Internet Supplement to Call centers with impatient customers: many-server asymptotics of the M/M/n+G queue

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1 Internet Supplement to Call centers with impatient customers: many-server asymptotics of the M/M/n+G queue Sergey Zeltyn and Avishai Mandelbaum Faculty of Industrial Engineering & Management Technion Haifa 32, ISRAEL s: May 23, 25 Acknowledgements. The research of both authors was supported by ISF (Israeli Science Foundation) grants 388/99, 26/2 and 46/4, by the Niderzaksen Fund and by the Technion funds for the promotion of research and sponsored research.

2 Contents Summary of the Internet Supplement 2 Summary of Baccelli-Hebuterne s results on the M/M/n+G queue 3 The M/M/n+G queue: summary of performance measures 2 3. Proofs of (3.8)-(3.25) Asymptotic behavior of integrals 7 4. Asymptotic results Proofs of Lemmata Some properties of the normal hazard-rate 6 QED operational regime 3 6. Formulation of results Main case: patience distribution with a positive density at the origin Patience distribution with density vanishing near the origin Delayed distribution of patience Patience with balking Patience with scaled balking Numerical experiments Proofs of the QED results Quality-Driven operational regime Formulation of results Proof of Theorem Numerical experiments Efficiency-Driven operational regime 5 8. Formulation of results Numerical Experiments Proofs of the ED results Economies of scale in the M/M/n+G queue QED regime QD regime ED regime Economies of Scale: conclusions

3 Some statistical applications to call centers 62. General description of the data set Model primitives Performance measures Fitting QED approximations Summary of our data analysis

4 Abstract of the main paper The subject of the present research is the M/M/n+G queue. This queue is characterized by Poisson arrivals at rate λ, exponential service times at rate µ, n service agents and generally distributed patience times of customers. The model is applied in the call center environment, as it captures the tradeoff between operational efficiency (staffing cost) and service quality (accessibility of agents). In our research, three asymptotic operational regimes for medium to large call centers are studied. These regimes correspond to the following three staffing rules, as λ and n increase indefinitely and µ held fixed: Efficiency-Driven (ED): n (λ/µ) ( γ), γ >, Quality-Driven (QD): n (λ/µ) ( + γ), γ >, and Quality and Efficiency Driven (QED): n λ/µ + β λ/µ, < β <. In the ED regime, the probability to abandon and average wait converge to constants. In the QD regime, we observe a very high service level at the cost of possible overstaffing. Finally, the QED regime carefully balances quality and efficiency: agents are highly utilized, but the probability to abandon and the average wait are small (converge to zero at rate / n). Numerical experiments demonstrate that, for a wide set of system parameters, the QED formulae provide excellent approximation for exact M/M/n+G performance measures. The much simpler ED approximations are still very useful for overloaded queueing systems. Finally, empirical findings have demonstrated a robust linear relation between the fraction abandoning and average wait. We validate this relation, asymptotically, in the QED and QD regimes.

5 Summary of the Internet Supplement The goal of this supplement is to elaborate on material presented in the main paper. To facilitate the reading, statements of results from the main paper are repeated here. (Note that some results are in fact expanded in the Supplement.) Table displays the correspondence between results; for example, Theorem 4. of the main paper is Theorem 6. in the Supplement. Table : Relation between the statements from the main paper and the Internet Supplement Main Paper Internet Supplement Theorem 4. Theorem 6. Theorem 4.2 Theorem 6.6 Theorem 4.3 Theorem 6.7 Theorem 5. Theorem 7. Theorem 6. Theorem 8. Lemma. Lemma 4. Lemma. Lemma 6. In Section 2 we briefly describe the results of Baccelli and Hebuterne [2] that are used in the following proofs. Sections 3 and 4 contain proofs of the results from Sections 9 and of the main paper, respectively. Section 5 discusses relevant properties of the hazard rate of the standard normal random variable. Sections 6-8 contain proofs and additional numerical experiments for the three operational regimes: the QED, Quality-Driven (QD) and Efficiency- Driven (ED), respectively. We also study two additional special cases in the framework of the QED regime. (See Subsections 6..2 and 6..3.) In both cases, the density of the patience distribution vanishes at the origin. Then Section 9 explores the Economies-of-Scale (EOS) problem for the three regimes. Specifically, assuming that the arrival rate increases by a factor m >, we apply the corresponding operational regime and check how the most important performance measures change in these circumstances. Finally, in Section our models are applied to call center data of a large bank in the USA. 2 Summary of Baccelli-Hebuterne s results on the M/M/n+G queue The analysis in [2] is based on a Markov process (N(t), η(t)), t, where N(t) is the number of busy agents and η(t) is the virtual offered waiting time (the offered wait of a virtual customer

