Cross-Selling in a Call Center with a Heterogeneous Customer … · 2007-11-19 · Vs(t) and Vq(t) be, respectively, the residual work for the customers in service at time t, i.e.

Cross-Selling in a Call Center with

a Heterogeneous Customer Population

Technical Appendix

Itay Gurvich† Mor Armony∗ Costis Maglaras ‡

A Introduction

This is the technical appendix accompanying the paper, “Cross-Selling in a Call Center with a

Heterogeneous Customer Population,” [3]. The organization of this appendix is as follows: we

begin in §B with the completion of the proof of Proposition 1, whose sketch was given in §A of

[3]. We continue in §C with some preliminaries required for the performance analysis of (S)-(C).

Specifically, we provide a sample-path construction that uses a collection of independent rate-1

Poisson processes. We also discuss some strong approximation tools. Finally, in §D we prove the

main performance-analysis results which are used in the proof in §A of the main paper [3]. Some

auxiliary results are proved in §E.

B A Detailed Proof of Proposition 1

This section is dedicated to the completion of the proof of Proposition 1 in [3]. Following our proof

sketch in §A of the main paper [3] we show that there exists δ0 ≥ 0 such that for any sequence of

initial states ξn ⊆ Ξ with |ξn| → ∞:

lim supn→∞

1

|ξn|Eξn [|Ξ(|ξn|(1 + δ0)|] = 0. (A1)

Whenever this holds we can always find a positive number K, such that for all |ξ| > K, (20) holds.

Toward that end, let V (t) =∑

j≥1 vj(t) be the amount of residual work in the system. Also, let

†Columbia Business School, 4I Uris Hall, 3022 Broadway, NY, NY 10027. ([email protected])∗Stern School of Business, NYU, 44 West 4th Street, NY, NY 10012. ([email protected])‡Columbia Business School, 409 Uris Hall, 3022 Broadway, NY, NY 10027. ([email protected])

1

Vs(t) and Vq(t) be, respectively, the residual work for the customers in service at time t, i.e. Vs(t) =∑

j≤N vj(t), and the residual work for the customers in queue at time t, i.e. Vq(t) = V (t) − Vs(t).

Set1

µ=

K∑

i=1

λi

Λ

(

1

µs+

1

µcsi

)

,

so that 1/µ is the mean customer handling time in case cross-selling is performed. The following

Lemma is the analogue of Lemma 4.3 in Dai [2]. Its proof is the same and is hence omitted.

Lemma 4 Fix a sequence of initial conditions |ξn| with |ξn| → ∞. Then,

(a) Almost surely, uniformly on compact sets,

lim supn

Vs(|ξn|t)|ξn| ≤ N(1 − t)+, (A2)

and

lim supn

Vq(|ξn|t)|ξn| ≤ 1 +

Λ

µt, (A3)

(b) For each t ≥ 0, the sequences

Vs(|ξn|t)|ξn| , n ≥ 1

and

Vq(|ξn|t)|ξn| , n ≥ 1

, are uniformly inte-

grable.

Finally, for any fixed t ≥ 0

lim supn

1

µs

Q(|ξn|t)|ξn| ≤ lim sup

n

Vq(|ξn|t)|ξn| ≤ lim sup

n

1

µ

Q(|ξn|t)|ξn| , (A4)

and

lim supn

1

µs

Eξn [Q(|ξn|t)]|ξn| ≤ lim sup

n

Eξn [Vq(|ξn|t)]|ξn| ≤ lim sup

n

1

µ

Eξn [Q(|ξn|t)]|ξn| . (A5)

For the following let D(t) be the cumulative number of customer departure from the system

up to time t. Also, let Wi(t) be the virtual waiting time for type-i customers at time t. Since

customers are assumed to be served FCFS we have that Wi(t) = Wj(t) for all i 6= j. Hence,

we can define W (t) to be the virtual waiting time for all the customers, regardless of their type.

Using the standard notation, we let Dd[0,∞) be the space of functions f(·) : [0,∞) 7→ Rd that

are Right Continuous with Left Limits (RCLL). A sequence of processes, Xn, in Dd is said to be

2

C-tight if, in addition to being tight (as random elements of Dd), every convergent subsequence

converges to a limit that is a.s. continuous. The following lemma establishes C-tightness of the

different processes in consideration and identifies important properties of the limit points. We let

µmax = maxµs, µcs1 , . . . , µcs

K.

Lemma 5 For any sequence of initial conditions ξn ⊆ Ξ with |ξn| → ∞ as n → ∞, and on

compact subsets of [1,∞), the sequence(

Q(|ξn|·)|ξn| , D(|ξn|·)

|ξn| ,Vq(|ξn|·)

|ξn| , W (|ξn|·)|ξn|

)

is C-tight. Moreover,

any limit point (Q(t), D(t), Vq(t), W (t)) satisfies

D(t) ≤ Nµmaxt, (A6)

W (t) ≥ Q(t)

Nµmax, (A7)

1

µQ(t) ≥ Vq(t), (A8)

for all t ≥ 0. Finally,

Vq(t) ≤[

Vq(1) + (t − 1)

(

Λ

µs− N

)]+

, (A9)

for all t ≥ 1.

Proof: The proof of C-tightness for the sequence(

Q(|ξn|t)|ξn| , D(|ξn|t)

|ξn|

)

is reminiscent of the proof of

Theorem 4.1 in Dai [2] and we omit it. The inequality (A6) is trivial to establish. We proceed then

to prove (A7). The virtual waiting time satisfies the representation

W (t) = infs ≥ 0 : D(t + s) − D(t) ≥ Q(t), (A10)

and in particular,

W (|ξn|t)|ξn| = inf

s ≥ 0 :D(|ξn|(t + s)) − D(|ξn|t))

|ξn| ≥ Q(|ξn|t)|ξn|

.

Considering a convergent subsequence nj of (Q(|ξn|t)/|ξn|,D(|ξn|t)/|ξn|), we can now apply the

corollary in [9], to conclude that W (|ξnj |t)/|ξnj | also converges to a limit W (t). Consequently, the

sequence (Q(|ξn|t)/|ξn|,D(|ξn|t)/|ξn|,W (|ξnj |t)/|ξnj ) is also C-tight. Moreover, any limit point

satisfies W (t) ≥ Q(t)/Nµmax. The inequality (A8) follows from Lemma 4. Finally, to establish

(A9), fix ǫ > 0 and set τn := inft ≥ 1 : Vq(|ξn|t) ≤ ǫ|ξn|. Then, for 1 ≤ t ≤ τn and by work

3

conservation

Vq(|ξn|t) + Vs(|ξn|t) = Vq(|ξn|1) + Vs(|ξn|1) − N |ξn|(t − 1) +

A(|ξn|t)∑

l=A(|ξn|)sl (W (τl)) . (A11)

Here τl is the time of the lth arrival. Consider a convergent subsequence

(

Q(|ξnj |t)|ξnj | ,

D(|ξnj |t)|ξnj | ,

W (|ξnj |t)|ξnj |

)

.

Then, we claim that uniformly on compact subsets of [1, τnj ),

ynj (t) :=

∑A(|ξnj |t)l=A(|ξnj |) sl (W (τl))

|ξnj | → Λ

µs(t − 1), in probability, as j → ∞. (A12)

To prove this claim, fix T > 1 and consider the set

Ωnj :=

ω ∈ Ω : inf1≤t≤T

W (|ξnj |t) ≤ 1

2

µVq(|ξnj |t)Nµmax

.

