Technical Appendix for: When Promotions Meet Operations: Cross-Selling and Its Effect on Call-Center Performance

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1 Technical Appendix for: When Promotions Meet Operations: Cross-Selling and Its Effect on Call-Center Performance In this technical appendix we provide proofs for the various results stated in the manuscript titled: When promotions meet operations: Cross-selling and its effect on call-center performance. We begin the technical appendix with a formal construction of the sample paths under the TP[K] control. A. Sample-path construction The sample path construction follows a strong approximation approach (see for example [7] and [8]). Let N i ( ), i = 1,...,11, be independent unit rate Poisson processes. Then, under the TP[K] control, one can write the system dynamics through the following equations: Q (t) + Z 1 (t) = Q () + Z 1 () + ÑA(t) ÑD 1 (t), (A1) Z2 (t) = Z2 () ÑD 2 (t) t ) + 1 {K } [N 7 (pµ s Z1 (u)1 {<Q (u) K }du ( t )] + N 5 pµ s Z1 (u)1 {Q (u)=}du ( t ) + 1 {K <}N 8 pµ s Z1 (u)1 {Y (u) N K }du. (A2) Here, we use a time change of the processes N 7 ( ) and N 5 ( ) to construct service completions with cross-selling candidates that are followed by actual cross-selling. These two processes are used if the threshold is non-negative. In this case, a service completion is followed by a cross-selling offer whenever the queue is equal to or when it is greater than but smaller than the threshold. We use N 8 ( ) similarly for the cases in which the threshold is negative. In this case, a service completion is followed by a cross-selling offer if the number of idle servers is less than K. The processes Ñ A (t), Ñ D1 (t) and ÑD 2 (t) will be defined shortly. 1

2 Also, we have that ( Z1 (t) = Z1 () + N 1 t 1 {Z1 (u)+z 2 }du ( (u)<n ) t + N 3 µ cs 1 {K } Z 2 (u)1 {Q (u)>}du [N 5 ( pµ s t Z 1 (u)1 {Q (u)=}du ( t 1 {K }N 7 pµ s Z 1 (u)1 {<Q (u) K }du ( t 1 {K <}N 8 pµ s 1 {K <} ) ) + N 6 ( ) Z 1 (u)1 {Y (u) N K }du ( [N t 9 pµ s Z 1 (u)1 {K <Y (u) N }du ) )] t (1 p)µ s Z 1 (u)1 {Q (u)=}du ) + N 1 ( )] t (1 p)µ s Z 1 (u)1 {Q (u)=}du Here, in addition to the previous processes, we use a time change of N 1 ( ) to construct the arrivals that occur when some servers are idle. Both processes N 6 ( ) and N 1 ( ) are used to construct service completions with customers that are not cross-selling candidates when the queue is empty. N 6 ( ) is used for non-negative thresholds and N 1 ( ) is used for negative ones. The process N 9 ( ) is used to construct service completions with cross-selling candidates when the queue is empty (for negative threshold), and, finally, the process N 3 ( ) is used to construct cross-selling completions when the queue is positive. Note that the event epochs, in which there is a service completion that is followed by admission of a customer to service, are not modeled above as these transitions keep Z 1 unchanged. We construct the aggregate-arrival process, Ñ A (t), and the process of cross-selling completions, Ñ D2 (t), as follows: t ) ( t ) Ñ A (t) := N 1 ( 1 {Z 1 (u)+z2 }du + N (u)<n 2 1 {Z 1 (u)+z2 }du. (u) N ( t ) t ) Ñ D2 (t) := N 3 µ cs Z2 (u)1 {Q (u) >}du + N 4 (µ cs Z2 (u)1 {Q (u)=}du. Here, we use N 2 ( ) to construct arrivals when all servers are busy and N 4 ( ) to construct cross- 2

3 selling completions when the queue is empty. Finally, we have ( t ) Ñ D1 (t) = 1 {K }N 5 pµ s Z1 (u)1 {Q (u)=}du ( t ) + 1 {K }N 6 (1 p)µ s Z1 (u)1 {Q (u)=}du ( t ) + 1 {K }N 7 pµ s Z1 (u)1 {<Q (u) K }du ( t ) + 1 {K <}N 8 pµ s Z1 (u)1 {Y (u) N K }du ( t ) + 1 {K <}N 9 pµ s Z1 (u)1 {K <Y (u) N }du ( t ) + 1 {K <}N 1 (1 p)µ s Z1 (u)1 {Q (u)=}du ( t ) + N 11 ˆ(u)du, (A3) where the rate function ˆ(t) is set to satisfy that the sum of the instantaneous rates of all the processes in (A3) equals µ s Z1 (t) at time t. This construction follows by noting that all input and output processes in the system can be generated through thinning of Poisson processes. By Lemma 9.4 in [7], there exists a probability space (Ω, F, P), a filtration {F t } t and an 11-dimensional Brownian Motion (B 1 ( ),..., B 11 ( )) such that the random variable E i := sup t N i (t) t B i (t), (A4) log(2 t) has a moment generating function in a neighborhood of the origin and in particular, there exist constants c 1, c 2 and Γ, such that, for all i = 1,..., 11, and all x, P {E i Γ + x} c 1 e c 2x. (A5) We let E := 11 i=1 E i. As we consider only cases in which N R + p µ cs, the value of the time change at time t in each of the equations (A1)-(A3) is bounded by ct for some positive constant c. 3

4 We now use (A4) to express the dynamics as follows: t Q (t) = Q () + Z1 () + t µ s Z1 (u)du Z1 (t) + MZ,Q(t) + O(log(2 ct)), (A6) Z 2 (t) = Z 2 () µ cs t Z 2 (u)du + pµ s t Z 1 (u)du t pµ s Z1 (u)1 {Y (u) N >K }du + M Z 2 (t) + O(log(2 ct)), (A7) and t t Z1 (t) = Z 1 () + 1 {Z 1 (u)+z2 }du µ (u)<n s Z1 (u)1 {Q (u)=}du t pµ s Z1 (u)1 {<Y (u) N K }du t + µ cs Z2 (u)1 {Q (u)>}du + M Z 1 (t) + O(log(2 ct)) (A8) Here, M Z,Q ( ), M Z 1 ( ) and M Z 2 ( ) are sums of time changed Brownian motions. For example, if K >, t ) MZ 1 (t) = B 1 ( 1 {Z 1 (u)+z2 }du (u)<n ( t ) + B 3 µ cs Z2 (u)1 {Q (u)>}du ( t ) t ) B 5 pµ s Z1 (u)1 {Q (u)=}du B 6 ((1 p)µ s Z1 (u)1 {Q (u)=}du ( t ) + B 7 pµ s Z1 (u)1 {<Y (u) N K }du, where B i ( ), i {1, 3, 5, 6, 7}, are standard Brnownian motions. Using the Brownian motion strong law of large numbers (see problem in [6]) and the bound ct on the time change values, we have that, uniformly on compact sets, ( M Z,Q (t) ), M Z 2 (t), M Z 1 (t) (,, ), as. (A9) 4

