adjacent

classes. These studies showed that chaining has the potential to achieve most of the benefits

of pooling. Hopp et al. (2004) investigated value of chaining in serial production lines, and

showed that the impact of forming a complete chain of skills sets can be substantial. Iravani

et al. (2004) developed a structural flexibility index for systems with cross-trained servers

that quantifies the flexibility inherent in an arbitrary system structure.

3 Pooling Strategies

We consider a call center that services N customer types. We assume that the system has

a departmental structure, so that customer type i is served by an agent in department i,

i = 1, 2, . . . , N . We further assume that customer type i arrives according to a Poisson

process with mean λi, and requires a service time drawn from an i.i.d. sequence with mean

Ti and squared coefficient of variation ν2i . Department i has ci servers. Hence, the initial

system can be modeled as N M/G/c queues in parallel.

We define the pooling of the queues defined by set K (K ⊆ {1, 2, . . . , N}) as the

merging of the k = |K| departments in set K by cross-training all servers in the pooled

departments to handle all customer types in K (see Figure 1 for an illustration of pooling

departments 1 and 2 in a system with N = 3). Our interest is in determining which k

departments to pool in order to achieve the largest reduction of average customer waiting

time. We denote the average waiting time of customers in the queue of department i,

i = 1, 2, . . . , N , by Wi. We assume that the original system is stable, which requires

ρi =λiTi

ci

< 1, i = 1, 2, . . . , N.

Because standard performance measures cannot be expressed in closed form for the M/G/c

queue, a variety of approximation methods have been proposed. A number of authors have

independently proposed the following approximation for the average delay in queue, denoted

5

λ1, Τ1, ν1 c1

servers

c2

servers

λ2, Τ2, ν2

c3

servers

λ3, Τ3, ν3

Specialist System

c1+c2

servers

c3

servers

Pooled System

Figure 1: Pooling of two departments

by Wi (Krampe et al. 1973, Stoyan 1976, Nozaki and Ross 1978, Hokstad 1978):

Wi =1 + ν2

i

2λi

ρi(ρici)ci

ci!(1− ρi)2p0(ci, ρi) =

1 + ν2i

2λi

g(ci, ρi) (1)

where

p0(ci, ρi) =

[ci−1∑n=0

(ρici)n

n!+

(ρici)ci

ci!(1− ρi)

]−1

(2)

is the probability of having no customers, and

g(ci, ρi) =ρi(ρici)

ci

ci!(1− ρi)2p0(ci, ρi)

is the average number of customers in queue for an M/M/ci queue with utilization ρi.

Equation (1) is a heavy-traffic approximation and is exact for the M/G/1 and M/M/c

queues. Whitt (1993) reported that it is usually an excellent approximation, especially when

ρ is high. In our analyses, we will use expression (1) to compute the average delay. We

denote the average system delay (i.e., across all customer types) for the original system as

follows:

D0 =1

λT

N∑i=1

(ν2i + 1)

2g(ci, ρi) (3)

where λT =∑N

i=1 λi.

If we pool departments in the set K = {i1, i2, . . . , ik} (where k ≤ N so that K ⊆{1, 2, . . . , N}), the average system delay is

DK =1

λT

[(∑i∈K

λi)WK +∑

i/∈K

λiWi

](4)

6

where WK is the average delay for the pooled departments K = {i1, i2, . . . , ik}. Our problem

is to choose the set K to minimize WK subject to the constraint |K| = k. That is, we seek

the maximum impact from pooling k departments for any given k.

Computing delay in the pooled queue, WK , requires specification of the policy used

to serve the k customer types. We consider two policies: First-come-first-serve (FCFS) and

nonpreemptive priority (NPP) service. Many service systems use FCFS either to achieve

fairness or because they cannot distinguish customer types until service has begun. However,

it is well known that FCFS can increase average waiting time. So, we also consider the NPP

discipline to represent systems in which customer types are identified prior to service (e.g., via

a phone menu) and this information is used to prioritize them. In oder to obtain analytical

conditions and insights, in the sequel we will use the above queueing approximations as

acceptably accurate models of system behavior.

3.1 First-Come-First-Served Service Discipline

Suppose departments in the set K = {i1, i2, . . . , ik} are pooled so that these customer types

are served in FCFS order by∑

i∈K ci servers. The squared coefficient of variation for the

aggregate service time distribution (ν2K) of the pooled department is given by:

ν2K =

λK

∑i∈K λiT

2i (ν2

i + 1)

(∑

i∈K λiTi)2− 1 , (5)

where λK =∑

i∈K λi. Using expression (1), we can (approximately) evaluate the average

delay for customers in the pooled department as:

W FCFSK =

ν2K + 1

2λK

g

(∑i∈K

ci,

∑i∈K ciρi∑i∈K ci

), (6)

and hence, average system delay can be expressed as:

DFCFSK =

1

λT

[ν2

K + 1

2g

(∑i∈K

ci,

∑i∈K ciρi∑i∈K ci

)+

∑

i/∈K

νi + 1

2g(ci, ρi)

]. (7)

Because customer arrivals are Poisson and service times are i.i.d., all customers within the

pooled department experience the same mean wait in a FCFS queue prior to service.