By inequalities (A7) and (A8), P (Ωnj) → 0 as j → ∞. Then,

P

sup1≤t≤T∧τnj

∣

∣

∣

∣

ynj (t) − Λ

µs(t − 1)

∣

∣

∣

∣

> ǫ

≤ P

sup1≤t≤T∧τnj

∣

∣

∣

∣

ynj(t) − Λ

µs(t − 1)

∣

∣

∣

∣

> ǫ, (Ωnj)c

+ PΩnj → 0, as j → ∞. (A13)

for any ǫ > 0, where the convergence

P

sup1≤t≤T∧τnj

∣

∣

∣

∣

ynj(t) − Λ

µs(t − 1)

∣

∣

∣

∣

> ǫ, (Ωnj )c

→ 0, as j → ∞,

follows from the definition of Ωnj and a careful application of the strong law of large numbers using

the fact that on (Ωnj )c, for t ≤ τnj and for any ǫ > 0 there exists j large enough so that for all

k ≥ j, E[sl(W (τl))] ≤ 1µs + ǫ. Now, Let

y(t) :=

[

Vq(1) + (t − 1)

(

Λ

µs− N

)]

∨ 3

2ǫ.

4

Combining (A11) and (A13) we then have that for any ǫ > 0,

P

sup0≤t≤τnj ∧T

[

Vq(|ξnj |t)|ξnj | − y(t)

]+

> ǫ

→ 0, as j → ∞. (A14)

Define now two random times as follows:

τn = inft ≥ τn : Vq(|ξn|t) > 2ǫ|ξn| and τ′n = supt ≤ τn : Vq(|ξn|t) ≤ ǫ|ξn|.

Then, we can extend the arguments we used above to show that for any ǫ > 0,

P

supτ′nj∧T≤t≤τnj∧T

[

Vq(|ξnj t)

|ξnj | − y(t)

]+

> ǫ

→ 0, as j → ∞.

Since for ǫ < ǫ/2,

τnj < T ⊂

supτ′nj∧T≤t≤τnj∧T

[

Vq(|ξnj t)

|ξnj | − y(t)

]+

> ǫ

,

we have that Pτnj < T → 0, as j → ∞. In particular, since

sup0≤t≤T

[

Vq(|ξnj t)

|ξnj | − y(t)

]+

> ǫ

⊆

sup0≤t≤τnj∧T

[

Vq(|ξnj t)

|ξnj | − y(t)

]+

> ǫ

⋃

τnj < T,

we conclude that for arbitrarily small ǫ,

P

sup0≤t≤T

[

Vq(|ξnj t)

|ξnj | − y(t)

]+

> ǫ

→ 0, as j → ∞. (A15)

The result of the Lemma now follows since ǫ was arbitrary.

To complete the proof of Proposition 1, note that with δ0 = (1 + Λ/µ)(N − R), and since by

Lemma 4, Vq(1) ≤ 1 + Λ/µ, we have that

Vq(1) + (δ0)

(

Λ

µs− N

)

≤ 0.

To conclude the argument, we fix an arbitrary sequence of initial conditions ξn. Consider a

convergent subsequence njj≥1, ofE[Vq(|ξn|(1+δ0))]

|ξn| . By Lemma 5 each subsequence njk ⊂ nj

5

such thatVq(|ξnjk |(1+δ0))

|ξnjk | converges satisfies

limk→∞

E[Vq(|ξnjk |(1 + δ0))]

|ξnjk | = 0.

In particular,

lim supn→∞

E[Vq(|ξn|(1 + δ0))]

|ξn| = 0.

Since the sequence ξn was arbitrary the argument is complete.

C Sample Path Construction, Strong Approximations and Other

Preliminaries

For the rest of our results we replace the sample path construction from Proposition 1 with a

different one that takes advantage of the properties of the Poisson process. This alternative con-

struction follows an approach that is, by now, quite common; see Whitt [4] for an overview. Having

constructed the sample paths using Poisson processes we can use strong approximations to bound

the distance of the underlying Poisson process from their respective rates. This section is divided

then to two subsections. First, in §C.1 we construct the sample path. Then, in §C.2 we introduce

the strong approximation tools that will be required for our proofs.

C.1 Sample-Path Construction

We base the sample-path construction on independent unit-rate Poisson processes A, S1 and Si,2,

i = 1, . . . , k, where k is the last customer class that is cross-sold under the control (C); see §3.1 in

[3]. We generate arrivals of customers using the Poisson process AΛ(t) := A(Λt). The arrival times

are registered in a dynamic array in which the customers are ordered in order of their arrival times.

Denote the value of this array at time t ≥ 0 by the vector T (t). Let T (t) be the process obtained by

setting Tk(t) = t−Tk(t). Whenever a customer leaves the queue to be served he is erased from this

array. We hold an NΛ dimensional vector w(t) = (w1(t), . . . , wNΛ(t)) with wj(t), j = 1, . . . , NΛ

standing for the registered waiting time of the customer currently in service with agent j. This

value is registered immediately when the customer is admitted to service.

We let S(t) be the set of agents giving service at time t, i.e., S(t) contains all the agents that are

6

not idle and not cross-selling at time t. We generate the process of phase 1 (service) completions

with type-i customers using a time change of a unit-rate Poisson process. Specifically, let DΛ1 (t) be

the number of phase 1 completion up to time t. Then, we write

DΛ1 (t) := S1

(

µs

∫ t

0ZΛ

1 (s)ds

)

, (A16)

where ZΛ1 (t) is the number of agents giving service at time t. Whenever the process DΛ

1 (t) jumps we

generate a discrete random variable with values in S(t−) to identify the number of the agent that

completed service. Formally, for a subset J of 1, . . . , NΛ we let es(J, l), l ∈ N be a sequence

of i.i.d random variables distributed uniformly over J . We will use these to determine the actual

agents that completes services. Also, we let ec(l), l ∈ N be a sequence of i.i.d discrete random

variables on 1, . . . ,K with Pec(1) = i = λi/Λ. Finally, we let ecs(l), l ∈ N be i.i.d uniform

random variables on [0, 1].

The number of service completions followed by a cross-selling phase with a type-i customer,

Di(t), for i ≤ k is then given by

DΛi (t) =

DΛ1 (t)∑

l=1

∑

j∈S(t−)

1es(S(t−), l) = j, ec(l) = i, ecs(l) ≤ qi(wj(t−)), QΛ(t−) ≤ ηiΛ.

Here we used the fact that an agent that completes service will attempt cross-selling to a type-i

customer only if at the time of completion the queue is less than the threshold ηΛi . The process

DΛi (t) is then a non-homogenous Poisson process with an instantaneous rate that is bounded from

above byk∑

i=1

λi

Λµsqi(0)Z

Λ1 (t)1QΛ(t) ≤ ηi.

Finally, for i ≤ k, we generate cross-selling completions by a non-homogenous Poisson process with

an instantaneous rate µcsi ZΛ

i,2(t) at time t, i.e, we write

DΛi,2(t) = Si,2

(

µcsi

∫ t

0ZΛ

i,2(s)ds

)

,

where ZΛi,2(t) is the number of type-i customers undergoing cross-selling at time t.

7

The system state is then captured by the multi-dimensional Markov process

ΞΛ(t) := (ZΛ1 (t), ZΛ

i,2(t),SΛ(t), wΛ(t), QΛ(t), T Λ(t); i = 1, . . . ,K). (A17)

We let X be the domain of this process. The following Lemma is a corollary of Proposition 1 and

its proof is easily obtained by expanding the state-space of the Markov process constructed in the

proof of Proposition 1. Although the proof would use a different sample path construction, the

uniqueness of the stationary distribution is invariant to this construction as the probability law is

the same under both constructions.

Lemma 6 Fix Λ. Then the process ΞΛ(t) admits a unique stationary distribution πΛ which coin-

cides with the unique limit distribution.

We let ξ be a general element in X , and for given ξ we let q(ξ), z1(ξ), and zi,2(ξ), i = 1, k be

respectively the queue length, the number of agents giving service and the number of agents cross-

selling to a type-i customer in state ξ. The notation q(ξ) should not be confused with the integer

q that we will use as a general power nor with the delay sensitivity functions qi(·).