5 Put T1 (t) := pµ t s Z 1 (u)1 {Y (u) N >K }du, T 2 (t) := t 1 {Z1 (u)+z 2 }du, (u)<n T 3 (t) := µ s t Z 1 (u)1 {Q (u)>}du, T 4 (t) := µ cs t Z 2 (u)1 {Q (u)>}du, T 5 (t) := pµ s t Z 1 (u)1 {<Y (u) N K }du, and re-write (A6)-(A8) as follows: t Q (t) = Q () + Z1 () + t µ s Z1 (u)du Z1 (t) + MZ,Q(t) + O(log(2 ct)), (A1) Z 2 (t) = Z 2 () µ cs t Z 2 (u)du + pµ s t Z 1 (u)du T 1 (t) + M Z 2 (t) + O(log(2 ct)), (A11) Z 1 (t) = Z 1 () + T 2 (t) µ s t Z 1 (u)du + T 3 (t) + T 4 (t) T 5 (t) + M Z 1 (t) + O(2 log(ct)). (A12) Notational conventions and organization of the appendix. Under a T P[K] policy, the process (Ξ (t), t ) defined by Ξ (t) := (Q (t), Z2 (t), Z1 (t)), is a Markov process. We denote X as the state-space of Ξ (t) and use the notation ξ for a general element in X. For a given ξ X we let q(ξ), z 2 (ξ) and z 1 (ξ) be its corresponding coordinates. We use P ξ { } = P ξ { Ξ () = ξ} and E ξ [ ] := E ξ [ Ξ () = ξ]. If ν is the steady-state distribution of Ξ (t) 1, we let P ν { } be the probability with respect to an initial condition that is distributed according to ν. E ν [ ] is the corresponding expectation. Finally, for x R, we let x := max{, x} and x + := max{, x}. The rest of this appendix is organized as follows. Each of the sections B, C, D is dedicated to the proof of Theorem 3.1 under one of the conditions 1, 2 and 3, respectively. E is dedicated to the proofs of the results in 4 of the main paper. The proofs of some auxiliary results are relegated to F. Finally, G provides some numerical examples that were omitted from 5 of the paper for space considerations. 1 In Lemma F.1 we prove the existence of a steady-state distribution under TP[K] assuming N > /µ s. 5

6 B. Asymptotic optimality under Condition 1 Asymptotic optimality under Condition 1 is proved in Corollary B.1. The main step is the following theorem. Theorem B.1 Consider a sequence of systems such that: (a) the th system uses N = R + βr + γ R + o( ) agents for some < β pµs µ cs and, (b) the th system uses TP[K ] for control with a non-negative sequence {K } that satisfies K / R ϱ as. Then, E[(Q K ) + ] R as, (A13) and E[I ] N R as. (A14) The main challenge in proving Theorem B.1 arises from the focus on the steady-state behavior rather than on the behavior on compact intervals. Most of the section is dedicated to the proof of this result. We first use Theorem B.1 to prove the asymptotic optimality result for this section: Corollary B.1 Suppose that Assumptions 3.1 and 3.2 hold. If, in addition, N2 R N1 R, then the following is asymptotically optimal in the sense of Definition 3.2: Staffing: Staff with N2 agents. Control: Use TP[K ] with K [, W ]. Proof: By Little s law: E[W ] W = E[Q ] W K + E[(Q K ) + ] W. (A15) Equation (A13) now implies that lim sup E[W ] W 1, (A16) and in particular that TP[K ] is asymptotically feasible. 6

7 To establish optimality, recall that E[Z 2 ] = N 2 E[Z 1 ] E[I ]. Since, by Little s law E[Z 1 ] = /µ s := R, (A14) implies that µ cs E[Z2 ] 1 as. (A17) µ cs (N2 R) For each, rµ cs (N 2 R) (C (N 2 ) C (R)) constitutes an upper bound for the (centered) optimal value of the cross-selling problem given by V () := sup π Π,N Z+ V (π, N); see formulation (6) and the discussion below it as well as Definition 3.2. Equation (A17) now implies that the upper bound is asymptotically achieved since, then, rµ cs E[Z 2 ] (C (N 2 ) C (R)) rµ cs (N 2 R) (C (N 2 ) C (R)) 1 as. Here we used the fact that, for three sequences {a }, {b } and {c } such that a and a /b 1 as, we also have that (a + c )/(b + c ) 1 as. Hence, the sequence of pairs {(N2, TP[K ])} asymptotically achieves the upper bound and is, in particular, asymptotically optimal. We proceed now to prove Theorem B.1. We prove the theorem in two steps. Proposition B.1 covers (A13) and Corollary B.2 covers (A14). Proposition B.1 Under the conditions of Theorem B.1, E[(Q K ) + ] as. (A18) We first provide an informal outline of the proof: 1. A constrained Lyapunov function argument: We start by examining the behavior of the Markov process (Ξ (t), t ) initialized, at time, within the set A ɛ := { ξ X : ( z 1 (ξ) ) } ɛ. (A19) µ s 7

8 We will show that, if Ξ () A ɛ and Q () > M for some constant M > large enough, then the process (Q (t), t ), decreases at a certain rate. In other words, we require a negative drift condition, reminiscent of the standard conditions used in Lyapunov function arguments; see e.g. [3]. In contrast to the standard requirement, which is imposed on all ξ X, we impose this requirement only for ξ A ɛ. Hence the term Constrained Lyapunov function. The (constrained) negative drift condition will allow us to obtain bounds on the stationary queue length. This bound will depend, however, on the behavior of the queue length when the process Ξ (t) is not inside the set A ɛ. 2. Bounding the behavior outside of A ɛ : We will bound the behavior of the queue length outside of the set A ɛ by: (a) showing that the stationary distribution is, in a sense, concentrated in A ɛ for all large enough see Lemma B.2, and (b) establishing a fluid scale bound on the stationary queue length see Lemma B.3. This fluid scale bound is weaker than the diffusion-scale one in (A13), but together with step (a) will allow us to bound the stationary queue length on states that are not in A ɛ. Proof of Proposition B.1: We prove the result for K. The proof is extended to arbitrary K > by replacing Q ( ) everywhere with (Q ( ) K ) +. We first establish bounds on the behavior of the queue length assuming that Ξ () A ɛ with A ɛ as in (A19). Fix constants Θ, T >, assume that Q () > 2Θ and define the random time 2 τ = inf{t : Q (t) Q () 3Θ/2} T. Define the set Ω (δ,, T, Θ) := { } ω Ω : 11 max sup B i (ct) + E i log(2 ct) δt Θ. (A2) i=1,...,11 t T We will be using the set Ω (,,, ) in various proofs in this appendix and set the parameters δ,, T, Θ according to the need of the specific proof. In any of these cases, we will choose the parameters in a way that will guarantee that the set of sample paths that we consider has sufficiently 2 Note that τ is a random time but not necessarily a stopping time. Our arguments are sample-path arguments so that whether or not τ is a stopping time is immaterial for our proofs. 8