6 that arrives at time t). Then the steady-state characteristics are defined by: v(x) PN(t) = n, x < η(t) x + ɛ = lim lim, x t ɛ ɛ π j = lim t PN(t) = j, η(t) =, j n (2.) Here v(x) is the density of the virtual offered waiting time. The unique solution of the steadystate equations is given by λ j π j = µ j! π, j n (2.2) x v(x) = λ π n exp λ Ḡ(u)du nµx, (2.3) where [ π = + λ ] λ n µ + + ( + λj), (2.4) µ (n )! x J = exp λ Ḡ(u)du nµx dx. (2.5) Moreover, probability to abandon can be calculated by PAb = ( nµ λ ) n π j j= + π n. (2.6) 3 The M/M/n+G queue: summary of performance measures Here we summarize exact formulae for M/M/n+G performance measures. Recall the definitions from the main paper. M/M/n+G primitives. The M/M/n+G model requires four input parameters: λ arrival rate, µ service rate, n number of agents, G patience distribution (Ḡ survival function). Building blocks. Define H(x) = x Introduce the integrals Ḡ(u)du. Note that H( ) = τ, where τ is the mean patience-time. J = exp λh(x) nµx dx, (3.) 2

7 In addition, let Finally, define E = J = x exp λh(x) nµx dx, (3.2) J H = H(x) exp λh(x) nµx dx. (3.3) J(t) J (t) J H (t) = = = n t t t ( λ j! µ j= ) n = (n )! ( λ µ exp λh(x) nµx dx, (3.4) x exp λh(x) nµx dx, (3.5) H(x) exp λh(x) nµx dx. (3.6) ) j e t ( + tµ λ ) n dt. (3.7) List of performance measures: Recall notation from the main paper: PAb probability to abandon, PSr probability to be served, Q queue length, W waiting time, V offered wait (time that a customer with infinite patience would wait). Then PV > = λj E + λj, (3.8) PW > = λj Ḡ(), E + λj (3.9) PAb = + (λ nµ)j, E + λj (3.) PAb V > = + (λ nµ)j, λj (3.) PSr = E + nµj, E + λj (3.2) E[V ] = λj E + λj, (3.3) E[V V > ] = J J, (3.4) λj H E[W ] = E + λj, (3.5) E[Q] = λ2 J H E + λj, (3.6) E[V Ab] = (λ nµ)j + J (λ nµ)j +, (3.7) 3

8 E[W Ab] = J + λj H nµj (λ nµ)j +, (3.8) E[V Sr] = E[W Sr] = nµj J E + nµj, (3.9) PV > t = λj(t) E + λj, (3.2) PW > t = λḡ(t)j(t) E + λj, (3.2) E[V V > t] = J (t) J(t), (3.22) E[W W > t] = J H(t) (H(t) tḡ(t)) J(t), Ḡ(t)J(t) (3.23) PAb V > t = λ nµ expλh(t) nµt +, λ λj(t) (3.24) PAb W > t = 3. Proofs of (3.8)-(3.25) λ nµ G(t) λḡ(t) Here we present the proofs of (3.8)-(3.25), one by one. (3.8). First, (2.2)-(2.5) and definition (3.7) imply the useful formula + expλh(t) nµt λḡ(t)j(t). (3.25) π n = n j= j! ( λ µ ) n ( λ (n )! µ ) j + λj ( λ (n )! µ ) n = E + λj. (3.26) Then use that n PV > = π j = j= where the last equality of (3.27) follows from (2.3) and (2.5). v(x)dx = λπ n J, (3.27) (3.9). Follows from PW > V > = Ḡ(). (3.). Formula (2.6) implies that Now substitute (3.8). PAb = ( nµ λ ) PV > + E + λj. (3.). Immediate consequence of (3.8) and (3.). (3.2). PSr = PAb. 4