We conclude this section with some additional notation. For fixed T > 0, a positive integer d

and a function y ∈ Dd[0,∞), we define ‖y(·)‖T := sup0≤t≤T

∑di=1 |y(t)|. We let (Ω,F , P ) be the

probability space which will remain fixed throughout and use the notation ω to denote an element

in Ω. We let Pξ be the probability distribution under which Pξ

(

ΞΛ(0) = ξ)

= 1, and put Eξ[·]be the expectation operator with respect to to the probability distribution Pξ. We let PπΛ be the

probability distribution under which ΞΛ(0) ∼ πΛ where πΛ is the unique stationary distribution

from Lemma 6 and we define EπΛ [·] accordingly. Finally for x, y ∈ R, we use the standard notations

x ∨ y = maxx, y and x ∧ y = minx, y as well as x+ = maxx, 0 and x− = max−x, 0.

C.2 Strong Approximations

Time changes of unit-rate Poisson processes, as the ones we have used above, can be approximated

by time-changed Brownian motion plus a logarithmic error term. We refer the reader to Mandel-

baum et. al. [8] and the references therein for a detailed discussion of strong approximations and

their application to Markovian queueing networks. For our purposes, it suffices to know that given

a unit-rate Poisson process N (·) and an instantaneous bounded rate function 0 ≤ λ(t) ≤ λ, for

8

some λ > 0, we have that

supt≥0

N(

∫ t0 λ(u)du

)

−∫ t0 λ(u)du − B

(

∫ t0 λ(u)du

)

log(λt ∨ 2)≤ C,

where C is a non-negative random variable with

PC > γ + βx ≤ c1e−c2x, (A18)

for some strictly positive constants γ, β, c1 and c2; see Lemma 9.4 in [8]. Since the rate of service

(or cross-selling completions) is bounded by NΛ(µs∨maxi=1,...,K µcsi ) and since the staffing rule (S)

dictates NΛ = R(1+ z) we can find m large enough so that all the instantaneous rates in the system

are bounded mΛ. We henceforth fix m to be such a value. We can then define a 3K-dimensional

Brownian motion, B(t), such that

∥

∥AΛ(·) − Λ·∥

∥

T+

∥

∥

∥

∥

DΛ1 (·) − µs

∫ ·

0ZΛ

1 (s)ds

∥

∥

∥

∥

T

+k∑

i=1

∥

∥

∥

∥

DΛi,2(·) − µcs

i

∫ ·

0ZΛ

i,2(s)ds

∥

∥

∥

∥

T

≤ ‖B(mΛ·)‖T + Clog(mΛT ∨ 2). (A19)

Finally, we will be using some basic facts about Brownian motion. Specifically, for a d-

dimensional Brownian motion and a constant m > 0, it can be easily shown that for all x ≥ 0, T > 0,

P

‖B(mΛ·)‖T ≥ x√

Λ

≤ c3

√

T

xe−c4

x2

T ,

for some strictly positive constants c3 and c4; see e.g. Problem 2.8.2 in [7]. Using (A18) we then

have that (A18) for all x > 0

P

‖B(mΛ·)‖T + C log(mΛT ∨ 2) ≥ x√

Λ

≤ c5

(

1 ∨√

T

x

)

e−c6 minx2

T, x√

T, (A20)

for some strictly positive constants c5, c6. Denote by

Ω∗(Λ, T, x) = ω ∈ Ω : ‖B(mΛ·)‖T + C log(mΛT ∨ 2) ≤ x . (A21)

9

Then, by (A20)

P

Ω∗(Λ, T, x√

Λ)

≥(

1 − c5

(

1 ∨√

T

x

)

e−c6 minx2

T, x√

T)+

.

D Performance Analysis for (S)-(C)

The aim of this section is two-fold. First, we want to establish that the queue length is, in some

sense, bounded by the smallest threshold ηΛk. Then, fixing ǫ > 0, we want to show that the

steady-state expected number of agents cross-selling type-i customers satisfies

∣

∣

∣

∣

E[ZΛi,2(∞)] − λiqi(0)

µcsi

∣

∣

∣

∣

≤ ǫΛ,

for all Λ large enough. Since ǫ is arbitrary, we will have that

E[ZΛi,2(∞)] =

λiqi(0)

µcsi

+ o(Λ),

which is what is used in the proof of Theorem 1 in §A of the main paper [3].

All the subsequent proofs share the same basic ideas. Using the strong approximation construc-

tion we examine the behavior of the system on a subset of the sample paths (such as Ω∗(Λ, T, x))

where the stochastic fluctuations generated by the Brownian Motion are bounded. This allows

us to examine a deterministic version of the system dynamics. For the deterministic version, and

with arguments reminiscent of Lyapunov function tools used for stability proofs, we show that the

system is in some sense attracted back into a certain domain. Finally, we remove the conditioning

and apply some arguments from [5] to obtain the steady state bounds. Some of the arguments are

common to several proofs. We abbreviate the proofs whenever this is the case. Throughout we fix

T > 0 and ǫ > 0.

Most of the Propositions in this section share a common structure. The first part of each such

proposition states a steady-state bound. The second part essentially states that if the process is

initialized at time 0 close enough to its steady-state distribution (in a sense to be made precise), it

actually stays there.

Our first result towards our stated goals shows that the number of agents busy giving service

(not cross-selling) is close to the load R. It also shows that the queue length will not exceed, in

10

some sense, the largest threshold, ηΛ1 . We will use these results to refine the bound on the queue

length process in Proposition 7. For the following we define

X1 := ξ ∈ X : |z1(ξ) − R| ≤ ǫΛ, q(ξ) ≤ ηΛ1 + ǫ

√Λ (A22)

Proposition 5 Fix ǫ > 0 and q ≥ 2. Then,

P∣

∣ZΛ1 (∞) − R

∣

∣ > ǫΛ

≤ c7

(√

Λ)q−1, (A23)

and

lim supΛ→∞

E

[

(

(QΛ(∞) − ηΛ1 )

+)q−1

]

≤ CQ, (A24)

for all Λ large enough and some strictly positive constants c7 and CQ Moreover,

supξ∈X1

Pξ

∥

∥ZΛ1 (·) − R

∥

∥

T≥ 2ǫΛ

≤ c9e−c10

√Λ, (A25)

and

supξ∈X1

Pξ

∥

∥(QΛ(·) − ηΛ1 )+

∥

∥

T≥ 2ǫ

√Λ

≤ c9e−c10ǫ

√Λ (A26)

for all Λ large enough and for some strictly positive constants c9 and c10.

Note that with the exception of the customer cross-selling probability being delay sensitive,

whenever the queue-length process exceeds the greatest threshold, ηΛ1 , it behaves just as the queue

length process in the single class model of Armony and Gurvich [1] when it exceeds the unique

threshold there. Moreover, when the queue is above ηΛ1 the delay sensitivity does not play any role

as by the control mechanism (C), no cross-selling will be attempted until the queue goes below ηΛ1 .

The proof of this result will then follow some of the results in [1]. A detailed proof of Proposition

5 would be obtained by expanding the proofs of Lemma B.1 and Proposition B.1 in [1] with the

unique threshold there replaced by ηΛ1 . We postpone the proof of this result to §E. We will also

need the following Proposition which establishes, the otherwise intuitive result, that the number

of agents busy cross-selling to class-i customers cannot exceed λiqi(0)/µcsi significantly. For the

following we define

X2 := ξ ∈ X1 : |zi,2(ξ) − λiqi(0)/µcsi | ≤ ǫΛ, i = 1, . . . , k, (A27)

11

where X1 was defined in equation (A22).

Proposition 6 Fix ǫ > 0. Then, for all Λ large enough and all x > 0,

P

(

ZΛi,2(∞) − λiqi(0)

µcsi

)+

> ǫΛ

≤ c11e−c12

√Λ, i = 1, . . . ,K.

for some strictly positive constants c7 and c8. Moreover,

supξ∈X2

Pξ

∥

∥

∥

∥

∥

(

ZΛi,2(·) −

λiqi(0)

µcsi

)+∥

∥

∥

∥

∥

T

≥ 2ǫΛ

≤ c11e−c12

√Λ, i = 1, . . . ,K, (A28)

for all Λ large enough and for some strictly positive constants c11 and c12.