9 large probability; see Lemma F.2. In the current proof it is important that we use T/ instead of T. That is we focus on a small time interval and characterize the behavior of the queue length there. To that end, plugging equation (A8) into equation (A6), and using the fact that Z2 (t) = N Z1 (t) whenever Q (t) >, we have that on Ω (δ,, T/, Θ/2) t τ Q (t τ ) Q ()+(t τ ) µ s Z1 (u)du µ cs t τ N Z1 (u)du+δt+θ/2. (A21) The following lemma provides a handle on the process Z 1 (t) that, in turn, allows us to characterize the behavior of Q (t). Lemma B.1 Suppose the conditions of Theorem B.1 hold and fix δ > small enough. Assume that Ξ () A ɛ. Then, for all ɛ >, there exist (ɛ) (independent of the initial conditions) such that, for all ω Ω (δ,, T/, Θ/2), ( sup Z1 (t) ) 2ɛ, t T/ µ s (A22) for all (ɛ). Using Lemma B.1 we have that on Ω (δ,, T/, Θ/2) and for all t τ, ) ) + ( ) µ s Z1 (t) µ csz2 (t) = µ s (Z 1 (t) 1µs µs (Z 1 (t) 1µs 1+β µcs µ s Z1 (t) ( ) ) + µ s ɛ µ( Z1 (t) 1 µ s + 1+β µ s Z1 (t), where µ = µ s µ cs. Hence, µ s ɛ µ β µ s, (A23) Q (t τ ) Q () + ( δ + µ s ɛ µ β ) (t τ ) + Θ/2. (A24) µ s Let η := (µ s ɛ + δ µ β µ s ), choose ɛ and δ small enough so that η > and let t := 2Θ/η. Then, as τ is the first time that Q (t) goes below Q () 3Θ/2 we must have that τ t / on Ω (δ,, T/, Θ/2). The arguments that lead to equation (A24) can be modified to show that, on Ω (δ,, T/, Θ/2) the queue will remain below Q () Θ after it reaches Q () 3Θ/2 for the first time. That is, 9

10 that Q (t) Q () Θ for all t / t T/. In particular, on Ω (δ,, T/, Θ/2), Q (2t /) 2 q(ξ) 2 sup ξ A ɛ :q(ξ)>2θ q(ξ) (q(ξ) Θ) 2 q(ξ) 2 sup ξ A ɛ :q(ξ)>2θ q(ξ) Θ, (A25) where A ɛ is as in (A19). Since As Q (t) Q () + A (t), it is also the case that (Q (t)) 2 (Q ()) 2 2Q ()A (t) + (A (t)) 2, (A26) so that, removing the condition that ξ A ɛ, we have that E ξ [Q (2t /) 2 ] q(ξ) 2 sup q(ξ)>2θ q(ξ) Θ+2E [( Θ + 2A (2t /) + (A (2t /) 2) 1 (Ω (δ,,t/,θ/2)) c ]. (A27) In Lemma F.2 we show that P {(Ω (δ,, T/, Θ/2)) c } c 5 e c 6(Θ/2 Γ)/log(2 ct), for some positive constants c 5 and c 6 and all large enough. We note that E[2A (2t /) + (A (2t /) 2 ] c 7 for some constant c 7 and all. Then, applying the Cauchy-Schwartz inequality, and re-choosing Θ large enough, we have that E ξ [Q (2t /) 2 ] q(ξ) 2 sup ξ A ɛ :q(ξ)>2θ q(ξ) Θ 2. (A28) The crude inequality (A26) also guarantees that sup E[Q (2t /) 2 q(ξ) 2 ] c 1, ξ A ɛ :q(ξ) 2Θ (A29) where c 1 := 4Kc 8 + c 9 for some constants c 8, c 9 that are independent of and Θ so that some simple manipulations lead to q(ξ) 2 E ξ [Q (2t /) 2 ] Θ ( 2 q(ξ) c 11 + c 11 Θ ) 2 q(ξ) E ξ[q (2t /) 2 q(ξ) 2 ] 1{ξ / A ɛ }, (A3) where c 11 = Θ 2 + c 1. By definition of stationarity we have that E ν [Q () 2 ] = E ν [Q (2t /) 2 ] 1

11 where ν is the steady-state distribution of the process (Ξ (t), t ), and in particular, When applied to (A3), this yields ( = q(ξ) 2 E ξ [Q (2t /) 2 ] ) ν (dξ). ξ Ξ E ν [Q ()] 2c 11 Θ + 2 ( [( ) ]) Θ E ν Θ 2 Q () c 11 + E Ξ ()[Q (2t /) 2 Q () 2 ] 1{Ξ () / A ɛ } (A31) To establish a bound on E ν [Q ()] we need to provide bounds for E[Q ()1{Ξ () / A ɛ }] which appears on the right hand side of (A31). The following two lemmas provide us with the necessary tools. Lemma B.2 Under the conditions of Proposition B.1, there exists T > such that P ν { Ξ () / A ɛ} c3 e c 4/log(2 ct), (A32) for all large enough. Lemma B.3 Under the conditions of Theorem B.1 for any integer m 1. lim sup E ν [( ) Q m ] () <, The proof of these lemmas are postponed to F and we now use them to complete the proof of the proposition. To that end, Using Lemmas B.2 and B.3 together with the Cauchy-Schwartz inequality, yields [ lim sup E ν (Q ()) m 1{ξ / A ɛ }] =. (A33) Applying (A33) and (A26) to (A31) we then have that E ν [Q ()] c 12, for some constant c 12 and all large enough and, in particular, that lim sup E ν [Q ()] =. This concludes the proof of Proposition B.1. 11

12 With the proof of Proposition B.1, we have established the first part of Theorem B.1 equation (A13). We turn now to prove (A14). First, we show that the number of idle servers does not exceed the negative part of the threshold. It applies to both cases β = and β > and will be used also in the proofs in C and D. Theorem B.2 Consider a sequence of systems such that: (a) the th system uses N = R + βr + γ R + o( ) agents for some β pµs µ cs, max(β, γ) > and, (b) the th system uses TP[K ] for a sequence {K } that satisfies K / R ϱ (, ) as. Then, E[ ( (N Z ) [K ] ) + ] N R as. (A34) Corollary B.2 below is a special case of Theorem B.2. Indeed, under the conditions of Theorem B.1 we have β > and K for all so that [K ] =. Corollary B.2 proves (A14) and hence completes the proof of Theorem B.1. Corollary B.2 Under the assumptions of Theorem B.1, E[I ] N R as. (A35) Proof of Theorem B.2: more involved and is proved in F. Here we prove the theorem only for the case β >. The case β = is We initialize the th system with Ξ () distributed according to its stationary distribution ν. The process (Ξ (t), t ) is then stationary. We will show that there exists t > such that E ν [I ( t)]/ as. Since, by stationarity, E ν [I (t)] = E ν [I ()] for all t, this will imply that E ν [I ()]/ as. Finally, since we assumed that β > we have that N R > c for some c > and all large enough. Consequently, we will conclude that E ν [I ()]/(N R) as. We now gradually fill-in the gaps in the above argument. First, we need some characterization of the fluid-level behavior of the system. Below, the processes Ti (t), i = 1,..., 5 are as defined in A. 12