3.2 Nonpreemptive Priority Service Discipline

In this section, we consider systems in which customers are served (non-preemptively) in

priority order in the pooled queue. For an M/G/1 queue with k customer types and a non-

7

preemptive priority (NPP) service discipline, one can compute the exact average delay of

each customer type. For multi-server systems, we can approximate the average delay of each

customer with priority n as follows:

WNPPn =

WM/G/c:FCFS

WM/G/1:FCFSWM/G/1:NPP

n (8)

The first term in equation (8) is the ratio by which the average delay changes by going from

one server to c servers for the FCFS service discipline. This approximation is based on the

assumption that this ratio is rather insensitive to the service discipline (see Buzacott and

Shanthikumar 1984 p. 88 for justification and further details.)

Now suppose we pool departments K = {i1, i2, . . . , ik}. Without loss of generality,

assume that K = {1, . . . , k} and customer type i is given non-preemptive priority over

customer type j whenever i < j. In order to use equation (8), we need to evaluate each

component, i.e., WM/G/1:FCFS, WM/G/c:FCFS, and WM/G/1:NPP . The first one is well known

to be

WM/G/1:FCFS =λE(S2)

2(1− ρ)

where λ is the arrival rate, E(S2) is the second moment of the service time and ρ is the

utilization (Kulkarni 1995 p. 379).

To compute WM/G/1:FCFS, first note that when departments K = {1, 2, . . . , k} are

pooled, the pooled system has an arrival rate of∑

i∈K λi and∑

i∈K ci servers. To model

this system as an M/G/1 queue, for each i ∈ K we define T′i = Ti/

∑n∈K cn and E(S

′i

2) =

E(S2i )/(

∑n∈K cn)2 as the new mean service times and corresponding second moments. Then

the second moment of the service time in the pooled queue is

E(S2K) =

∑i∈K

λi

λK

E(S′i

2) =

∑i∈K λiE(S2

i )

λK(∑

i∈K ci)2=

∑i∈K λiT

2i (ν2

i + 1)

λK(∑

i∈K ci)2.

Hence, the single server pooled system has average delay

WM/G/1:FCFS =

∑i∈K λiT

2i (ν2

i + 1)

2(1− ρ′)(∑

i∈K ci)2

where

ρ′=

∑i∈K λiTi∑

i∈K ci

.

From Equations (1) and (5), we can compute the multi-server average delay as

WM/G/c:FCFS =

∑i∈K λiT

2i (ν2

i + 1)

2(∑

i∈K λiTi)2g(

∑i∈K

ci, ρ′).

8

Finally, to compute WM/G/1:NPPn , we note that the average delay for a customer with priority

n is (see Gelenbe and Mitrani 1980 pp. 35–40)

WM/G/1:NPPn =

W0

(1− ρHn)(1− ρHn − ρ′n)

where ρHn = ρ′1 + ρ

′2 + . . . + ρ

′n−1,

ρ′n =

λnTn∑i∈K ci

,

and

W0 =

∑i∈K λiT

2i (ν2

i + 1)

2(∑

i∈K ci)2.

Combining these terms in Equation (8) yields:

WNPPn =

1− ρ′

2(1− ρHn)(1− ρHn − ρ′n)

∑i∈K λiT

2i (ν2

i + 1)

(∑

i∈K λiTi)2g(

∑i∈K

ci, ρ′) . (9)

Hence, the average system delay under the NPP service policy is:

DNPPK =

1

λT

[∑n∈K

λn

(1− ρ′)∑

i∈K λiT2i (ν2

i + 1)

2(1− ρHn)(1− ρHn − ρ′n)(∑

i∈K λiTi)2g(

∑i∈K

ci, ρ′) +

∑

n/∈K

ν2n + 1

2g(cn, ρn)

].

(10)

4 Analysis

We now make use of the above models to examine the impact of various system parameters

on the choice of which departments to pool: (1) mean service times (Ti), (2) the number of

servers (ci) for each department, (3) squared coefficient of variation of service times (ν2i ), (4)

the combined effects of mean service time and the number of servers, and (5) arrival rates

(λi). The system utilization has a first order effect on performance, so we must carefully

control for this effect. We require that all N departments be adequately staffed so that their

utilizations are all equal to ρ. This scenario is reasonable in light of the many applications

in which managers strive for a high utilization of all agents. Koole and Mandelbaum (2002)

report that most call centers operate at a service level of approximately 90-95%. Given

uniform utilization of departments, we investigate the impact of T, c, λ and ν on the pooling

decision. Finally, we consider the impact of different department utilizations by varying the

arrival rate. We perform all of our analyses for both FCFS and NPP service disciplines.

9

4.1 Effect of Service Time

To investigate the role of service time, we consider systems where cρ = λiTi and νi = ν,

i = 1, 2, . . . , N . Note that we are forced to scale the arrival rates so that the utilization

is held constant even though service times vary between departments. Using Equation (6),

we can write the average delay of customers in the pooled queue under the FCFS service

discipline as:

W FCFSK =

ν2 + 1

2k2cρ

(∑i∈K

Ti

)g(kc, ρ)

From Equation (7), the average system delay is given by:

DFCFSK =

1

λT

ν2 + 1

2

[1

k2

(∑i∈K

1

Ti

) (∑i∈K

Ti

)g(kc, ρ) + (N − k)g(c, ρ)

](11)

Smith and Whitt (1981) showed that when service rates are different, pooling can be

counterproductive. By considering an example where two M/M/1 queues are pooled, they

demonstrate that the average waiting time can be arbitrarily large if the two service rates

differ greatly from each other. In Proposition 1, we provide a sufficient condition for pooling

to be advantageous in systems with uniform utilization and service time variability.

Proposition 1 If ρ = λiTi/c and νi = ν for i = 1, 2, . . . , N , given the mean service times

Tn, n = 1, 2, . . . , N , if there exists a k-tuple for which k3 ≥ (∑i∈K 1/Ti

) (∑i∈K Ti

), then

DFCFSK ≤ D0.