9 (3.3). Results from (3.26) and E[V ] = λπ n (3.4). Formulae (3.8) and (3.3). x expλh(x) nµxdx. (3.5). According to formula (2.3), the survival function of the virtual wait is given by PV > t = Hence, the average wait is equal to E[W ] = and integrating by parts V (t) = λπ n exp λh(x) nµx dx. Ḡ(t) V (t)dt = λπ n Then use formula (3.26) and definition (3.3). t Ḡ(t) exp λh(x) nµx dxdt t E[W ] = λπ n H(t) exp λh(t) nµt dt. (3.6). Follows from (3.5) and Little s formula. (3.7). where from (2.3) and (3.26) E[V Ab] = E[V τ V ] PAb v(x) = = λ expλh(x) nµx E + λj xv(x)g(x)dx PAb Integration by parts implies that x[λḡ(x) nµ] expλh(x) nµxdx and = xd [expλh(x) nµx] =, (3.28). (3.29) expλh(x) nµxdx = J, xg(x) expλh(x) nµxdx = (λ nµ)j + J λ which, combined with (3.28), (3.29) and (3.) implies (3.8). Similar to the previous calculation E[V Ab] = (λ nµ)j + J (λ nµ)j +., (3.3) E[W Ab] = E[τ τ V ] PAb = x V (x)dg(x) PAb. 5

10 Note that d[xg(x) + H(x) x] = xdg(x). (3.3) Then x V (x)dg(x) (use (3.29) and (3.)) PAb = v(x) [xg(x) + H(x) x]dx PAb = λ [xg(x) + H(x) x] expλh(x) nµxdx + (λ nµ)j = J + λj H nµj + (λ nµ)j, where the last equality follows from (3.3) and the definitions of J and J H. (3.9). This formula for E[V Sr] can be checked via Since the event Sr is equivalent to W=V, E[V ] = E[V Sr] PSr + E[V Ab] PAb. E[V Sr] = E[W Sr]. (3.2). Follows from (3.29). (3.2). Consequence of PW > t = PV > t Pτ > t. (3.22). Follows from definitions of J(t) and J (t). (3.23). E[W W > t] = t xw(x)dx PW > t, (3.32) where w(x) is the waiting-time density. The denominator of (3.32) is equal to PW > t = Calculating the numerator of (3.32): xw(x)dx = t Ḡ(t) V (t) = λπ n Ḡ(t) t t expλh(x) nµxdx. (3.33) xv(x)ḡ(x)dx + x V (x)dg(x) [ = λπ n xḡ(x) expλh(x) nµxdx + + x t t ( x ) ] expλh(u) nµudu dg(x) Use (3.3) to show that the double integral in (3.34) is equal to [xg(x) + H(x) x] expλh(x) nµxdx t 6 t (3.34)

11 [tg(t) + H(t) t] expλh(x) nµxdx. (3.35) t After some terms cancel, we get from (3.32), (3.33), (3.34) and (3.35) that E[W W > t] = t [H(x) + tḡ(t) H(t)] expλh(x) nµxdx Ḡ(t) t expλh(x) nµxdx (3.24). PAb V > t = = = J H(t) (H(t) tḡ(t)) J(t). Ḡ(t)J(t) t Pτ V ; V > t G(x)v(x)dx = PV > t t v(x)dx t G(x) expλh(x) nµxdx. (3.36) J(t) Using integration by parts [λḡ(x) nµ] expλh(x) nµxdx = expλh(t) nµt. Hence, t t G(x) expλh(x) nµxdx = (λ nµ)j(t) + expλh(t) nµt λ. (3.37) Now (3.36) and (3.37) imply (3.25). PAb V > t = λ nµ λ PAb W > t = = G(x) V (x) t + t G(x)v(x)dx Ḡ(t) V (t) = + Pτ V ; τ > t Pmin(V, τ) > t = = λ nµ G(t) λḡ(t) expλh(t) nµt λj(t) t. V (x)dg(x) Ḡ(t) V (t) t G(x) expλh(x) nµxdx G(t)J(t) Ḡ(t)J(t) + expλh(t) nµt λḡ(t)j(t). 4 Asymptotic behavior of integrals Here we prove Lemma 4. (Lemma. from the main paper) and two additional lemmata that will be needed in the following proofs. 7