13 Lemma B.4 (fluid Limits) Consider a finite interval [, T] and suppose that ( Q (), Z 1 (), Z 2 () ) ( Q(), Z1 (), Z 2 () ). ( Q (t) ) Then, under the assumptions of Theorem B.1, the sequence ; Z 1 (t) ; Z 2 (t) ; T i (t), i = 1,...,5 is tight in D[, T] and every subsequence { k } k 1 contains a further subsequence that converges in probability to some limit uniformly on compact sets. Moreover, any such limit process ( Q(t); Z1 (t); Z 2 (t); T i (t), i = 1,...,5 ), satisfies the following equations: Z 1 (t) + Q(t) = Z 2 (t) = Z 1 (t) = Q() + Z 1 () + t Z 2 () µ cs t t Z 1 () + T 2 (t) µ s t µ s Z1 (u)du, (A36) t Z 2 (u)du + pµ s Z1 (u)du T 1 (t), (A37) Z 1 (u)du + T 3 (t) + T 4 (t) T 5 (t), (A38) T 1 (t)1 { Q(t)>} = pµ s Z1 (t), (A39) T 2 (t)1 { Z1 + Z 2 < 1+β µs } = 1, (A4) T 3 (t)1 { Q(t)>} = µ s Z1 (t), (A41) T 4 (t)1 { Q(t)>} = µ cs Z2 (t), (A42) T 5 (t)1 { Q(t)>} =. (A43) Lemma B.5 Fix ɛ > and assume < β pµs µ cs. Let ( Q(t); Z1 (t); Z 2 (t); T i (t), i = 1,...,5 ), be a non-negative process that satisfies equations (A36)-(A43). Then, there exists t (ɛ) (independent 13

14 of Z 1 () and Z 2 () ), such that, for all t t (ɛ), Z 1 (t) 1 ɛ. µ s (A44) Moreover, there exists t t (ɛ), such that for all t t. Ī(t) := 1 + β µ s Z 1 (t) Z 2 (t) ɛ, (A45) Lemmas B.4 and B.5 are proved in F. We now use them to complete the proof of Theorem B.2 under the assumption that β >. To this end, initialize the th system according to its stationary distribution. Then, using Proposition B.1 and the fact that Z 1 + Z 2 N /µ s + p/µ cs, we have that the sequence of steady-state random variables ( Q /, Z 1 /, Z 2 / ) is tight and every limit point is of the form (, Z 1 (), Z 2 () ). By Lemma B.4, the sequence of processes ( Q (t)/, Z 1 (t)/, Z 2 (t)/ ) is tight and every limit point ( Q(t), Z 1 (t), Z 2 (t)) satisfies equations (A36)-(A43). We can thus apply Lemma B.5 to conclude the existence of t such that Ī(t) ɛ, for all t t. Since this holds for every limit point, we have that { I (t) lim sup P ν } > 2ɛ =. (A46) Since I (t) N /µ s p/µ cs we have that lim sup E ν [I (t)] 3ɛ, (A47) for all t t. Finally, since E ν [I (t)] = E ν [I ()], for all t, we have that lim sup E ν [I ()] 3ɛ. (A48) Since ɛ was arbitrary, this concludes the proof of the theorem for the case β >. The case β = is proved in F. 14

15 C. Asymptotic optimality under Condition 2 The asymptotic optimality result for this section is stated in the following theorem. Theorem C.1 Suppose that Assumptions 3.1 and 3.2 hold. Also, assume that µ cs µ s, and lim inf N 2 R N 1 R 1. Then, the following is asymptotically optimal in the sense of Definition 3.2: Staffing: Staff with N 2 agents. Control: Use TP[]. Proof: Proposition C.1 below guarantees the asymptotic feasibility of pairs (N2, TP[]). The asymptotic optimality argument is exactly the same as in the proof of Corollary B.1 using Theorem B.2. Proposition C.1 Assume µ cs µ s. Consider a sequence of systems such that: (a) the th system uses TP[] for control, and (b) the th system uses N agents so that lim inf N R N 1 R 1. Then, lim sup E[W ] W 1. Proof: Recall that W,µ FCFS s (N) is the steady-state waiting time in an M/M/N system with service rate and service rate µ s. Also, let W be the steady-state waiting time under TP[] for a cross-selling system with N agents, arrival rate, service rate µ s and cross-selling rate µ cs. The following is a straightforward result. Lemma C.1 Fix. Assume N R. If µ cs µ s and TP[] is used then, E[W W > ] E[W,µ FCFS s (N) W,µ FCFS s (N) > ] = 1 Nµ s (A49) 15

16 We continue with the proof of the proposition. Fix a sequence of staffing levels {N, } with lim inf N R N 1 R 1. Then, by Lemma C.1, E[W W > ] W 1 W (N µ s ). (A5) By the definition of N 1, lim sup 1/( W ( N 1 µ s )) 1. Hence, lim sup E[W W > ] W lim sup 1 W ( N 1 µ s ) W ( N 1 µ s ) W (N µ s ) 1, (A51) Noting that E[W ] E[W W > ], the proof is complete 16

17 D. Asymptotic optimality under Condition 3 Our optimality results under Condition 3 are closely related to the results of Gans and Zhou [4]. While [4] considers a system that is essentially different from the cross-selling system, we prove that in this asymptotic regime the two systems are, in some sense, equivalent. Specifically, we prove that the optimal solution in [4] constitutes an upper bound on the expected profit for the cross-selling model and that this upper bound is asymptotically achieved under the appropriate staffing and control. To simplify the presentation of the results in which we use this asymptotic equivalence, we give first a brief description of the model considered in [4]: Consider a call center with two types of jobs: Type-H and Type-L. Type-H jobs arrive at rate H, are processed at rate µ H and served FCFS within their class. A constraint of the form E[W] W limits the expected delay that these jobs may face. An infinite backlog of type-l jobs awaits processing at rate µ L. A pool of homogeneous servers process all jobs, and a system controller tries to maximize the rate at which type-l jobs are processed, subject to the service-level constraint for the type-h jobs. Given a fixed number of agents, the problem of finding the optimal control is formulated as a constrained average-cost Markov Decision Process (MDP) and the structure of effective routing policies is determined. When µ H = µ L, the suggested policies are globally optimal and have a very simple threshold structure. We refer to this model as the G&Z model. To create a basis for comparison of the two models (Cross-Selling vs. G&Z) one may consider cross-selling transactions against processing of type-l jobs and service transactions against processing of type-h jobs. Clearly, the dynamics of the two models are different. In the cross-selling system, rather than having an infinite backlog of cross-selling jobs, these become available only upon a completion of a service job, and if they are not processed right away they disappear. Hence, the processing rate of type-l jobs in the G&Z model constitutes an upper bound on the cross-selling rate in the cross-selling model. We prove this formally in Lemma D.1. These differences also illustrate the relative technical complexity of the cross-selling model. While in the G&Z model there is an infinite backlog of type-l jobs, the availability of cross-selling jobs is tightly related to the number of customers in the service phase in our model. The technical implication of this difference, is that any description of the system dynamics of the cross-selling system must be at least two-dimensional, regardless of whether µ s = µ cs or not. Our asymptotic analysis, however, allows us to reduce the dimensionality of the problem whenever µ s = µ cs and prove that, under TP, the upper bound, as given by the G&Z model, is asymptotically achieved. 17