Proof: From Equations (3) and (11),

D0 −DFCFSK =

1

λT

ν2 + 1

2

[kg(c, ρ)− 1

k2

(∑i∈K

1

Ti

)(∑i∈K

Ti

)g(kc, ρ)

]

≥ 1

λT

ν2 + 1

2g(kc, ρ)

[k − 1

k2

(∑i∈K

1

Ti

) (∑i∈K

Ti

)]

where the inequality follows from g(c, ρ) ≥ g(kc, ρ) (i.e., average number of customers in an

M/M/c queue is decreasing in the number of servers when ρ is fixed). Therefore, D0 ≥ DFCFSK

if

k − 1

k2

(∑i∈K

1

Ti

)(∑i∈K

Ti

)≥ 0 (12)

10

which proves the result. 2

As an example, consider k = 2 and assume that we pool departments i and j. Condition

(12) reduces to 2− (Ti + Tj)2/4TiTj which implies that T 2

i − 6TiTj + T 2j ≤ 0. Without loss

of generality, we can assume Ti > Tj and define Ti = aTj where a ∈ R and a ≥ 1. Then the

last inequality can be written as a2−6a+1 ≤ 0. The root that satisfies the constraint a ≥ 1

for this quadratic function is a = 3 + 2√

2, and the inequality is satisfied when a ≤ 3 + 2√

2.

Thus, if there exist two departments for which the expected service time of one does not

exceed approximately six times that of the other, pooling is always advantageous.

Assuming that there is at least one k-tuple that satisfies the condition in Proposition

1, the k departments to pool to minimize the average system delay given by Equation (11)

are the ones that minimize(∑

i∈K Ti

) (∑i∈K 1/Ti

). Note that this expression is minimal

when all service times are equal, and it implies that the k departments to pool should be

the ones with mean service times close to each other.

For instance, for k = 2, the departments to pool are (i, j) = arg minij{Tj/Ti :

Tj ≥ Ti, i = 1, 2, . . . , N, j 6= i}. The intuition behind this result is the following. If two

departments, one with very short and the other with very long service times, are pooled,

then under FCFS the customers with short processing times will wait longer than they did

in the original system because of the customers with long service times who arrive before

them.

To examine the “equity” (fairness) issues involved in pooling, we can compare the

difference in average delay experienced by customer type n, n ∈ K, without and with

pooling; given by:

Wn −W FCFSK =

ν2 + 1

2cρ

[Tng(c, ρ)− 1

k2(∑i∈K

Ti)g(kc, ρ)

]

≥ ν2 + 1

2cρg(kc, ρ)

[Tn − 1

k2(∑i∈K

Ti)

].

Hence, under FCFS, if k2 ≥ (∑

i∈K Ti)/Tn, average delay for customer type n declines.

Otherwise, pooling may increase the average delay of customer type n.

To see the impact of pooling on customer type n under the NPP service discipline,

without loss of generality assume that K = {1, 2, . . . , k}, and note that

WNPPn =

ν2 + 1

2

1− ρ

(1− (n− 1)ρ/k)(1− nρ/k)

∑i∈K Ti

k2cρg(kc, ρ) (13)

11

DNPPK =

1

λT

ν2 + 1

2

[∑i∈K Ti

k2

∑n∈K

1− ρ

Tn(1− (n− 1)ρ/k)(1− nρ/k)g(kc, ρ) + (N − k)g(c, ρ)

]

(14)

Equation (14) shows that average delay in the system is minimized when priority is given

to the customer type with smaller mean service time. Pooling k departments and serving

the customers in the pooled queue in NPP order may also result in a longer average system

delay than in the specialist system.

Proposition 2 If ρ = λiTi/c and νi = ν for i = 1, 2, . . . , N , given mean service times Tn,

n = 1, 2, . . . , N , if there exists a k-tuple for which

k3 ≥(∑

i∈K

Ti

)(∑n∈K

1− ρ

Tn(1− (n− 1)ρ/k)(1− nρ/k)

), (15)

then DNPPK ≤ D0.

Proof: Under conditions of the proposition,

D0 −DNPPK =

ν2 + 1

2λT

[kg(c, ρ)−

(∑i∈K

Ti

)(∑n∈K

1− ρ

Tn(k − (n− 1)ρ)(k − nρ)

)g(kc, ρ)

]

≥ ν2 + 1

2λT

g(kc, ρ)

[k −

(∑i∈K Ti

k2

) (∑n∈K

1− ρ

Tn(1− (n− 1)ρ/k)(1− nρ/k)

)].

The result follows. 2

Let us define B(n) = (1 − (n − 1)ρ/k)(1 − nρ/k) for n = 1, 2, . . . , k. If there is at

least one k-tuple that satisfies the condition in Proposition 2, the k departments to pool to

minimize the average system delay given by Equation (14) are the ones that minimize

(∑i∈K

Ti

)(∑i∈K

1/B(i)Ti

). (16)

Hence, under NPP, the decision of which k departments to pool depend on the mean service

times as well as the utilization. Expression (16) also implies that the mean service times in

the pooled system should be similar. On the other hand, pooling may increase the average

delay of customer types as shown below:

Wn −WNPPn ≥ ν2 + 1

2cρg(kc, ρ)

[Tn − 1− ρ

(1− (n− 1)ρ/k)(1− nρ/k)

∑i∈K Ti

k2

]. (17)

12

5

10

15

20

25

1 2 3 4

FCFSNPP

T1=1 T2=1.2 T3=50 T4=55 ρ =0.8 c=1 ν=1

FCFSk=2 Pool 3,4k=3 Pool 2,3,4

NPPk=2 Pool 1,2k=3 Pool 2,3,4

DK

k

Figure 2: Average delay in queue for ρ = λiTi/c for i = 1, 2, 3, 4

If

k2 ≥ 1− ρ

(1− (n− 1)ρ/k)(1− nρ/k)

∑i∈K Ti

Tn

, (18)

then Wn ≥ WNPPn . Hence, if conditions given in Equations (15) and (18) hold, pooling

departments improves service for all customer types.