The proof of Proposition 6 is very similar to the proofs of the estimate for ZΛ1 (t) that were

stated in Proposition 5. The detailed proof is omitted, and we turn now to the finer analysis of the

queue length and waiting time processes.

Towards than end, we let WΛi (t) be the virtual waiting time for type-i customers at time t.

Since customers are served in a FCFS fashion, we have that WΛi (t) = WΛ

j (t) for all i 6= j. Hence,

we may let WΛ(t) be the common virtual waiting time. Since, by Proposition 1, steady state

exists, the PASTA property guarantees that waiting time as seen by arriving customers is equal in

distribution to the steady-state virtual waiting time. The following proposition shows that using

(S)-(C) the queue length and the waiting time are small and in particular the queue exceeds the

threshold ηk at most by an amount that is o(√

Λ).

We now turn to a more refined analysis of the queue length process. Towards this end, define

the set

X3 := ξ ∈ X2 : q(ξ) ≤ ηΛk + ǫ

√Λ,

where X2 was defined in equation (A27). The following proposition is the most complicated one in

this appendix as it deals with a refined analysis of the queue length behavior and in particular one

that considers the behavior of the queue at an O(1) level. The other results are o(Λ) result and

are hence much simpler. The more refined analysis for the queue length is however necessary for

the other proofs as well as for the asymptotic feasibility result for the constrained case, as given in

Theorem 3 of the main paper [3].

12

Proposition 7 Fix an integer q ≥ 2. Then,

lim supΛ→∞

E

[

(

(QΛ(∞) − ηΛk )

+)q−1

]

≤ Cq, and lim supΛ→∞

√ΛE[WΛ(∞)] < ∞, (A29)

for some constant Cq. In particular, for all x > 0 and all Λ large enough,

P(QΛ(∞) − ηΛk )+ > x

√Λ ≤ Cq

(x√

Λ)q−1. (A30)

Also,

supξ∈X3

Pξ

∥

∥(QΛ(·) − ηΛk )+

∥

∥

T≥ 2ǫ

√Λ

≤ c11e−c12ǫ

√Λ, (A31)

and

supξ∈X3

Pξ

‖WΛ(·)‖T ≥ M2 + ǫ√Λ

≤ c11e−c12ǫ

√Λ, (A32)

for all Λ large enough and some strictly positive constants c11, c12 and M2.

Remark 3 (Lemma 1 and Theorem 3 in [3]) Note that Lemma 1 in the main paper [3] is

covered as a special case of Proposition 7. Moreover, when the threshold ηΛk

is chosen according to

the recommendation in §3.2 in the main paper [3], Theorem 3 there is a consequence of Proposition

7 above.

Proof of Proposition 7: Fix a constant K > 1 and define a function Φ(x) : R+ 7→ R+ as

follows:

Φ(x) = (x − ηΛk )+ + K.

The proof now proceeds as follows: We first fix the integer q ≥ 2 and establish that there exist

t∗ > 0 and M > 0 so that

supξ∈X1:q(ξ)>ηΛ

k+M

Eξ

[

Φ(QΛ(t∗/Λ))q]

− Φ(q(ξ))q ≤ −γΦ(q(ξ)q−1, (A33)

for some γ > 0. We will then use Lemma 8 from §E and adapt the argument used in the proof

of Theorem 5.1 in [5] to obtain a bound for the steady-state queue length process. The bound for

the steady-state waiting time will then follow from an application of Little’s law. Finally, we will

establish the bound in (A31) and (A31).

We start, then, by establishing (A33). Towards this end, fix M > 0 and assume that QΛ(0) >

13

ηΛk

+ M . Fix 0 < η ≤ M/2 and define the random time τΛ = inf

t ≥ 0 : QΛ(t) ≤ QΛ(0) − η

.

Note that on [0, τΛ ∧ T ], the queue length process QΛ(t) satisfies

QΛ(t) > ηΛk , and QΛ(t) = QΛ(0) + AΛ(t) −

k∑

i=1

DΛi,2(t) − DΛ

0 (t),

where DΛ0 (t) is the process of service completions not followed by a cross-selling phase, i.e, DΛ

0 (t) =

DΛ1 (t) −∑k

i=1 DΛi (t). On [0, τΛ] the instantaneous rate of DΛ

0 (t) can be bounded from below by

µsZΛ1 (t)

k−1∑

i=1

λi

Λ(1 − qi(0)) +

K∑

i=k

λi

Λ

,

which corresponds to the fact that, by the control (C), as long as the queue length is greater than

ηk all service completions of customers from type i ≥ k are followed by the admission to service of

a customer that is waiting in the queue. Fix now ǫ > 0 and δ > 0 and define the set

Ω(Λ) =

ω ∈ Ω :∥

∥ZΛ1 (·) − R

∥

∥

T≤ 2ǫΛ,

∥

∥

∥

∥

ZΛi,2(·) −

λiqi(0)

µcsi

∥

∥

∥

∥

T

≤ 2ǫΛ, i = 1, . . . , k

⋂

Ω∗(Λ, T/Λ, δ),

where Ω∗ is as defined in (A21). Then, on Ω(Λ) and for t ≤ τΛ ∧ T/Λ,

QΛ(t) ≤ QΛ(0) + Λt −k∑

i=1

µcsi

∫ t

0ZΛ

i,2(u)du − µs

k−1∑

i=1

λi

Λ(1 − qi(0)) +

K∑

i=k

λi

Λ

∫ t

0ZΛ

1 (u)du + δ,

Observe that for all t ≤ τΛ all servers are busy and consequently∑k

i=1 ZΛi,2 = NΛ − ZΛ

1 . Since, by

definition, NΛ − R =∑k

i=1 λiqi/µcsi , we have, after some straightforward algebra, that on Ω(Λ),

QΛ(t ∧ τΛ) ≤ QΛ(0) + CΛǫ(t ∧ τΛ) − λkqk(0) · t ∧ τΛ + δ, (A34)

for t ≤ T/Λ and for some constant C > 0. Equation (A34) is the crucial one. It reflects the fact

that once cross-selling to class k is stopped, the capacity of the system is large enough to attract

the aggregate queue back to ηk, and it will do so at a rate of approximately λkqk(0). We can now

choose ǫ small enough so that

QΛ(t ∧ τΛ) ≤ QΛ(0) + δ − 1

2Λλkqk(0) · t ∧ τΛ, (A35)

14

where for all i ≤ K, λi := λi/Λ. In particular, on Ω(Λ) we have that

ΛτΛ ≤ t∗ :=η + δ

12 λkqk(0)

.

Choosing δ ≤ η/2, the same argument shows that on Ω(Λ) and for all τΛ ≤ t ≤ t∗/Λ, QΛ(t) ≤QΛ(0) − η/2. Indeed, along the arguments leading to (A35) one can establish that on Ω(Λ), the

queue length is a linearly decreasing process as long as QΛ(t) > ηΛk. We conclude then that if

QΛ(0) ≥ ηΛk

+ M ,

Φ(QΛ(t∗/Λ)) − Φ(QΛ(0)) ≤ −η/2 (A36)

on Ω(Λ). Since QΛ(t) ≤ QΛ(0) + AΛ(t), it is straightforward to show that

lim supΛ→∞

supξ∈X1

Eξ[Φ(QΛ(t∗/Λ))1Ωc] − Φ(QΛ(0)) = 0.

Consequently, we have from (A36) that for all Λ large enough

supξ∈X1:Φ(q(ξ))>M

Eξ[Φ(QΛ(t∗/Λ))] − Φ(QΛ(0)) ≤ −η

4.