12 4. Asymptotic results Lemma 4. Let b, k, l, l 2 be positive numbers and let b 2, k 2, m be non-negative. In addition, assume that l and l 2 are integers. Consider a function r = r (λ), λ > such that r (λ) λ k, λ. Finally, assume that k l > k 2 l 2. (4.) Then and = Γ m+ l = Γ m+ l l b l [b r (λ)] m+ l m+ l b 2Γ x m exp b r (λ)x l b 2 λ k 2 x l 2 dx λ k (m+) l ) + o (λ k (m+) l, λ. (4.2) x m exp b r (λ)x l b 2 λ k 2 x l 2 dx m+l2 + l m+l 2 + l l b λ k 2 k (m+l 2 +) l ) + o (λ k 2 k (m+l 2 +) l. (4.3) Lemma 4.2 In addition to assumptions of Lemma 4., let k > k 2 and assume that the function r 2 = r 2 (λ), λ > satisfies r 2 (λ) = o(λ k 2 ), λ. Then x m exp b r (λ)x l b 2 r 2 (λ)x l 2 dx = Γ m+ l l b m+ l λ k (m+) l ) + o (λ k (m+) l. (4.4) Remark 4. Note that r 2 (λ) does not need to be positive, which is in contrast to the corresponding term λ k 2 in Lemma 4.. Remark 4.2 We can generalize (4.2) to x m exp n b r (λ)x l b i λ k i x l i dx = Γ m+ l m+ λ k (m+) l l i=2 l b ) + o (λ k (m+) l, λ, as long as k l > k i l i prevails for 2 i n. 8

13 Lemma 4.3 Let b, k, l, δ >, integer m, and < n <. Assume that the function r(λ) λ k, and assume λ. Define a function S(λ) = δλ n x m exp br(λ)x l dx, λ >, nl + k >. (4.5) Then there exists ν > such that S(λ) = o(e λν ). (4.6) 4.2 Proofs of Lemmata Proof of Lemma 4.. Define and I A I = = x m exp b r (λ)x l b 2 λ k 2 x l 2 dx x m exp b r (λ)x l [ b 2 λ k 2 x l 2 ]dx. Formula (.2) from the main paper and straightforward calculations imply Γ m+ l I A = b m+l2 + ) 2Γ l l [b r (λ)] m+ m+l 2 + λ k 2 k (m+l 2 +) l + o (λ k 2 k (m+l 2 +) l. l l l b Now ) I I A = o (λ k 2 k (m+l 2 +) l (4.7) will imply Lemma 4.. If x > and λ k 2 x l 2, then there exists C > such that Define δ = λ k 2/l 2 = δ Cλ 2k 2 l [b r (λ)] m+2l 2 + l exp b 2 λ k 2 x l 2 ( b 2 λ k 2 x l 2 ) Cλ 2k 2 x 2l 2. and note that the condition λ k 2 x l 2 is equivalent to x δ. Now x m exp b r (λ)x l exp b 2 λ k 2 x l 2 ( b 2 λ k 2 x l 2 ) dx C m + 2l2 + Γ l λ 2k 2 x m+2l2 exp b r (λ)x l dx ) = O (λ 2k 2 k (m+2l 2 +) l ) = o (λ k 2 k (m+l 2 +) l, where the last equality follows from (4.). In order to complete the proof, we show that the remainder δ of the integrals can be ignored. Specifically, there exists ν > such that δ x m exp b r (λ)x l b 2 λ k 2 x l 2 dx = o (e ) λν 9