We use the results of this section to provide managerial insights via two examples.

First, we consider a four department system (N = 4) in which two customer types have

short mean service times, while the others require much longer service times. For example,

type 1 calls may be arriving from remote repair technicians at independent shops who are

requesting simple firmware information, type 2 calls are for product registration, and types

3 and 4 are complex technical support and problem solving tasks. Figure 2 presents the

minimal average system delay as we successively pool departments. We use k to represent

the number of pooled departments and for each k find the “optimal” set of departments to

pool, so that average system delay is minimized. The case for k = 1 corresponds to the

specialist system. Observe that for k = 2 pooling decreases average system delay under both

FCFS and NPP. However, when we further increase the number of pooled departments to

k = 3, the performance becomes much worse, and the resulting average delay far exceeds even

than that of the specialist system. We note that the sufficiency conditions of Propositions

13

0

5

10

15

20

25

30

35

1 2 3 4 5 6

FCFS rho=0.9NPP rho=0.9FCFS rho=0.95NPP rho=0.95

(4,5)

(4,5,6)

(2,3,4,5)

(2,3,4,5,6)

(1,3)

(2,3,6)

(2,3,4,6)

(2,3,4,5,6)

(3,4,5,6)

(2,3,5)

(1,2)

(2,3,4,5,6)

T1=1 T2=3 T3=4.5 T4=10 T5=12 T6=24 c=2 ν=1DK

k

Figure 3: Average delay in queue for ρ = λiTi/c for i = 1, 2, . . . , 6

1 and 2 are not satisfied. Finally, when all departments are pooled (k = 4), the average

delay is less than in the specialist system, but only under NPP. Under FCFS, the delay from

pooling all departments is better than pooling three departments, but it is still worse than

that of the specialist system. Hence, whether or not pooling is beneficial always depends

on how many departments are pooled and the mean service times. In the case of NPP, the

question also depends on utilization. As our example graphically illustrates, it is important

for managers to understand that the effect of pooling can fluctuate from better to worse, or

vice versa, as the number of pooled departments increases.

Second, we consider a system with heterogenous mean service times where the

sufficiency conditions are always satisfied as we successively pool departments i = 1, 2, . . . , 6.

Figure 3 shows that this results in a decrease in the average system delay as k increases for

both FCFS and NPP. For each k value in the figure, the optimal set of departments to pool

is also indicated. As k increases, the optimal set of departments to pool does not change in

a greedy way. For instance, under FCFS when ρ = 0.95, it is optimal to pool departments 4,

5 and 6 if k = 3, but it is optimal to pool departments 2, 3, 4 and 5 if k = 4. Moreover, as

utilization increases, the incremental benefit of pooling declines more rapidly with k. Note

that, under FCFS, the optimal set of departments to pool does not change with utilization.

14

However, under NPP, the optimal set does depend on the utilization, as we expect from

Equation (14).

4.2 Effect of Department Size

To consider the impact of department size, we begin with the case where all service times

are drawn from the same distribution. We set T = Ti and ν = νi, and ρ/T = λi/ci,

i = 1, 2, . . . , N . That is, departments see different arrival rates but are initially staffed such

that utilization is the same. Average customer delay in the pooled queue and the overall

system under FCFS are given by:

W FCFSK =

ν2 + 1

2

T

ρ∑

i∈K ci

g(∑i∈K

ci, ρ) (19)

DFCFSK = DNPP

K =1

λT

ν2 + 1

2

[g(

∑i∈K

ci, ρ) +∑

i/∈K

g(ci, ρ)

](20)

The average delay in the system is the same under FCFS and NPP because the mean service

times are equal for all departments (see Lemma 1 in the Appendix for a proof of this intuitive

result).

Pooling decreases the average system delay for both FCFS and NPP since

D0 −DK =ν2 + 1

2λT

[∑i∈K

g(ci, ρ)− g(∑i∈K

ci, ρ)

]> 0

and g(c, ρ) is a non-increasing function of c when ρ is fixed. It can also be shown that

g(c, ρ) is convex in c when ρ is fixed (see Dyer and Proll 1977). This enables us to prove the

following.

Theorem 1 If ρ = λiT/ci and νi = ν for i = 1, 2, . . . , N , then pooling the k smallest

departments minimizes the average delay in the system under both FCFS and NPP.

Proof: Without loss of generality, assume that ci ≤ cj whenever i ≤ j. Let DK denote

the average system delay when the k smallest departments are pooled. Let DK′ denote the

average system delay when the k−1 smallest departments and department k +1 are pooled.