18 The following is an adaptation of Definition 7 from [4]. Definition D.1 Fix. A randomized threshold reservation policy with threshold K and probability p acts as follows at each event epoch in which there are no type-h calls waiting to be served: 1. A type-h customer will enter service immediately upon arrival if there are any idle agents. 2. Upon service completion (of either a type-l or a type-h job): If there are K or fewer idle agents, the policy does nothing. If there are K + 1 or more idle agents, then with probability 1 p the policy puts enough type-l jobs into service so that exactly K agents are idle, and with probability p the policy puts enough type-l jobs into service so that exactly K 1 agents are idle. Note that, if one removes the randomization components from the above definition, the threshold reservation policy in the G&Z model can be thought of as the TP control adapted to the G&Z model. Denote by TP (N, p ) the randomized threshold policy of G&Z with threshold K determined through (A52) and with a randomization probability p. The following is a version of the optimality result of [4] for the case µ s = µ cs. We only cite the parts of the Theorem that are relevant for our results. Theorem D.1 (Theorem 1 - Gans and Zhou:) Consider a G&Z model with arrival rate, service rates µ H = µ L = µ s = µ cs, N agents and average delay bound W. Then, either 1. the problem is infeasible, or 2. A randomized threshold reservation policy with a threshold K and probability p is optimal, for some p [, 1]. Moreover, the optimal threshold K is chosen so that { K (N ) = max k [ N, ] } ξ k (N ) N µ s W. (A52) Here ξ k (N ) = P {Z,µ FCFS s (N ) = N Z,µ FCFS s (N ) N + k} and Z,µ FCFS s (N ) is the steadystate number of busy servers in an M/M/N system with arrival rate and service rate µ s. 18

19 Given two random variables X and Y, we use the notation X st Y to denote that a random variable X is stochastically greater than Y. Let CS π (t) be the cumulative cross-selling completions up to time t when the control π is used. Also, let TH π (t) be the cumulative completion of type-l jobs up to time t in the G&Z model when the control π is used. Letting Z,π be the steadystate number of busy agents in the G&Z model under the control π, we have that the steady-state throughput rate of type-l jobs equals µ cs (E[ Z,π ] R). This is a consequence of a straightforward application of Little s law as in 2 of the main paper. Lemma D.1 Fix, µ s, µ cs, N and W. Let πg&z be the optimal control in the G&Z system with µ H = µ s and µ L = µ cs. Then, for any policy π Π(N) we have that TH π g&z (t) st CS π (t), t. (A53) In particular, if π Π(N) admits a steady-state distribution for the cross-selling system, then µ cs (E[Z,π ] R) µ cs (E[ Z,π g&z ] R). (A54) Proof: Note that this lemma does not require that µ s = µ cs. The result follows a sample path coupling argument that is independent of the specific values of µ s, µ cs and. We will show that under our sample path construction the inequality (A53) holds a.s. This, in turn, implies the stochastic ordering in (A53). We construct the coupled sample paths as follows: fix a common sample path of arrivals, service times and cross-selling times for both systems. Specifically, let {t n } n=1, {s n} n=1, and {c n } n=1 be, respectively, the sequence of arrival times, service times and potential cross-selling times (that is, if cross-selling is exercised on customer n, his cross-selling time will be c n ). Then, our sample path construction uses the same sequences, {t n }, {s n } and {c n } for both systems. For simplicity of notation label the cross-selling system by 1 and the G&Z system by 2. Fix a scheduling policy π 1 for system 1 and use the same scheduling policy for system 2. This is clearly possible because whenever system 1 can schedule a customer to cross-sell system 2 can schedule a type-l job to service. It is now straightforward to show by induction on the event epochs (arrival, or service completion of any type) that both systems will have exactly the same sample paths, and 19

20 we would have that pathwise TH π1 (t) = CS π1 (t), t, (A55) and µ cs (E[ Z,π 1 ] R) = µ cs (E[Z,π 1 ] R). (A56) Since we fixed the scheduling policy for system 1 we have, in particular, that µ cs (E[ Z,π g&z ] R) sup µ cs (E[Z,π 1 ] R), π 1 Π(N) for any policy π 1 that admits a steady-state distribution for the cross-selling system. For future reference let V (N ) = rµ cs (E[ Z,π g&z ] R) (C (N ) C (R)), so that V (N ) is the optimal throughput rate in the G&Z model with N agents. Now, let {N, } be a sequence with N N 1 for all and such that N R R ˆγ > as. (A57) The existence of such a sequence is guaranteed since, by 9 of [2], we have that N 1 R R γ, (A58) for some γ >. Let Ȳ,p be the steady-state overall number of customers in a G&Z system with N agents and using the control TP (N, p ). Also, let Y be the steady-state overall number of customers in a cross-selling system with N agents and using TP[K ] with K determined through (A52). Accordingly, we let Z and Z be the number of busy agents in the above two systems. Define the scaled variables X,p = Ȳ,p N N R, and X = Y N N R. (A59) For the following result, let D := D[, ) be the space right continuous processes with left limits endowed with the J 1 Skorohod topology. We say that a sequence of processes x ( ) x( ) in D if the convergence holds in D[s, T] for each < s < T <. We let Ȳ,p (t), t be the process representing the overall number of customers in a G&Z system with N agents and using 2

21 the control TP (N, p ). Also, let (Y (t), t ) be the process representing the overall number of customers in a cross-selling system with N agents and using TP[K ] with K determined through (A52). Define the scaled processes X,p (t) = Ȳ,p (t) N N R, and X (t) = Y (t) N N R. (A6) Proposition D.1 below is the main component in the proof of asymptotic optimality of our proposed solution under Condition 2. The proposition shows that the G&Z system operated under their optimal threshold reservation policy is asymptotically equivalent to the cross-selling system operated with the TP rule. Since we established that the G&Z throughput serves as an upper bound for the cross-selling rate, the asymptotic equivalence will allow us to prove that the upper bound is achieved. Proposition D.1 (Diffusion Limits:) Suppose that {N, } is a sequence that satisfies (A57). If, in addition, X,p () X(), and X () X() as, (A61) then, and X,p ( ) X( ) in D as, X ( ) X( ) in D as. (A62) (A63) Here, X( ) is a diffusion process with infinitesimal drift function βµ s, x m(x) = (β + x)µ s, δ x (A64) and infinitesimal variance term σ 2 = 2µ s and δ := δ(ˆγ, Ŵ) is the limit from Lemma D.3. We prove proposition D.1 at the end of this section and proceed towards the proof of asymptotic optimality. Since our profits are measured in terms of steady-state performance we first have the following corollary by which the convergence on compact intervals can be extended to convergence of the corresponding steady-state variables. 21