Let

LΛ(t∗) := supξ∈X1

Φ−(q−2)(ξ)Eξ

[

(Φ(QΛ(t∗/Λ)) − Φ(q(ξ)))2(Φ(q(ξ)) + |Φ(QΛ(t∗/Λ)) − Φ(q(ξ))|)q−2]

.

(A37)

Using again the fact that QΛ(t) ≤ QΛ(0) + AΛ(t) as well as some basic properties of the Poisson

process we have that lim supΛ→∞ LΛ(t∗) ≤ C1 for some constant C1. Hence, we have by Lemma 8

in §E that there exists a constant C2 such that

supξ∈X1:Φ(q(ξ))>C2

Eξ[Φ(QΛ(t∗/Λ))q] − Φ(q(ξ))q ≤ −ηq

8Φ(q(ξ))q−1. (A38)

Using again the fact that QΛ(t) ≤ QΛ(0) + AΛ(t), we have that

supξ∈X

Eξ

[

Φ(QΛ(t∗/Λ))q]

− Φ(q(ξ))q

Φ(q(ξ))q≤ C3, (A39)

15

for some C3. In particular,

supξ∈X :Φ(q(ξ))≤C2

Eξ

[

Φ(QΛ(t∗/Λ))q]

≤ C4,

for some C4 > 0. We now adapt arguments from the proof of Theorem 5.1 in Gamarnik and Zeevi

[5], to establish bounds for (QΛ(∞) − ηΛk)+. Specifically, by the definition of stationarity we have

that

EπΛ

[

Φ(QΛ(0))q]

= EπΛ

[

Φ(QΛ(t∗/Λ))q]

. (A40)

and, in particular,

0 =

∫

ξ∈ΞΛ

(

Φ(q(ξ))q − Eξ[Φ(QΛ(t∗/Λ))q ])

πΛ(dξ). (A41)

Combining equations (A38) and (A39) we have that

Eξ[Φ(QΛ(t∗/Λ))q ] − Φ(q(ξ))q ≤ −ηq8 Φ(q(ξ))q−11ξ ∈ X1 + C4 + C3Φ(q(ξ))q1ξ /∈ X1,

and, consequently, that for all ξ ∈ X ,

Φ(q(ξ))q−Eξ[Φ(QΛ(t∗/Λ))q] ≥ ηq

8Φ(q(ξ))q−1−C4−C3Φ(q(ξ))q1ξ /∈ X1+

ηq

8Φ(q(ξ))q−11ξ /∈ X1.

Plugging back into (A41), we have that

∫

ξ∈ΞΛ

(ηq

8Φ(q(ξ))q−1 − C4 − C3Φ(q(ξ))q1ξ /∈ X1 +

ηq

8Φ(q(ξ))q−11ξ /∈ X1

)

πΛ(dξ) ≤ 0.

(A42)

Now, using the bounds on the steady-state queue length from Proposition 5 and applying the

Cauchy-Schwarz inequality yields that both

EπΛ

[

Φ(QΛ(0))q−1(1ΞΛ(0) /∈ X1]

→ 0, as Λ → ∞, (A43)

and

EπΛ

[

Φ(QΛ(0))q(1ΞΛ(0) /∈ X1]

→ 0, as Λ → ∞. (A44)

Consequently, (A42) implies that for all Λ large enough

∫

ξ∈ΞΛ

(ηq

8Φ(q(ξ))q−1 − 2C4

)

πΛ(dξ) ≤ 0.

16

so that

E[Φ(QΛ(∞))q−1] ≤ 16C2

ηq.

Note that with q = 2 we have that E[(QΛ(∞) − ηΛk)+] ≤ C4, for some C4 > 0 and for all Λ large

enough. Consequently, lim supΛ→∞ E[(QΛ(∞) − ηΛk)+]/

√Λ = 0, so that the first part of (A29)

is established. Note that we have actually established that E[(QΛ(∞) − ηΛk)+] = O(1), which is

stronger than the statement of the Proposition which requires only that E[(QΛ(∞)−ηΛk)+] = o(

√Λ).

The statement about the steady-state waiting time now follows from Little’s law.

We now turn to the proof of equation (A31). To analyze the behavior of the queue length

process over the interval [0, T ], fix x > 0 and re-define the set

Ω(Λ) =

ω ∈ Ω :∥

∥ZΛ1 (·) − R

∥

∥

T≤ 2ǫΛ,

∥

∥

∥

∥

(

ZΛi,2(·) − λiqi(0)

µcsi

)+∥

∥

∥

∥

T

≤ 2ǫΛ, for all i = 1, . . . ,K

⋂

Ω∗∗(Λ, T, δΛ),

where

Ω∗∗(Λ, T, δΛ) :=

ω ∈ Ω : sup0≤η≤T

supt−s≤η

(|B(mΛt) − B(mΛs)| − δΛ(t − s) + O(log(mΛt ∨ 2))) ≤ ǫ

4

√Λ

.

(A45)

One can easily show that PΩ∗∗(Λ, T, δΛ)c ≤ C5e−C6ǫ

√Λ for all Λ large enough and for some

constants C5 and C6. Indeed, this probability bound follows from a combination of equation (A18)

and a basic result for Brownian motion (see Example 4.3.12 in page 264 of [7]). We now define the

random times

τ ′Λ = sup

t ≤ T : QΛ(t) ≤ ηΛk + ǫ

√Λ

, and τ ′′Λ = inf

t ≥ τ ′Λ : QΛ(t) ≥ ηΛk + 2ǫ

√Λ

.

By the same arguments preceding equation (A34), we have for all t ≥ τ ′Λ and by choosing ǫ

small enough that

QΛ(t ∧ τ ′′Λ) ≤ QΛ(τ ′Λ) − 1

2λkqk(0) ·

(

t ∧ τ ′′Λ − τ ′Λ)

+ δΛ(

t ∧ τ ′′Λ − τ ′Λ)

+ǫ

4

√Λ. (A46)

Choosing small enough δ, we have by that on Ω(Λ), QΛ(·) is smaller than QΛ(τ ′Λ) + ǫ/4√

Λ for all

17

t ≥ τ ′Λ4. Consequently, τ ′′Λ > T on Ω(Λ) so that sup0≤t≤T QΛ(t) > ηΛk

+ 2ǫ√

Λ. Hence,

supξ∈X3

Pξ

sup0≤t≤T

QΛ(t) > ηΛk + 2ǫ

√Λ

≤ PΩ(Λ)c ≤ C7e−C8

√Λ,

for all Λ large enough where the last inequality follows from Propositions 5 and 6 as well as the

probability bound for Ω∗∗(Λ, T, δΛ). This establishes equation (A31). The proof of (A32) uses the

following representation for the virtual waiting time:

WΛ(t) = inf

s ≥ 0 : DΛ(t + s) − DΛ(t) ≥ QΛ(t)

,

where DΛ(t) is the aggregate number of depletions of customers from the queue up to time t.

Let τΛ(t) = infs ≥ 0 : QΛ(t + s) = 0. Since while the queue is non-empty every cross-selling

completion is followed by an admission of a customer from the queue, we have that on [t, τΛ(t)],

DΛ(t + s)−DΛ(t) ≥k∑

i=1

DΛi,2(t + s)−DΛ

i,2(t) ≥k∑

i=1

µcsi

∫ t+s

tZΛ

i,2(u)du− |B(mΛ(t + s))−B(mΛt)|.

Then, on the set Ω(Λ) and for s ≤ τΛ(t),

DΛ(t + s) − DΛ(t) ≥k∑

i=1

µcsi

∫ t+s

tZΛ

i,2(u)du − δΛs ≥ sk∑

i=1

µcsi

λiqi(0)

µcsi

− C9ǫΛs − δΛs − ǫ

4

√Λ,

for some constant C9 > 0. Consequently, for all t ≥ 0,

WΛ(t) ≤ Cw√Λ

+‖QΛ(·)‖T

∑ki=1 λiqi(0) − CǫΛ − δΛ

,

for some constant Cw. The bound for the waiting time in (A32) now follows by choosing δ and ǫ

small enough and using the bound in (A31). Note that the virtual waiting time actually depends

on the behavior of the departures slightly after time T . This problem is easily overcome, however,

by re-defining Ω(Λ) using the interval [0, 2T ] instead of [0, T ].