Then we can write the difference between the average delays for these two systems as follows:

DK′ −DK =1

λT

ν2 + 1

2

[g(

k−1∑i=1

ci + ck+1, ρ) + g(ck, ρ)− g(k∑

i=1

ci, ρ)− g(ck+1, ρ)

]

15

Since g(c, ρ) is a non-increasing convex function of c when ρ is fixed, g(ck, ρ)− g(ck+1, ρ) ≥g(

∑ki=1 ci, ρ) − g(

∑k−1i=1 ci + ck+1, ρ). Hence, DK

′ ≥ DK . This inequality remains valid

whenever we replace any of the k smallest departments with a larger department. Hence,

the result follows. 2

This result is rather intuitive because g(c, ρ) is a convex non-increasing function of c

and the slope of this curve is very steep at small values of c. To gain insight into what type

of priority policy should be used, we define C(n) =∑n

i=1 ci, and note that average delay of

a customer with priority n under NPP is:

WNPPn =

(ν2 + 1)T

2ρC(k)

1− ρ

(1− ρC(n− 1)/C(k))(1− ρC(n)/C(k))g(

∑i∈K

ci, ρ) (21)

By a simple interchange argument in Equation (21), we see that priority should be given to

the customer with the smallest arrival rate in order to minimize delay.

Under both FCFS and NPP, the average delay for each customer type in the pooled

queue is less than or equal to the delay in the specialist system since

Wn −W FCFSK =

ν2 + 1

2

T

ρ

[g(cn, ρ)

cn

− g(∑

i∈K ci, ρ)∑i∈K ci

]> 0

and

Wn −WNPPn =

ν2 + 1

2

T

ρ

[g(cn, ρ)

cn

− (1− ρ)/C(k)

[1− ρC(n− 1)/C(k)][1− ρC(n)/C(k)]g(

∑i∈K

ci, ρ)

]

>ν2 + 1

2

T

ρ

[g(cn, ρ)

cn

− 1

[1− C(n− 1)/C(k)]C(k)g(

∑i∈K

ci, ρ)

]

=ν2 + 1

2

T

ρ

[g(cn, ρ)

cn

− g(∑

i∈K ci, ρ)

C(k)− C(n− 1)

]

>ν2 + 1

2

T

ρcn

[g(cn, ρ)− g(

∑i∈K

ci, ρ)

]> 0

To summarize, when ρ = λiT/ci and νi = ν for i = 1, 2, . . . , N , pooling is always

beneficial. It is optimal to pool the k smallest departments. The average system delay and

the average delay for each customer type are decreasing in the number of pooled departments

(k) as can be seen from Equations (19), (20) and (21) .

4.3 Effect of Service Time Coefficient of Variation

To see how variability of service times affects the decision of which departments to pool, we

consider the case where departments differ only in terms of their service time cv’s (νi). Since

16

λ = λi, T = Ti, c = ci for i = 1, 2, . . . , N , it follows that the average delay in the pooled

queue under FCFS is:

W FCFSK =

∑i∈K(ν2

i + 1)

2λk2g(kc, ρ)

DFCFSK = DNPP

K =1

λT

[∑i∈K(ν2

i + 1)

2kg(kc, ρ) +

∑

i/∈K

(ν2i + 1)

2g(c, ρ)

](22)

As in the previous section, DFCFSK = DNPP

K (see Lemma 2 in the Appendix for a proof).

Furthermore, since

D0 −DK =1

λT

(∑i∈K

ν2i + 1

2

)[g(c, ρ)− g(kc, ρ)/k] > 0,

pooling is always advantageous under both FCFS and NPP when departments differ only in

service time cv. The following result shows how to pool for maximum effect.

Theorem 2 If ρ = λT/c for all departments and νi 6= νj for i = 1, 2, . . . , N , i 6= j, pooling

the k departments with the highest service time cv’s minimizes the average system delay.

Proof: By adding and subtracting the term

1

λT

∑i∈K

ν2i + 1

2g(c, ρ)

to (from) Equation (22) we obtain the following equality:

λT DK =g(kc, ρ)− kg(c, ρ)

k

∑i∈K

ν2i + 1

2+

N∑i=1

ν2i + 1

2g(c, ρ) (23)

We want to choose a set K to minimize DK given in Equation (23). Note that the last

term in equation (23) is a constant and hence does not affect the minimization. In the first

term, g(kc, ρ) − kg(c, ρ) < 0 because g(c, ρ) > g(kc, ρ). DK is minimized when the value∑

i∈K(ν2i + 1)/2 is maximal. Hence, we conclude that the set of departments to be pooled

are the ones with the highest cv’s. 2

Under this principle, the average delay in the system decreases as the number of pooled

departments (k) increases. To observe this relabel the departments so that νi ≥ νj whenever

i < j, and denote K = {1, . . . , k} and K′= {1, . . . , k + 1}. Then using Equation (22), we

can show that DK ≥ DK′ (see Lemma 3 in the Appendix for a proof).

17

However, since

Wn −W FCFSK =

ν2n + 1

2λg(c, ρ)−

∑i∈K(ν2

i + 1)

2λk2g(kc, ρ)

≥ g(kc, ρ)

2λ

[(ν2

n + 1)−∑

i∈K(ν2i + 1)

k2

]

average waiting times of customers in the pooled departments will decrease under FCFS if

k2 ≥ ∑i∈K(ν2

i + 1)/(ν2n + 1). For example, if ν2

i ≤ 1 for i ∈ K, this condition is satisfied for

all customer types served by the pooled department.

In contrast, under NPP

WNPPn =

1− ρ

2λ(k − (n− 1)ρ)(k − nρ)

(∑i∈K

(ν2i + 1)

)g(kc, ρ) (24)

and

Wn −WNPPn ≥ g(kc, ρ)

2λ

[(ν2

n + 1)− 1− ρ

(k − (n− 1)ρ)(k − nρ)

∑i∈K

(ν2i + 1)

](25)

Hence, pooling does not increase the average delay for customer with priority n if the term

in the brackets in Equation (25) is non-negative.