22 Corollary D.1 Assume that µ s = µ cs and that {N, } is a sequence that satisfies (A57). Then, for any p [, 1], X,p X, as, (A65) and X X, as, (A66) where X has the steady-state distribution of the diffusion process X( ) in Proposition D.1. Furthermore, the convergence in (A66) also holds in expectation. The proof of Corollary D.1 is given in F. Note that under TP(N, ) (i.e. when setting p = ), the steady-state number of busy agents in the G&Z system, denoted by E[ Z ], satisfies E[ Z ] = E[Z,µ FCFS s (N ) Z,µ FCFS s (N ) N + K ]. (A67) Since the limit in Corollary D.1 is independent of the precise value p, the fact that Z = N (Ȳ,p N ) implies that E[Z,µ FCFS s (N ) R Z,µ FCFS s (N ) N + K ] (E[ Z ] R) N R as, (A68) provided that the sequence of staffing levels {N, } satisfies (A57). This observation is formalized in the following Corollary. Corollary D.2 Assume µ s = µ cs and consider a sequence of cross-selling systems such that (a) the th system uses N agents with {N, } satisfying equation (A57), and (b) the th system uses TP[K ] for control with K determined through equation (A52). Then, lim sup E[W ] W 1, (A69) and V (N, TP[K ]) V (N ) 1, as. (A7) The following lemma will help us to translate the result of Corollary D.2 to the more general asymptotic optimality result that we need. 22

23 Lemma D.2 Assume µ s = µ cs in addition to Assumptions 3.1 and 3.2. Let N and K be determined through (13) and (14) and assume that lim sup (N 2 R)/( N 1 R) < 1. Then, lim inf N R R >, (A71) and lim sup N R R <. (A72) Lemma D.2 then shows that the sequence of staffing recommendations N given by (13) satisfies that the sequence (N R)/ R is bounded. Consequently, it has convergent subsequences. We can then apply Corollary D.2 to any convergent subsequence to show that TP[K ] will outperform asymptotically any other routing rule over this subsequence. This reasoning can be applied to any convergent subsequence to conclude the asymptotic optimality of (N, TP[K ]), as stated in the following corollary which concludes the proof of asymptotic optimality for this section. Corollary D.3 Assume that µ s = µ cs in addition to Assumptions 3.1 and 3.2. Also, assume that lim sup N 2 R N 1 R < 1. Then, the following is asymptotically optimal for the cross-selling system in the sense of Definition 3.2: Staffing: Staff with N agents where N is given by equations (13) and (14). Control: Use TP[K (N )] where K (N ) is given by equation (14). We end this section with the proof of the diffusion-limits result. Proof of Proposition D.1: We begin with the sequence of processes {X ( )}. We will first prove the convergence of a certain Birth and Death (B&D) process. We will then show, via coupling, that this B&D process is asymptotically equivalent to X ( ). This will allow us to apply the convergence together theorem to conclude the convergence of {X ( )}. 23

24 To this end, consider the B&D process with birth rates rates: ˆ i = for all i, where i is the number of customers in the system, and ˆµ i = (N + K + i)µ s 1 i K 1 N µ s i K otherwise, (A73) where one should recall that K is negative in this setting. We denote this process by Ŷ ( ) and define its scaled version by ˆX ( ) = Ŷ ( )+K. We now have the following lemma: N R Lemma D.3 Assume that µ s = µ cs, N satisfies (A57) and K is defined through (A52). Then, there exists a function δ(, ) such that for all (γ, Ŵ), K δ(ˆγ, Ŵ) < as. (A74) R Lemma D.3 is proved in F. We apply it now in the proof of the proposition. To this end, initializing the th B&D process at state ˆX () ( K ) and using the convergence of K / R in Lemma D.3, the sequence ˆX ( ), converges weakly to X( ) with the diffusion parameters given in equation (A64); see for example [8]. To complete the proof we will show that for all T > d T ( ˆX, X ) P, as, (A75) where d T (, ) = sup <t T ˆX (t) X (t). The convergence of {X ( )} will then follow from the convergence together theorem (see e.g. Theorem of [1]). In order to evaluate d T ( ˆX, X ), we use a coupling argument and deduce that d T ( ˆX, X ) sup <t T [Z (t) (N + K )]. (A76) N R In Remark F.1 we show that the right-hand side above converges to in probability. Hence, to complete the proof, it only remains to provide the coupling argument between the cross-selling system and the B&D process that we constructed. 24

25 To this end, fix (and omit it from the notation). Initialize the cross-selling system with all agents busy and no customer in queue and we initialize the B&D process with K customers in system. We generate arrivals from the same Poisson process. We generate the departures from the same Poisson process with thinning. Let Ŷ (t) be the value of the state dependent M/M/1 process at time t. Y (t), as before, is the number of customers in the cross-selling system at time t. We prove by induction that Ŷ (t) Y (t) (N + K), for all t. Ŷ (t) (Y (t) (N + K)) sup s t[z(t) (N + K)], for all t By our initial conditions the assumption holds at the first departure from the system. Assume that it holds for the first n 1 departures and consider the n th, let the time of this departure be t n. By our inductive assumption the n th departure will be a departure in both systems if Ŷ (t n ) = Y (t n ) (N + K) while preserving the ordering. It will be a departure in the M/M/1 system, and not in the cross-selling system, only if Ŷ (t n ) > and Ŷ (t n ) > Y (t n ) (N + K) thus preserving the ordering. It will be a departure in the cross-selling system and not in the M/M/1 queue only if = Ŷ (t n ) > Y (t n ) (N + K) again preserving the ordering. Also, whenever Ŷ (t n ) > Y (t n ) (N + K) > the difference between the two processes cannot increase, since every departure will necessarily be a departure in the M/M/1 system. The difference can only increase when = Ŷ (t n ) > Y (t n ) (N+K), in which case Ŷ (t n) (Y (t n ) (N+K)) = [Y (t n ) N +K] = [Z(t n ) N +K], where the last equality follows from the fact Y (t) = Z(t) whenever Y (t) N. Thus, the second part of the inductive assumption is preserved. Note that the result would still hold as long as Ŷ () = (Y () (N K))+. The proof for the sequence X,p ( ) is much simpler. It is trivial to show through a coupling argument that, for any p [, 1], one can construct the sample path of the processes X, ( ), X,p ( ), X,1 ( ), so that X, (t) X,p (t) X,1 (t), t. Note that for p = the overall number of customers in the G&Z system has exactly the same law as the state dependent M/M/1 defined through equation (A73) above. For p = 1 the same holds with K replaced with K + 1. But, by [8] the scaled versions of these two M/M/1 systems will have the same limit X( ). The proof is completed by applying the convergence together theorem. 25