In order to analyze the number of cross-sold customers from each class a finer analysis is required.

In particular, we need a handle of the waiting time of the customers that are present in the system

at time 0 (which is assumed to be distributed according to the stationary distribution). For the

following results, we let ZΛ(t) be the number of customers in the first phase of service or in queue

18

that found upon arrival a virtual waiting time that is longer than 2M2/√

Λ, where M2 is as given

in Proposition 7. Formally,

ZΛ(t) =∑

j∈S(t)

1wj(t) > 2M2

√Λ. (A47)

We then have the following Lemma where we use

X4 := ξ ∈ X3 : z(ξ) ≤ ǫΛ, (A48)

and X3 is as defined in (A29).

Lemma 7 Fix an integer q ≥ 2 and ǫ > 0. Then,

P

ZΛ(∞) > 2ǫΛ

≤ c131

√Λ

(q−1). (A49)

Moreover,

supξ∈X3

Pξ

‖ZΛ(·)‖T > 2ǫΛ

≤ c13e−c14ǫ

√Λ, (A50)

for all Λ large enough and for some constant c13 and c14.

Proof: We define QΛ(t) to be the number of customers in queue that found a virtual waiting time

longer than 2ηΛk/Λ upon arrival. Then, focusing on the number of customers with waiting times

longer then 2ηΛk/Λ in queue or in service we have the equation,

QΛ(t) + ZΛ(t) = QΛ(0) + ZΛ(0) +

∫ t

01WΛ(t−) > 2M2

√ΛdAΛ(t)

−DΛ

1 (t)∑

l=1

∑

j∈S(t−)

1es(S(t−), l) = j, wj(t−) > 2M2

√Λ. (A51)

Note that∑DΛ

1 (t)l=1

∑

j∈S(t−) 1es(S(t−), l) = j, 2M2

√Λ is a non-homogeneous Poisson process with

instantaneous rate equal to µsZΛ(u).

Consider the differential equation (initialized at ZΛ(0)),

¯ZΛ(t) = ¯ZΛ(0) − µs

∫ t

0

¯ZΛ(u)du. (A52)

19

Since ZΛ(0) ≤ NΛ, there exists a finite time t∗ after which ¯ZΛ(t) ≤ ǫΛ. Fix the set

Ω(Λ) := Ω∗(Λ, T, ǫΛ)⋂

ω ∈ Ω : ‖WΛ(·)‖T ≤ M2 + ǫ√Λ

⋂

ω ∈ Ω : ‖QΛ(·)‖T ≤ ηΛk + 2ǫ

√Λ

.

Subtracting ZΛ(t) from ¯ZΛ(t), using the fact that∫ t0 1WΛ(t−) > 2ηΛ

k/ΛdAΛ(t) = 0 on Ω(Λ) and

finally applying Gronwall’s inequality (see, e.g., Problem 5.2.7 on page 287 of Karatzas and Shreve

[7]) we have that

‖ZΛ(·) − ¯ZΛ(·)‖T ≤ CeCT (‖B(mΛ·)‖T + ‖QΛ(·)|T ), (A53)

on Ω(Λ). We the have that

PπΛZΛ(t∗) > 2ǫΛ ≤ PπΛΞΛ(0) /∈ X3 + supξ∈X3

PξC ′Λ > ǫΛ,1Ω(Λ)

+ supξ∈X3

Pξ1Ω(Λ)c ≤ C13e−C14ǫ

√Λ, (A54)

where C ′Λ is the right hand side in equation (A53) and the last inequality follows from the definition

of Ω(Λ), Proposition 7 and the properties of Brownian Motion enlisted in §C.2. Since under the

steady state distribution ZΛ(t∗) has the same distribution as ZΛ(∞), we can use Proposition 7

and the bounds from §C.2 to establish equation (A49). The transient bound in equation (A50)

follows a similar argument in which one replaces in equation (A54) the initial distribution πΛ with

an arbitrary initial condition within the set X4.

We now turn to the analysis of the number of agents busy cross-selling to type-i customers. We

first focus on types i ≤ k − 1. Type k requires a separate analysis which is given in Proposition 9.

We define

X5 :=

ξ ∈ X4 :

∣

∣

∣

∣

zi,2(ξ) −λiqi(0)

µcsi

∣

∣

∣

∣

≤ ǫΛ, i = 1, . . . , k − 1

(A55)

and X4 is as defined in (A48).

Proposition 8 Fix an integer q ≥ 2 and ǫ > 0. Then,

P

∣

∣

∣

∣

ZΛi,2(∞) − λiqi(0)

µcsi

∣

∣

∣

∣

≥ ǫΛ

≤ c151

(√

Λ)(q−1), i = 1, . . . , k − 1, (A56)

and

E

[∣

∣

∣

∣

ZΛi,2(∞) − λiqi(0)

µcsi

∣

∣

∣

∣

]

≤ ǫΛ, (A57)

20

for all Λ large enough and for some constant c15. Moreover,

supξ∈X5

Pξ

∥

∥

∥

∥

ZΛi,2(·) −

λiqi(0)

µcsi

∥

∥

∥

∥

T

≥ 2ǫΛ

≤ c16e−c17ǫ

√Λ, i = 1, . . . , k − 1, (A58)

for all Λ large enough and for some constant c16 and c17.

Remark 4 (Proof of Theorem 1 in [3]) Since ǫ is arbitrary, Proposition 8 implies that for

i ≤ k − 1, E[ZΛi,2(∞)] = λiqi/µ

csi + o(Λ). In particular, since E[ZΛ

1 (∞)] = R + o(Λ) and N =

R +∑k

i=1 λiqiµcsi , it suffices to show that E[IΛ(∞)] = o(Λ), where IΛ(t) is the number of idle

agents at time t, to conclude that also for k:

E[ZΛk,2(∞)] =

λkqk

µcsk

+ o(Λ).

The required result for IΛ(∞) is established in Proposition 9, which together with Proposition 7

establish (22) and complete the proof of Theorem 1 in the main paper [3].

Proposition 9

E[IΛ(∞)] = o(Λ), and E[ZΛi,2(∞)] =

λiqi

µcsi

+ o(Λ), ∀i ≤ k. (A59)

The proof of Proposition 9 is postponed until after the proof of Proposition 8.

Proof of Proposition 8: The argument is very similar in nature to the one used in the proof

of Lemma 7. Define first the set

Ω(Λ) := Ω∗(Λ, T, x√

Λ)⋂

ω ∈ Ω : ‖WΛ(·)‖ ≤ M2 + 2ǫ√

Λ√Λ

⋂

w ∈ Ω : ‖ZΛ(·)‖T ≤ (ǫ + x)Λ

⋂

w ∈ Ω : ‖ZΛ1 (·) − R‖T ≤ 2ǫΛ

⋂

w ∈ Ω : ‖QΛ(·)‖T ≤ ηk + ǫ√

Λ

, (A60)

where we choose M so that M < (ηk−1 − ηk)/2. As ZΛi,2(t) = ZΛ

i,2(0)−DΛi,2(t) + DΛ

i (t), we have on

Ω(Λ) that

ZΛi,2(t) ≥ ZΛ

i,2(0)−µcsi

∫ t

0ZΛ

i,2(u)du+µs λi

Λqi

(

2M2/√

Λ)

∫ t

0ZΛ(u)−ZΛ(u)du−‖B(mΛ·)‖T +O(log(mΛT∨2)).