4.4 Effect of Department Size and Service Times

We now consider environments with fixed utilization, but varying department size and service

times so that ρ/λ = Ti/ci ,i = 1, 2, . . . , N . Average delay for customers in the pooled queue

under FCFS is:

W FCFSK =

ν2 + 1

2λ

∑i∈K c2

i(∑i∈K ci

)2 g(∑i∈K

ci, ρ)

Average system delay under FCFS is:

DFCFSK =

1

λT

(ν2 + 1)

2

[k

∑i∈K c2

i(∑i∈K ci

)2 g(∑i∈K

ci, ρ) +∑

i/∈K

g(ci, ρ)

](26)

Since

D0 −DFCFSK =

1

λT

ν2 + 1

2

[∑i∈K

g(ci, ρ)− k∑

i∈K c2i

(∑

i∈K ci)2g(

∑i∈K

ci, ρ)

]

>1

λT

ν2 + 1

2

[∑i∈K

g(ci, ρ)− kg(∑i∈K

ci, ρ)

]> 0,

18

c1 = 1 c3 = 50 λλλλ = 1 ρ ρ ρ ρ = 0.75

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 5 10 15 20 25 30 35 40 45 50

c2

Wa

itin

g t

ime

in

th

e q

ue

ue

Pool 1 and 2Pool 1 and 3Pool 2 and 3

Figure 4: Average delay for ρ/λ = Ti/ci

c1 = 1 c3 = 50 λλλλ = 1 ρρρρ = 0.95

9

9.5

10

10.5

11

11.5

12

12.5

13

13.5

0 5 10 15 20 25 30 35 40 45 50

c2

Wa

itin

g t

ime

in

th

e q

ue

ue

Pool 1 and 2Pool 1 and 3Pool 2 and 3

Figure 5: Average delay for ρ/λ = Ti/ci

pooling is always beneficial regardless of the set K. Furthermore, the average delay of each

customer type in the pooled queue is also reduced, as can be seen by the following equation:

Wn −W FCFSK ≥ ν2 + 1

2λg(

∑i∈K

ci, ρ)

[1−

∑i∈K c2

i

(∑

i∈K ci)2

]> 0.

However, unlike the other environments, there is no simple pooling rule for this case.

To see this, consider a system initially composed of three departments with c1 < c2 < c3

which also implies T1 < T2 < T3. Figures 4 and 5 show that pooling any of the three

combinations can be the best depending on how far c1, c2 and c3 are from each other. In

Figures 4 and 5, we set c1 = 1, c3 = 50, and vary c2 from 2 to 49 for utilization ρ = 0.75 and

ρ = 0.95, respectively. For ρ = 0.75, plotting DFCFS{1,2} , DFCFS

{1,3} , and DFCFS{2,3} , we observe that

pooling 1 and 2 is optimal only for c2 = 2. For every other value of c2, pooling 1 and 3 is

optimal. However, when we increase utilization to 0.95, pooling 1 and 2 is optimal for small

values of c2 but pooling 2 and 3 (the two largest departments) minimizes average system

delay for larger values of c2. Moreover, depending on the ρ and c values, DFCFSK can be a

convex or a concave function of c2.

To consider the NPP case, let C(n) =∑n

i=1 ci as before and assume that K =

{1, 2, . . . , k}. The average delay for a customer with priority n and the average system

delay under NPP are:

WNPPn =

(ν2 + 1)

2λ

(∑i∈K

c2i

)1− ρ

(C(k)− ρC(n− 1))(C(k)− ρC(n))g(C(k), ρ). (27)

19

0

5

10

15

20

25

30

1 2 3 4 5 6

FCFS rho=0.9NPP rho=0.9FCFS rho=0.95NPP rho=0.95

(4,5)

(3,4,5)

(2,3,4,5)

(2,3,4,5,6)

(1,2)

(1,2,3)

(1,2,3,4)

(1,2,3,4,5)

(1,2,3,4)

(2,3,4)

(2,3)

(1,2,3,4,6)

c1=2 c2=3 c3=5 c4=8 c5=10 c6=13 λ=0.5 ν=1DK

k

(1,2)

(1,2,3)

(1,2,3,4)(1,2,3,4,5)

Figure 6: Average delay in queue for ρ/λ = Ti/ci

DNPPK =

1

λT

ν2 + 1

2[

(∑i∈K

c2i

) ∑n∈K

1− ρ

(C(k)− ρC(n− 1))(C(k)− ρC(n))g(C(k), ρ)

+∑

i/∈K

g(ci, ρ)] (28)

From Equations (27) and (28), it follows that priority should be given to the customer with

smallest mean service time.

D0 −DNPPK =

ν2 + 1

2λT

[∑i∈K

g(ci, ρ)−(∑

i∈K

c2i

) ∑n∈K

(1− ρ)g(C(k), ρ)

(C(k)− ρC(n− 1))(C(k)− ρC(n))

]

>ν2 + 1

2λT

[∑i∈K

g(ci, ρ)−(∑

i∈K c2i

)

C(k)

(∑n∈K

1

C(k)− C(n− 1)

)g(C(k), ρ)

]

>ν2 + 1

2λT

[∑i∈K

g(ci, ρ)− k(∑

i∈K c2i

)

ckC(k)g(C(k), ρ)

]

>ν2 + 1

2λT

[∑i∈K

g(ci, ρ)− kg(C(k), ρ)

]≥ 0 (29)

Hence, pooling under NPP always reduces the average system delay. Pooling also reduces

the average delay for each customer type (i.e, Wn ≥ WNPPn for n ∈ K). The proof is similar

to that of (29) and hence omitted for brevity.