26 E. Proofs for 4 E.1 Proof of Lemma 4.1 The argument is straightforward and we only provide a sketch of the proof. For any policy π Π(N) we construct the corresponding sample paths as follows: We generate arrivals from a Poisson stream. In addition, we generate an infinite sequence of service times {s i } i 1 and cross-selling times {c i } i 1. When constructing the sample paths, the service s i will be assigned to the i th customer to begin service and the cross-selling time c i will be assigned to the i th customer to begin cross-selling. Under this construction the process (Z 2 (t), Y (t)) is invariant to the order in which customers are admitted from the queue. In particular, π which is obtained from π by admitting customers to service in a FCFS manner induces the same sample path under this construction. This invariance guarantees (through Little s Law) that if π is feasible so will be π. Moreover, both controls will admit the same cross-selling rate since the steady-state number of customers in cross-selling, Z 2, has the same distribution probability law under both π and π. E.2 Proof of Lemma 4.2 We use a coupling argument to prove this assertion. Consider two cross-selling systems with the same number of agents, N, in both systems. Let system 1 be the system that uses the policy π and system 2 be the system that uses π. The latter is the work conserving system. We assume that both systems are initialized empty and we let {t i } i 1 and {s i } i 1 and {c i } i 1 be, respectively, the sequences of arrival times, service times and cross-selling times in system 1. Specifically, customer i arrives at time t i and requires a service time of s i. If cross-selling is exercised at customer i the cross-selling will require c i units of time. If cross-selling is not exercised on customer i we set c i =. We construct the sample path of system 2 from system 1 as follows: we use the same stream of arrivals, services and cross-selling times. We cross-sell to customer i in system 2 if and only if we cross-sell to him in system 1. To differ from system 1, upon service completion of customer i, if cross-selling is not exercised, a customer from the queue will be admitted to service (unless the queue is empty). Let b j i, j = 1, 2, be the time at which customer i begins service in system j (In particular, the waiting time of customer i in system j is given by t i b j i ). Let Q j (t), j = 1, 2, be the queue length at time t in system j. Also, let CS j (t) be the number 26

27 of customers that left system j up to time t after cross-selling was exercised on them. In order to prove that the assertion of the lemma holds it suffices to show the following: 1. Q 1 (t) Q 2 (t), for all t. 2. CS 1 (t) CS 2 (t). Indeed, if Q 1 (t) Q 2 (t), the assumed feasibility of system 1 will imply the feasibility of system 2. Moreover, if CS 1 (t) CS 2 (t), then system 2 performs at least as well as system 1 in terms of cross-selling rate. Since we exercise cross-selling on customer i in system 2 only if we exercise cross-selling on this customer in system 1, it suffices to prove that for all i 1, b 2 i b 1 i. That is, in system 2 all the customers begin service earlier. We will now proceed by induction on the customer number to prove that indeed i 1, b 2 i b 1 i. The conditions clearly holds for the first customer since both systems are initialized empty. Assume the condition holds up to customer n 1 and consider customer n. Specifically consider the following cases: If at time t n there are idle agents in system 2 the customer will be admitted to service immediately upon arrival (in system 2) and the inductive assumption will be kept. Otherwise, let us consider the time b 2 n 1 at which customer n 1 will begin service in system 2 (while he might still be waiting for service in system 1). Let r j i time of customer i n 1 in system j at time b 2 n 1. That is, be the remaining handling r j i = [ s i + c i [b 2 n 1 bj i ]+] +. (A77) In particular, by our inductive assumption ri 2 r1 i, i n 1, and by work conservation and the fact that customer n had to wait in queue, we have that t n < b 2 n 1 +min i n 1{r 2 i r2 i > }) and b 2 n = b2 n 1 + min i n 1{r 2 i r2 i > }. If we can show that for system 1 b 1 n b2 n 1 + min i n 1 {r 1 i r1 i > }, then we are done. To see that this indeed the case note that since b 2 i b 1 i for all i n 1 and since the handling times are common for both system, we have that at time b 2 n 1 the overall number of customers in system 1 is at least as large as the overall number of customers in system 2. Now, recall that we assumed that all agents are busy in system 2 at time t n, this implies that on the interval [b 2 n 1, t n) all agents are busy (otherwise customer n would not have to wait by 27

28 work conservation). Hence, at time b 2 n 1 the number of customers in system 2 (and then in both systems) will be at least N. For system 1 this implies that the number of idle agents at time b 2 n 1 is smaller than the queue length. Formally, if Z1 (t) is the number of busy agents in system 1 at time t, then we just argued that Z 1 (b 2 n 1 ) + Q1 (b 2 n 1 ) N, and in particular I 1 (b 2 n 1 ) Q1 (b 2 n 1 ), where I1 (t) is the number of idle agents in system 1 at time t. Hence, even if at time b 2 n 1 system 1 admits all waiting customers to service, by the assumption that t n < b 2 n 1 + min i n 1{r 2 i r2 i > } b 2 n 1 + min i n 1{r 1 i r1 i > }, customer n must find either a non-empty queue or an empty queue but with all agents busy. If he finds an empty queue with all agents busy he will enter at time b 2 n 1 + min i n 1{r 1 i r1 i > }, otherwise he will have to wait more. In any case we have that b 1 n b2 n 1 + min i n 1{r 1 i r1 i > }. E.3 Proof of Proposition 4.1 Since we fix we omit the superscript from the all the notation. Let π be any feasible policy for the original cross-selling problem which exists by our assumption that N N 1. Define π L to be an adaptation of π to a system where there is a limited number of trunk lines, L. The adaptation of π to the system with finite buffer is straightforward. Note that π (i, j) defines what action to take in an event epoch when the system is in state i, j. Then, we take π L (i, j) = π (i, j), i, j : j L, z 2 N. Also, from any feasible policy, π L, in the finite buffer the system we can construct a corresponding policy, πl for the infinite buffer system by setting π L (i, j) =, i, j : j > L. To establish the result of the proposition, it suffices to show that: (a) starting from a work conserving policy π, the sequence that we construct π L (which is also work conserving and hence within the set of possible solutions for the LP), achieves asymptotically the same value, as L, and (b) that starting from a sequence of policies {π L }, the sequence of adapted policies for the infinite buffer system, {πl }, achieves asymptotically the same value for the infinite buffer system. Formally, we want to show that, given ɛ >, Ṽ (N, π ) ˆV (N, L, π L ) ɛ, (A78) and Ṽ (N, π L ) ˆV (N, L, π L ) ɛ., (A79) 28

29 for all L large enough. Here, Ṽ (N, π ) and ˆV (N, L, π L ) are, respectively, the cross-selling rates in the infinite and finite buffer systems, equipped with π and π L, N agents and L trunk lines (in the finite buffer system). Then, by definition V LP(N, L) = sup π L ˆV (N, L, π L ) and V (N) = sup π Ṽ (N, π), where the supremum is taken over feasible policies for each system. Recalling that the cross-selling rate under any policy π equals µ cs E[Z2 π ], in order to prove (A78) it suffices to [ ] show that E[Z2 π ] = lim L E Z L,πL 2 B, where Z L,πL 2 B is the steady-state number of agents busy cross-selling in the finite buffer system with L trunk lines and using a control π L. We first fix a feasible policy π for the infinite buffer system and prove (A78). Consider the truncation of the resulting Markov chain to the subspace of the domain in which {j L}. Then, the restricted Markov chain has the same law as the finite buffer system with π L. Hence, [ ] E[Z2 π ] = E Z L,πL 2 B P {Y π L} + E[Z2 π 1 {Y π >L}]. (A8) The feasibility of π implies that E[Q π ] W. Using Markov s inequality we have P {Y π > L} = P {Q π > L N} W L N (A81) Using the Cauchy-Schwartz inequality we have that E[Z π 2 1 {Y π >L}] E[(Z π 2 ) 2 ]P {Y π > L}. (A82) Since, by definition, Z π 2 N, we then have E[Z π 2 1 {Y π >L}], as L. (A83) Plugging (A81) and (A83) back into equation (A8) we have that ] E[Z2 π ] = lim E [Z L,πL 2 L B. (A84) This completes the proof (A78). The proof of (A79) is very similar and is omitted. 29