(A61)

Here we used the definition of ZΛ as well as the fact that while QΛ(t) ≤ ηk−1 all service completions

21

with type-i customers are followed by a cross-selling offer if the customer agrees to listen. Also,

ZΛi,2(t) ≤ ZΛ

i,2(0)−µcsi

∫ t

0ZΛ

i,2(u)du+µs λi

Λqi(0)

∫ t

0ZΛ

1 (u)−ZΛ(u)du+‖B(mΛ·)‖T +O(log(mΛT∨2)).

(A62)

Consider now the differential equation (initialized at ZΛi,2(0)):

ZΛi,2(t) = ZΛ

i,2(0) + µs λi

Λqi(0)Rt − µcs

i

∫ t

0ZΛ

i,2(u)du. (A63)

Then, subtracting ZΛi,2(t) from ZΛ

i,2(t), using the inequalities (A61) and (A62) as well as equation

(A63) and applying Gronwall’s inequality we have that

‖ZΛi,2(·) − ZΛ

i,2(·)‖T ≤ CeCT [qi(‖WΛ(·)‖T )‖ZΛ1 (·) − R‖T ∨ ‖ZΛ(·)‖T

∨ (‖B(mΛ·)‖T + O(log(mΛT ∨ 2)))].(A64)

The argument now follows as in the proof of Lemma 7. Re-define the function Φ(x) = |x −λiqi(0)/µ

csi |. We first consider the differential equation (A63). Observe that since ZΛ

i,2(0) ≤ NΛ,

there exists t∗ so that

Φ(ZΛi,2(t

∗)) ≤ ǫΛ.

Hence,

PπΛΦ(ZΛi,2(t

∗)) > 2ǫΛ ≤ PπΛΞΛ(0) /∈ X4 + supξ∈X4

PξC ′′Λ > 2ǫΛ, Ω(Λ)

+ supξ∈X4

PξΩ(Λ)c ≤ C

(√

Λ)q−1

2

, (A65)

where C ′′Λ is the right hand side in (A64) and X4 is as defined in (A48). Since Φ(ZΛi,2(t

∗)) has the

same distribution as Φ(ZΛi,2(∞)) when starting with the steady distribution we have established

equation (A56). Equation (A57) follows a similar argument noting that

EπΛ [Φ(ZΛi,2(t

∗))] ≤ NΛPπΛΞΛ(0) /∈ X3 + supx∈X3

Eξ[Φ(ZΛi,2(t

∗))]

and applying the bounds we have from the previous propositions to show that

NΛPπΛΞΛ(0) /∈ X4 → 0, as Λ → ∞

22

as well as

supξ∈X4

Eξ[Φ(ZΛi,2(t

∗))] ≤ NΛ supξ∈X4

PξΩc + 2ǫΛ.

Replacing ǫ with ǫ/2 throughout one has the result. The transient bound is easily obtained using

similar arguments.

Proof of Proposition 9: Re-define the set

Ω(Λ) = Ω∗∗(Λ, T, δΛ)⋂

ω ∈ Ω :∥

∥ZΛ1 (·) − R

∥

∥

T≤ 2ǫΛ

∥

∥

∥

∥

ZΛi,2(·) −

λiqi

µcsi

∥

∥

∥

∥

T

≤ 2ǫΛ, ∀i = 1, . . . , k − 1;

‖ZΛ(·)‖T ≤ 2ǫΛ; ‖WΛ(·)‖T ≤ (M2 + ǫ)/√

Λ

, (A66)

where Ω∗∗ is defined in (A45). Assume that IΛ(0) ≥ 2ǫΛ and set τΛ = inft ≥ 0 : IΛ(t) ≤IΛ(0) − ǫΛ. Then, on Ω(Λ),

IΛ(t ∧ τΛ) ≤ IΛ(0) − Λ(t ∧ τΛ) +

k−1∑

i=1

µcsi

∫ t∧τΛ

0ZΛ

i,2(u)du

+ µs(

1 − q(

2M2/√

Λ))

∫ t∧τΛ

0ZΛ(u) − ZΛ(u)du

+ µcsi

∫ t∧τΛ

0

N − IΛ(u) − ZΛ(u) −k−1∑

i=1

ZΛi,2(u)

du + δΛt + ǫ√

Λ. (A67)

Some algebra yields that

IΛ(t ∧ τΛ) ≤ IΛ(0) + ǫ√

Λ − CΛt, (A68)

for some constant C > 0. Starting at IΛ(0) > 2ǫΛ, then, there exists t∗ at which IΛ(t∗) ≤ ǫΛ.

If, on the other hand, IΛ(0) ≤ 2ǫΛ, then we claim that IΛ(t) ≤ 3ǫΛ for all t ≤ T . Indeed let

τ ′Λ := inft ≥ t∗ : IΛ(t) ≥ ǫΛ and τ ′′Λ := inft ≥ τ ′Λ : IΛ(t) ≥ 2ǫΛ. Then, using (A68) we have

that that on Ω(Λ), τ ′′Λ > T . Consequently, on Ω(Λ), there exists t∗ wich IΛ(t∗) ≤ 3ǫΛ regardless

of the initial condition. We then have that

EπΛ [IΛ(t∗)] ≤ NΛPΞΛ(0) /∈ X4 + 3ǫΛ + supξ∈X5

PΩ(Λ)c,

23

and the proof is completed by noting that

NΛPΞΛ(0) /∈ X5 + supξ∈X5

PΩ(Λ)c → 0, as Λ → ∞.

As explained in Remark 4, with Proposition 9 we conclude the proof of Theorem 1 in the main

paper [3]. We conclude this appendix with Lemma 8 that was used in the proof of Proposition 7.

E Auxiliary Results

In this section we prove Proposition 5 as well as one simple general result that we used in the proof

of Proposition 7. We begin with the latter.

The following Lemma is an adaptation of a result that and appears in [6] and is due to Gamarnik

and Zeevi. We state and prove it here for completeness. Fix Λ and consider the process Ξ(t) with

the domain X and Let Φ(x) be a function Φ(x) : X 7→ R+. We fix a subset X ⊂ X . We let

L(t) = supξ∈X

Φ−(q−2)(ξ)Eξ

[

Φ(Ξ(t)) − Φ(ξ))2(Φ(ξ) + |Φ(Ξ(t)) − Φ(ξ)|)q−2]

. (A69)

Lemma 8 Fix an integer q ≥ 2. Assume that there exists K > 0, γ > 0 and t∗ > 0, so that

supξ∈X :Φ(ξ)>K

Eξ[Φ(Ξ(t∗))] − Φ(ξ) ≤ −γ, (A70)

and that L(t∗) is finite. Then,

supξ∈X :Φ(ξ)>K ′

Eξ[Φ(Ξ(t∗))q] − Φ(ξ)q ≤ −γq

2Φ(ξ)q−1, (A71)

with K ′ = maxK,L(t∗)(q − 1)/γ.

Proof: Using second order Taylor’s expansion of the function xp around Φ(ξ) we obtain for every

24

ξ ∈ X such that Φ(ξ) > k,

Eξ[Φq(Ξ(t∗))] − Φq(ξ) = qΦq−1EξEξ[Φ(Ξ(t∗)) − Φ(ξ)]

+q(q − 1)

2Eξ[(Φ(ξ) + Z(Φ(Ξ(t∗)) − Φ(ξ)))q−2(Φ(Ξ(t∗)) − Φ(ξ))2]

≤ −γqΦq−1(ξ) +q(q − 1)

2Eξ[(Φ(ξ) + |Φ(Ξ(t∗)) − Φ(ξ)|)q−2(Φ(Ξ(t∗)) − Φ(ξ))2]

≤ −γqΦq−1(ξ) +q(q − 1)

2L(t∗)Φq−2(ξ) (A72)

where Z is a random variable with support in [0, 1]. When Φ(ξ) > L(t∗)(q − 1)/γ, we obtain that

Eξ[Φq(Ξ(t∗))] − Φq(ξ) ≤ −γq

2Φq−1(ξ).

Proof of Proposition 5: We now prove the estimates given in Proposition 5 for the number of

agents busy giving service and the queue length. Some of the steps are very similar to those in [1].