As under FCFS, it is not possible to identify a simple rule to decide which k departments

20

to pool under NPP. Hence, the k departments to pool should be determined by evaluating

Equations (26) and (28) for FCFS and NPP, respectively, for all possible combinations.

Figure 6 shows a numerical example for ρ/λ = Ti/ci, i = 1, 2, . . . , 6. The optimal set

of departments to pool depends on the utilization for both FCFS and NPP. Under NPP,

pooling the smallest departments results in the minimum average system delay in almost all

cases.

4.5 Effect of Utilization

Finally, we consider pooling when department utilizations are different. We investigate this

case by varying the arrival rates (λ) and setting other parameters equal for all departments.

W FCFSK =

ν2 + 1

2

1∑i∈K λi

g(kc,

∑i∈K ρi

k)

DFCFSK = DNPP

K =1

λT

ν2 + 1

2

[g(kc,

∑i∈K ρi

k) +

∑

i/∈K

g(c, ρi)

]

(See Lemma 4 in the Appendix for a proof of W FCFSf(K) = WNPP

f(K) .) Pooling always reduces

average system delay since

D0 −DK =ν2 + 1

2λT

[∑i∈K

g(c, ρi)− g(kc,∑i∈K

ρi/k)

]> 0.

However, when a high utilization department is pooled with a low utilization department,

the average delay for the low utilization department increases. Therefore, Wn > W FCFSK is

not always true.

Under NPP, the average delay for a customer with priority n is:

WNPPn =

ν2 + 1

2

1−∑i∈K ρi/k

(1− 1k

∑n−1i=1 ρi)(1− 1

k

∑ni=1 ρi)(

∑i∈K λi)

g(kc,∑i∈K

ρi/k).

Hence, giving priority to the lowest utilization customer type decreases the average customer

delay. It is also straightforward to show Wn ≥ WNPPn , so pooling with prioritization reduces

delay for all customer types.

For both FCFS and NPP, Table 1 shows how the average delays for individual customer

types changes in k. Note that this study allows heterogeneity in the departmental utilizations

of the initial, unpooled system. In Table 1, the numbers in bold in each column show the

21

k=1 k=2 k=3 k=4 k=5 k=6FCFS Dept. 1 10.00 11.41 10.52 8.98 10.00 5.97

Dept. 2 56.67 56.67 56.67 56.67 15.65 5.97Dept. 3 66.92 66.92 66.92 66.92 15.65 5.97Dept. 4 90.00 90.00 90.00 8.98 15.65 5.97Dept. 5 132.86 132.86 10.52 8.98 15.65 5.97Dept. 6 240.00 11.41 10.52 8.98 15.65 5.97

NPP Dept. 1 10.00 4.11 2.57 1.82 10.00 1.07Dept. 2 56.67 56.67 56.67 56.67 1.85 1.39Dept. 3 66.92 66.92 66.92 66.92 2.82 2.02Dept. 4 90.00 90.00 90.00 2.80 4.91 3.26Dept. 5 132.86 132.86 4.91 5.87 11.11 6.32Dept. 6 240.00 15.21 20.10 21.51 53.97 18.37

Table 1: Average delay for each department when ρ = (.5, .85, .87, .9, .93, .96), Ti = 10,νi = 1 for i = 1, 2, . . . , 6.

average delay for the departments that are pooled into a single department. If there are

some departments with much lower utilization than others, it is optimal to pool the high

utilization departments with low utilization departments. This situation is illustrated in

Table 1 for the case k = 2, in which call type 1 (the lowest utilization, at 0.5) is merged

with type 6 (the highest utilization, at 0.96). On the other hand, if all departments have

high utilization, then it is optimal to pool the highest utilization departments. The table

also illustrates this property in the sense that we see in pure form for k = 5. The columns

for k = 2, 3, 4 reveal hybrid patterns that merge both the very lowest utilization queue, 1,

with the k − 1 most highly utilized queues.

Under FCFS , the first row reveals that pooling may increase the average delay of a

department with low utilization, an insight important for managers to realize. In general,

particularly as k increases, the average delay for each department decreases. The NPP

service discipline results in a substantial reduction in average delay for the low utilization

departments. On the other hand, under FCFS offers simplicity and fairness by ensuring equal

average delays across all pooled customer types. Consequently, FCFS provides a relatively

greater benefit to the high utilization stations, while NPP provides a pooling benefit to all

the merged call types.

22

5 Conclusion

In this paper, we examined pooling strategies for call centers consisting of multiple

departments. For different scenarios of system parameters, we compared average system

delay for specialist and pooled systems, and investigated the impact of different system

parameters such as arrival rates, service times, service time variability, and size of the

departments in deciding which departments to pool. In addition to analytical sufficient

conditions derived from well-accepted queueing approximations, we are in some cases able

to distill the mathematics into straightforward managerial principles. When mean service

times differ greatly, pooling departments with a high ratio of mean service times may actually

result in worse performance than not pooling, even if with very high utilization.

The pooling structure treated here is a common and relatively tractable type of cross-

training structure. More generally, each server is given a set of skills, and a clean pooling

structure may not exist. Previous studies suggest that partial cross-training can be nearly as

effective as full pooling (see e.g., Hopp et al. 2004, Jordan et al. 2004). Thus, an important

future research goal is to see whether analogies to the simple pooling principles identified

here can be derived for systems using more elaborate patterns of cross-training. In such cases

without simple departmental structures, customer routing becomes a concern and the simple

routing policies of this paper may need to be replaced by more complex control policies.