14 and δ x m exp b r (λ)x l [ b 2 λ k 2 x l 2 ]dx = o e λν. The last two statements follow from Lemma 4.3. (Condition (4.5) applies due to (4.).) Proof of Lemma 4.2. The proof is similar to the proof of Lemma 4.. The integration domain is again divided by δ = λ k 2/l 2. For large λ the inequality x δ implies x l 2 r 2 (λ), which, in turn, implies exp b 2 r 2 (λ)x l 2 ( b 2 r 2 (λ)x l 2 ) C[r 2 (λ)] 2 x 2l 2 for some C >. Then one shows that δ δ x m exp b r (λ)x l b 2 r 2 (λ)x l 2 dx x m exp b r (λ)x l [ b 2 r 2 (λ)x l 2 ]dx and x m exp b r (λ)x l ) = o (λ k (m+) l, [ b 2 r 2 (λ)x l 2 ]dx = Γ m+ l The last step is to prove exponential bounds : and δ δ m+ l l b λ k (m+) l ) + o (λ k (m+) l. x m exp b r (λ)x l b 2 r 2 (λ)x l 2 dx = o e λν, ν >, (4.8) x m exp b r (λ)x l [ b 2 r 2 (λ)x l 2 ]dx = o e λν, ν >. In order to get (4.8), the condition k > k 2 is needed. It enables us to find < C <, such that for x > δ and λ large enough, exp b r (λ)x l b 2 r 2 (λ)x l 2 < exp b C r (λ)x l, and exp b r (λ)x l [ b 2 r 2 (λ)x l 2 ] < exp b C r (λ)x l, Now we can apply Lemma 4.3. (Its proof appears below.) Proof of Lemma 4.3. We perform a change of variables z = br(λ)x l, x = z /l, dx = dz z /l, br(λ) br(λ) br(λ)

15 getting S(λ) = C 2 r(λ) m+ l e z z m+ l dz, C r(λ)λ nl (4.9) where C and C 2 are positive constants. Under condition (4.5), the lower bound C r(λ)λ nl of the integral in (4.9) converges to infinity. Therefore, there exists α > such that for λ large enough, S(λ) C 2 r(λ) m+ l e αz dz = C r(λ)λ nl C 2 r(λ) m+ l exp C 3 r(λ)λ nl, where C 3 is a positive constant. Since r(λ)λ nl λ nl+k, we can easily find ν > such that (4.6) is satisfied. 5 Some properties of the normal hazard-rate In the sequel, we use some properties of the hazard rate function of the standard normal distribution: h(x) = φ(x) Φ(x) = φ(x) Φ(x), (5.) where Φ(x) is its cumulative distribution function, Φ(x) = Φ(x) is the survival function and φ(x) = Φ (x) is the density. Figure : Normal hazard rate hazard rate The derivative of the normal hazard rate is equal to h (x) = h(x) (h(x) x) (5.2) and, consequently, h(x) x = [ln h(x)].

16 The second derivative is Theorem.3 from Durrett [4] states that h (x) = h(x) (2h 2 (x) 3xh(x) + x 2 ). (5.3) ( x ) x 3 φ(x) Φ(x) φ(x), for x >. x Then it follows that: and h(x) x, x, h(x) x3 x 2, x >, h(x) x, as x. (5.4) It is well-known that h is an increasing function (see Gupta and Gupta [7] for a general treatment of multivariate normal case). Surprisingly, we have not found anywhere a proof that h is convex and we shall need this fact. So we constructed an indirect proof, based on the convexity of the Erlang-B formula [2], in the following way. Define the function B(s, a) = For a > and integer s > it can be shown that [ a e at ( + t) dt] s. B(s, a) = [ s i= a i ] as i! s!. The last expression is equal to the Erlang-B blocking probability in the M/M/s/s system with a = λ. It has been proved in [2] that B(s, a) is convex in s in [, ), for all a >. Now define µ B(β, a) = a B(a + β a, a). Obviously B(β, a) is also convex in β over ( a, ). The QED result for the Erlang-B system, derived by Jagerman [], implies that B(β, a) h( β) (a ). The pointwise limit of a sequence of convex functions is convex as well, implying that h is convex. Finally, formula (5.4) and the convexity of h imply h (x) <, < x <. (5.5) 2

17 6 QED operational regime 6. Formulation of results 6.. Main case: patience distribution with a positive density at the origin The first lemma continues Lemma. from the main paper. Lemma 6. (Building blocks) Under the assumptions of Theorem 4. from the main paper, the building blocks J, E and J, defined in Section 3, are approximated by: a. b. c. d. Define Then J 2 J = n µg J = [ n ˆβ µg h( ˆβ) = E = n h( ˆβ) + o n. (6.) ] ( + o. (6.2) n) h( β) + o( n). (6.3) x x 2 exp λ G(u)du nµx dx. (6.4) J 2 = [ ] n 3/2 ˆβ2 + (µg ) 3/2 ˆβ + o h( ˆβ) n 3/2. (6.5) Theorem 6. (Performance measures) Under the assumptions of Theorem 4. from the main paper, the performance measures of the M/M/n+G queueing system in the QED regime are approximated by: a. The delay probability converges to a constant that depends on β and the ratio g /µ: [ g PW > + µ h( ˆβ) ], (6.6) h( β) In addition, if λ and PW > α, with < α <, then where α = [ + g µ h( ˆβ) ]. h( β) n = λ µ + β λ µ + o( λ), (6.7) b. The probability to abandon of delayed customers decreases at rate n : PAb V > = g [ n µ h( ˆβ) ˆβ ] + o n. (6.8) 3