30 F. Proofs of auxiliary results For the following recall that Ξ (t) := (Q (t), Z 2 (t), Z 1 (t)). Lemma F.1 Fix. Assume the system is staffed with N > R agents and that TP[K] is used for control with some K N. Then, the Markov process (Ξ (t), t ) admits a steady-state distribution. Consequently, TP is admissible in the sense that E[Q (t)] lim t t =. (A85) Proof: It is immediate to see that the chain is irreducible. Because the rates are bounded we can use uniformization and define a related Discrete Time Markov Chain (DTMC). Define the set C = {(i, max{n N + K}) : i N}. Let τ C be the first hitting time in the set C. Accordingly, E x [τ C ] is the expected hitting time given that S() = x. C is a compact set and it is easy to prove that sup x C E x [τ C ] < M C < (an elaborate derivation of the bound, M C, would be similar to the proof of Lemma 8 in [4] and we omit the detailed argument). Stability is now established by applying theorem from [9]. Equation (A85) follows directly from stability. Lemma F.2 Let Ω (δ,, T, Θ) := { } ω Ω : 11 max sup B i (ct) + E i log(2 ct) δt Θ. i=1,...,11 t T Then, ( P {(Ω (δ,, T, Θ)) c } 11 2c 1 e c2(θ Γ) + e δθ 2c ), where Γ, c 1, c 2 are as in (A5). 3

31 Proof: Fix i = 1, Note that { P {(Ω (δ,, T, Θ)) c } P E i log(2 ct) > Θ } 2 { + P sup B i (ct) δt > Θ } t T 2 (A86) Now we treat each element on the right-hand side separately. From (A5) it now follows that P { E i log(2 ct) > Θ } c 1 e c 2( Θ 2 Γ)/(log(2 ct). 2 The second element on the right hand side of (A86) is bounded by e δ Θ 2c by a well known result for negative-drift Brownian motions; see e.g. Exercise in [6]. The following auxiliary lemma is used later in the proofs of Lemmas B.1 and B.2. Lemma F.3 Consider a sequence of systems such that: (a) the th system uses N = R + βr + γ R + o( ) agents for some β pµs µ cs, max(β, γ) > and, (b) the th system uses TP[K ] for a sequence {K } that satisfies K / R ϱ (, ) as. Then, for all ɛ >, there exist t (ɛ), (ɛ) (independent of the initial state Ξ ()), such that, ( sup Z1 (t) ) ɛ, t (ɛ) t T µ s (A87) for all ω Ω (,, T, ɛ/8) and (ɛ). Here, t (ɛ) = whenever Ξ () A ɛ/2. Consequently, P { sup t (ɛ) t T ( Z1 (t) ) } > ɛ c 3 e c4/log(2 ct), µ s (A88) for two positive constants c 3 and c 4 and, finally, E [ sup t (ɛ) t T (Z 1 (t) µs ) ] 2ɛ. (A89) 31

32 Proof: We fix ω Ω (,, T, ζ). Assume that Z 1 () < µ s ɛ. The other case is treated at the end of this proof. Define τ = inf {t : Z 1 (t) /µ s ɛ 2 }. Fixing an interval [s, t) with t τ and such that Q (u) = for all u [s, t) we have, by equation (A6), that Z 1 (t) Z 1 (s) (t s) µ s ( ɛ ) µ s 2 ɛ (t s) ζ = µ s (t s) ζ. 2 On the other hand for intervals [s, t) with t τ and Q (u) > K, for all u [s, t), we have by equation (A8) that Z 1 (t) Z 1 (s) µ cs t s (N Z1 (u))du ζ µ ɛ cs (t s) ζ, 2 and finally, on intervals [s, t) with t τ and such that < Q (u) K for all u [s, t), we have by equation (A6) that Z 1 (t) Z 1 (s) (K )+(t s) µ s ( µ s ɛ 2 ) ζ (K ɛ )+µ s (t s) ζ. 2 By assumption, K / R ϱ (, ) as. In particular, there exists k > such that K k for all large enough. For such values of we then have that Z 1 (t τ ) Z 1 () + µ s µ cs ɛ 2 (t τ ) ζ. (A9) Choosing ζ = ɛ/8 and since Z 1 (), we have that Z 1 (t τ ) µ s µ cs ɛ 2 (t τ ) ɛ 8, so that, by the definition of τ, τ 1 µ s ɛ 4 (µ s µ cs )ɛ/2, (A91) 32

33 and this holds for all ω Ω (,, T, ɛ/8). Define now τ = sup {t τ : Z 1 (t) µs ɛ2 } T, and τ = inf { t τ : Z1 (t) < } ɛ T. µ s We note that, on Ω (,, T, ɛ/8), Z 1 (t) Z 1 (t) ct+ɛ/8 for some c >. This follows from (A8) and the definition of Ω (,, T, ɛ/8). Hence, τ > τ on Ω (,, T, ɛ/8). Repeating closely the arguments that lead to (A9) we have that Z1 (t) Z1 ɛ (s) + µ s µ cs 2 (t s) ɛ, (A92) 8 for all τ s < t τ. There are now two cases to consider: if τ = T, then Z1 (t) µ s ɛ/2, for all t τ. If, on the other hand, τ < T, then by (A92), we must have that τ (t) = T so that Z1 (t) µ s ɛ for all t τ. Consequently, we conclude that, on Ω (,, T, ɛ/8), Z1 (t) µ s ɛ for all t τ. In particular, choosing t (ɛ) = 1 µ s ɛ 4 (µ s µ cs )ɛ/2, we have by (A91) that, ( sup Z1 (t) ) ɛ, t (ɛ) t T µ s (A93) on Ω (,, T, ɛ/8). Using Lemma F.2 together with Z 1 N /µ s + p/µ cs we also have that E [ sup t (ɛ) t T (Z 1 (t) µs ) ] ɛ + cc 3 e c 4ɛ/log(2 ct), (A94) so that there exists large enough to guarantee that the above is smaller than 2ɛ. Finally, the last statement of the lemma follows from the above by noting that, if Ξ () A ɛ then τ = τ = so that Z 1 (t) /µ s ɛ for all t. Proof of Lemma B.1: Lemma B.1 is obtained as a special case of Lemma F.3, and specifically the first part of this lemma, by assuming that β, δ >, replacing ɛ/2 and assuming Ξ () A ɛ so 33