23

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27

6 Appendix

Lemma 1 When ρ = λiT/ci and νi = ν for i = 1, 2, . . . , N , DNPPK = DFCFS

K .

Proof: Under the assumptions of Lemma 1, using equation (10), we can write:

DNPPK =

1

λT

[ν2 + 1

2G(K)g(

∑i∈K

ci, ρ) +∑

n 6=K

ν2 + 1

2g(cn, ρ)

]

where

G(K) = (1− ρ)

(∑n∈K

cn

) ((1− ρ)cn

(∑

i∈K ci − ρ∑n−1

i=1 ci)(∑

i∈K ci − ρ∑n

i=1 ci)

).

We will show that G(K) = 1. Define A(n) :=∑n

i ci and without loss of generality, let

K = {1, 2, . . . , k}. Then,

G(K) = (1− ρ)A(k)k∑

n=1

cn

(A(k)− ρA(n− 1))(A(k)− ρA(n))

= (1− ρ)A(k)k∑

n=1

(A(n)/A(k)

A(k)− ρA(n)− A(n− 1)/A(k)

A(k)− ρA(n− 1)

)

= (1− ρ)k∑

n=1

(A(n)

A(k)− ρA(n)− A(n− 1)

A(k)− ρA(n− 1)

)

= (1− ρ)k∑

n=1

A(n)

A(k)

1

1− ρA(n)/A(k)− A(n− 1)

A(k)

1

1− ρA(n− 1)/A(k)

= (1− ρ)k∑

n=1

[A(n)

A(k)

∞∑j=0

(ρA(n)

A(k)

)j

− A(n− 1)

A(k)

∞∑j=0

(ρA(n− 1)

A(k)

)j]

= (1− ρ)k∑

n=1

∞∑j=1

Aj(n)− Aj(n− 1)

Aj(k)ρj−1

= (1− ρ)∞∑

j=1

ρj−1

Aj(k)

k∑n=1

(Aj(n)− Aj(n− 1))

= (1− ρ)∞∑

j=1

ρj−1 = 1.

The result follows. 2

Lemma 2 When ρ = λT/c for i = 1, 2, . . . , N and departments may have different cv’s for

service times, DNPPK = DFCFS

K .

28

Proof: We examine two algebraic relationships:

DNPPK =

1

λT

[(∑n∈K

k(1− ρ)

(k − (n− 1)ρ)(k − nρ)

)(∑i∈K(ν2

i + 1)

2k

)g(kc, ρ) +

∑

n/∈K

(ν2n + 1)

2g(c, ρ)

].

(30)

The result follows from

∑n∈K

k(1− ρ)

(k − (n− 1)ρ)(k − nρ)=

k∑n=1

(k − (n− 1)

k − (n− 1)ρ− k − n

k − nρ

)

=k−1∑n=0

k − n

k − nρ−

k∑n=1

k − n

k − nρ

= 1.

2

Lemma 3 When ρ = λT/c for all departments, and departments may have different service

time cv’s, the average system delay (DK) decreases as the number of pooled departments (k)

increases.

Proof: Assume that we relabel the departments so that νi ≥ νj whenever i < j. Let

K = {1, . . . , k} and K′= {1, . . . , k + 1}. Then using Equation (22), we have the following:

DK −DK′ =

1

λT

[

∑ki=1(ν

2i + 1)

2kg(kc, ρ) +

N∑

i=k+1

(ν2i + 1)

2g(c, ρ)

−∑k+1

i=1 (ν2i + 1)

2kg((k + 1)c, ρ)−

N∑

i=k+2

(ν2i + 1)

2g(c, ρ)]

=1

λT

[∑ki=1(ν

2i + 1)

2kg(kc, ρ)−

∑k+1i=1 (ν2

i + 1)

2kg((k + 1)c, ρ) +

(ν2i + 1)

2g(c, ρ)

]

>1

λT

[∑ki=1(ν

2i + 1)

2k−

∑k+1i=1 (ν2

i + 1)

2k

]g((k + 1)c, ρ) +

(ν2i + 1)

2g(c, ρ)

=1

λT

[∑ki=1(v

2i + 1)− k(v2

k+1 + 1)

2k(k + 1)

]+

(ν2i + 1)

2g(c, ρ) > 0

where the result follows from g(kc, ρ) > g((k + 1)c, ρ) and∑k

i=1(v2i + 1)− k(v2

k+1 + 1) ≥ 0.

2

Lemma 4 When ρi = λiT/c and νi = ν for i = 1, 2, . . . , N , DNPPK = DFCFS

K .

29

Proof: The result follows by algebra as seen in two steps:

DNPPK =

1

λT

ν2 + 1

2

(1− 1k

∑i∈K ρi)∑

i∈K ρi

[∑n∈K

ρn

(1− 1k

∑n−1i=1 ρi)(1− 1

k

∑ni=1 ρi)

g(kc,

∑i∈K ρi

k) +

∑

i/∈K

g(c, ρi)] .

We conclude by observing that

∑n∈K

ρn

(1− 1k

∑n−1i=1 ρi)(1− 1

k

∑ni=1 ρi)

= k∑n∈K

[1

1− 1k

∑ni=1 ρi

− 1

1− 1k

∑n−1i=1 ρi

]

= k

[1

1− 1k

∑i∈K ρi

− 1

]

=

∑i∈K ρi

1− 1k

∑i∈K ρi

.

2

30