The similar parts will be abbreviated and the reader will be referred to the technical appendix of

[1] for the details. The first step is the following Lemma, the first part of which is an analogue of

Lemma B.1 in [1].

Lemma 9 Fix ǫ > 0. Then, there exists t0(ǫ) (independent of the initial conditions), such that

P

supt0(ǫ)≤t≤T

(

ZΛ1 (t) − R

)−> ǫΛ

≤ c−c10√

Λ9 , (A73)

for all Λ large enough and for two positive constants c9 and c10. Consequently,

P

(ZΛ1 (∞) − R)− > ǫΛ

≤ c18e−c19

√Λ,

and

supξ∈X1

Pξ

‖(ZΛ1 (·) − R)−‖T ≥ 2ǫΛ

≤ c9e−c10

√Λ. (A74)

for all Λ large enough and for some strictly positive constants c9 and c10.

Proof: The first part of the Lemma is proved just as in [1] and its proof is omitted. Since the

bound is independent of the initial state, we can in particular initialize the system with its steady

25

state distribution in which case we get the bound on this steady state distribution.

The next step establishes a crude bound for the queue-length process. It shows that the queue

length is essentially of order Λ. This step is required to obtain later the more refined bound.

Lemma 10 Fix ǫ > 0. Then,

lim supΛ→∞

E

[

exp

(

QΛ(∞)

Λ

)]

≤ CQ

for some constant CQ.

Proof: We use a Lyapunov type of argument along the lines of Gamarnik and Zeevi [5]. First,

assume that QΛ(0) > 2MΛ for some constant M > 0 and define the random time

τΛ = inft ≥ 0 : QΛ(t) ≤ QΛ(0) − MΛ.

Since the largest threshold, ηΛ1 satisfies ηΛ

1 = η1

√Λ, there exists Λ large enough such that MΛ > ηΛ

1 .

Consequently, on [0, τΛ], all service or cross-selling completions will be followed by an admission of

a customer from the head of the queue. The queue length process satisfies then the equation

QΛ(t ∧ τΛ) = QΛ(0) + AΛ(t) − DΛ1 (t) −

k∑

i=1

DΛi,2(t). (A75)

Using the strong approximation we have that

QΛ(t∧ τΛ) ≤ QΛ(0) + Λt− µs

∫ t

0ZΛ

1 (s)ds−k∑

i=1

µcsi

∫ t

0ZΛ

i,2(s)ds + ‖B(mΛ·)|T + O(log(mΛT ∨ 2)).

(A76)

Let

Ω(Λ) = Ω∗(Λ, T, ǫΛ)⋂

ω ∈ Ω : ‖ZΛ1 (·) − R‖T ≤ ǫΛ,

with Ω∗(Λ, T, ǫΛ) as defined in (A21). We then have after some algebraic manipulation that on

Ω(Λ),

QΛ(t ∧ τΛ) ≤ QΛ(0) + ǫΛ − CΛ(t ∧ τΛ). (A77)

26

We claim now that with t∗ = 3M/(2C) we have that

Eξ∈X :q(ξ)>ǫΛEξ[exp(QΛ(t∗)/Λ)]

exp(q(ξ)/Λ)≤ exp(−M/4). (A78)

We omit the simple argument. We now use equation (A78) to establish a bounds for the steady state

queue length using a result from [5]. Towards that end, note first that since QΛ(t) ≤ QΛ(t)+AΛ(t),

we have that

supξ∈X

Eξ

[

eQΛ(t∗)/Λ]

eq(ξ)/Λ≤ C1, (A79)

for some constant C1 > 0 and for all Λ. We can now apply Theorem 5 of [5] to obtain that

E

[

exp

(

QΛ(∞)

Λ

)]

≤ CQ, (A80)

for some constant CQ and all Λ large enough.

In the next Lemma we show that the queue hardly exceeds the greatest threshold, ηΛ1 , as stated

in Proposition 5.

Lemma 11 ǫ > 0 and q ≥ 2. Then,

lim supΛ→∞

E

[

(

(QΛ(∞) − ηΛ1 )

+)q−1

]

≤ CQ (A81)

for some strictly positive constant CQ. Moreover,

supξ∈X1

Pξ

∥

∥(QΛ(·) − ηΛ1 )+

∥

∥

T≥ 2ǫ

√Λ

≤ c9e−c10ǫ

√Λ (A82)

for all Λ large enough and for some strictly positive constants c9 and c10.

Proof: The proof follows almost exactly the proof of the Lemma 10 up to equation (A77). The

main difference is that here, like in the proof of Proposition 7, one replaces the time interval

[0, T ] with the time interval [0, T/Λ]. Once the dynamics of the system above the level of ηΛ1 are

established, the proof follows almost exactly as the proof of Proposition 7 with two exceptions:

First, we replace X1 there with

X ′1 := ξ ∈ X : (z1(ξ) − R)− ≤ ǫΛ.

27

Then, the modification of equations (A43) and (A44) is established here using Lemma 10 (rather

than 5). Specifically, by the Cauchy-Schwarz inequality that

EπΛ

[

Φ(QΛ(0))q−1(1ΞΛ(0) /∈ X ′1]

≤√

EπΛ [(Φ(QΛ(∞)))2(q − 1)]√

PπΛΞΛ(0) /∈ X ′1. (A83)

By Lemma 10,

lim supΛ→∞

E[(Φ(QΛ(∞)))2(q−1)] ≤ CΛq−1,

for some constant C > 0. By Lemma 9

PπΛΞΛ(0) /∈ X ′1 ≤ c18e

−c19√

Λ.

Plugging back into (A83) we have that

EπΛ

[

Φ(QΛ(0))q−1(1ΞΛ(0) /∈ X ′1]

→ 0, as Λ → ∞.

The modification of equation (A44) follows similarly. The remainder of the proof follows almost

exactly the remainder of proof of Proposition 7 with the obvious modifications required by the

replacement of the smallest threshold ηΛk

with the largest threshold ηΛ1 .

The last component required to complete the proof of Proposition 5 is to establish a two-sided

bound for the difference ZΛ1 (·) − R, rather then the one sided bound from Lemma 9. The result is

given in the following Lemma.

Lemma 12 Fix ǫ > 0 and q ≥ 2. Then,

P∣

∣ZΛ1 (∞) − R

∣

∣ > ǫΛ

≤ c7

(√

Λ)q−1, (A84)

and

supξ∈X1

Pξ

∥

∥ZΛ1 (·) − R

∥

∥

T≥ 2ǫΛ

≤ c9e−c10

√Λ, (A85)

for all Λ large enough and for some strictly positive constants c9 and c10.

28

Proof: The proof is very similar in nature to the proof of Proposition 8 and is actually simpler.

First, note that QΛ(t) and ZΛ1 (t) satisfy the equation

QΛ(t) + ZΛ(t) = QΛ(0) + ZΛ(0) + AΛ(t) − DΛ1 (t),

or using the strong approximation decomposition,

QΛ(t)+ZΛ(t) = QΛ(0)+ZΛ(0)−Λt−µs

∫ t

0ZΛ

1 (s)ds+B

(

Λt + µs

∫ t

0ZΛ

1 (s)ds

)

+O(log(mλt∨2)).

Consider the differential equation (initialized at ZΛ1 (0))

ZΛ1 (0) = ZΛ

1 (0) + Λt − µs

∫ t

0ZΛ

1 (s)ds.

Then, subtracting ZΛ1 (t) − ZΛ

1 (t) and using Gronwall’s inequality we have that

‖ZΛ1 (·) − ZΛ

1 (·)‖T ≤ CeCT[

‖QΛ(·)‖T + ‖B(mΛ·)|T + O(log(mΛT ∨ 2))]

.

The proof now proceeds as in the proof of Proposition 8 using the bounds for the Brownian

motion from §C.2 and the bound for QΛ(·) from Lemma 11 above.

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