18 The probability to abandon PAb also decreases at rate product of (6.6) with (6.8). c. The average offered wait of delayed customers decreases at rate n and can be approximated by the n : E[V V > ] = [ n g µ h( ˆβ) ˆβ ] + o n. (6.9) The average offered wait E[V ] also decreases at rate n and can be approximated by the product of (6.6) and (6.9). d. The average waiting time is of the same order as the average offered wait: E[W ] E[V ]; E[W W > ] E[V V > ]. (6.) e. The ratio between the probability to abandon and average wait converges to the (positive) value of patience density at the origin: PAb E[W ] = PAb W > E[W W > ] g. (6.) f. The average offered wait and the average actual wait of abandoning customers decrease at rate : n or, in other words, E[V Ab] = n E[W Ab] = n [ g µ [ 2 g µ ] h( ˆβ) ˆβ ˆβ ] h( ˆβ) ˆβ ˆβ + o n. (6.2) + o n, (6.3) E[W Ab] E[V Ab]. (6.4) 2 Moreover, the following inequality prevails: [ ] 2 h( ˆβ) ˆβ < h( ˆβ ˆβ) ˆβ < h( ˆβ) ˆβ ˆβ, < ˆβ <. (6.5) (See also Remark 4.7 from the main paper.) g. The distribution of wait, given delay in queue, is asymptotically equal to W P E[S] > t W > n ( ) g Φ ˆβ + µ t Φ( ˆβ), t. (6.6) 4

19 h. The probability to abandon, given delay in queue, is asymptotically equal to P Ab W E[S] > t = [ ] g n n µ g h ˆβ + t µ ˆβ + o n. (6.7) i. The average wait, given delay in queue, is asymptotically equal to [ E W t ] = [ ( n n g µ h W E[S] > Parts h and i together imply a generalization of part e: ) ] g ˆβ + t µ ˆβ + o n. (6.8) P Ab W > t/ n E [W W > t/ n] g, t. (6.9) 6..2 Patience distribution with density vanishing near the origin We would like to cover models where customers are going through several stages of (im)patience before reneging. (See, for example, Ishay [] or Baccelli and Hebuterne [2]; the latter fit an Erlang distribution with 3 phases to patience, using real data.) In such models, we cannot expect significant abandonment near the origin, which suggests patience distributions with density vanishing near the origin. Lemma 6.2 (Building blocks) Assume that the density of patience time at the origin g = ; that the first (k ) derivatives vanish as well: g (i) () =, i k, and that the k-th derivative is positive: g (k) () = g k >. For β (positive or negative) let the QED staffing level be n = λ µ + β λ µ + o( λ). (6.2) If β = let n = λ µ + o (λs ), (6.2) for some s < k + 2. The asymptotic expression for E coincides with formula (.3) from the main paper for all the theorems of Section 6. The approximations for J and J are given by the following formulae: a. If β > J = J = ( nµ λ λg k (β λµ) k+3 + o (nµ λ) 2 (k + 3) λg k (β λµ) k+4 + o 5 ) λ (k+)/2, (6.22) λ (k+2)/2. (6.23)

20 b. If β = c. If β < J = J = k + 2 k + 2 J exp [ (k + 2)! λg k [ (k + 2)! λg k k + k + 2 ] /(k+2) ( Γ k + 2 ] 2/(k+2) ( 2 Γ k + 2 ) ( + o ) + o [ ] (k + )! /(k+) (λ nµ) (k+2)/(k+) λg k ) λ /(k+2), (6.24) λ 2/(k+2). (6.25) 2πk! (λg k ) /(2k+2) ((k + )!(λ nµ)) k/(2k+2), (6.26) J ( ) /(k+) β µ(k + )! J. (6.27) g k λ Remark 6. Expression (6.26) increases exponentially due to the (λ nµ) (k+2)/(k+) term in the exponent. Theorem 6.2 (Performance measures) Under the assumptions of Lemma 6.2, the performance measures of M/M/n+G are approximated by: a. Delay probability. If β >, the delay probability coincides (asymptotically) with the Erlang-C approximation from Halfin and Whitt [9]: PW > [ + β ]. (6.28) h( β) If β =, the probability to get service immediately converges to zero at rate PW = = π n k/(2k+4) 2 k + 2 Γ ( k+2 [ ) g k µ k+ (k + 2)! ] k+2 + o ( n k/(2k+4) n k/(2k+4) : ). (6.29) If β <, the probability to get service immediately decreases to zero at an exponential rate: PW = exp k + [ ] (k + )! /(k+) k + 2 (λ nµ) (k+2)/(k+) b. Probability to abandon. λg k ( β(k + )!) k/(2k+2) λ k/(4k+4) µ (k+2)/(4k+4) 2πk! h( β). (6.3) g /(2k+2) k If β > PAb V > = n (k+)/2 6 g k (βµ) k+ + o n (k+)/2. (6.3)

21 If β = If β < PAb V > = n (k+)/(k+2) k + 2 Γ ( k+2 [ ) g k µ k+ (k + 2)! ] k+2 + o ( n (k+)/(k+2) ). (6.32) PAb V > = β + o n. (6.33) n c. Average offered waiting time. If β >, the average offered wait is given by the Erlang-C approximation [9]: If β = If β < E[V V > ] = E[V V > ] = d. Average waiting time. E[V V > ] n /(k+2) Γ 2 [ k+2 (k + 2)! Γ µg k k+2 n /(2k+2) βµ n. (6.34) ] k+2 + o ( [ ] β(k + )! /(k+) ( + o g k n /(k+2) n /(2k+2) ). (6.35) ). (6.36) E[W ] E[V ]; E[W W > ] E[V V > ]. (6.37) Remark 6.2 The value β/ n in formula (6.33) is the minimal reneging rate that is required to avoid queue explosion. Indeed, one can check that β/ n is asymptotically equivalent to the fluid limit of the probability-to-abandon [6] /ρ, given n. Remark 6.3 We do not study the case when all derivatives at the origin are zero but the density is positive near the origin. We think that the answers here would depend on the specific distribution (e.g. lognormal). In general, the case above is intermediate between those described in Theorems 6.2 and 6.4. Example. Phase-type patience times. An important special case of distributions, relevant to Theorem 6.2, is phase-type (see Asmussen [] or Ishay []). Here we study the behavior of the phase-type density near the origin, which is essential if one is to apply Theorem 6.2. Definition. Consider a continuous-time Markov process X = X t, t with a finite statespace, 2,..., k,, where, 2,..., k are transient states and is the absorbing state. The distribution of X is characterized by: 7

22 Initial distribution q = (q,..., q k ), where q i = PX = i, i k (the process cannot start from the absorbing state). Phase-type generator R, a k k matrix of transition rates between the transient states. k We know that R ii <, R ij for i j, and R ij, where i, j k. Absorption intensities r = (r,..., r k ). Overall, the generator of X can be written as ( ) R r Q =,,..., where every row in Q sums up to zero: r = R. (Here is the vector with all components equal to.) j= Let T = inft > : X t = denote the absorption time. Then F T (t) = P q T t is a phase-type distribution with parameters ( q, R). is, The cumulative distribution function of the phase-type distribution with parameters ( q, R) F T (t) = q exprt, and it has a density f T (t) = q exprt r. (6.38) In order to apply Theorem 6.2, we must calculate the density at the origin and its derivatives. From (6.38), the density at the origin is f T () = q r and its n-th derivative (for convenience, we denote also f () T (t) = f T (t)) f (n) T () = qrn r, n. (6.39) Theorem 6.3 (Phase-Type patience) Represent the transient states of the underlying Markov process of a phase-type distribution by a directed graph. Two states j and k are connected if and only if R jk >. For any initial state j (q j > ), let L j denote the number of states in a minimal path that connects j with the absorbing state. Define L = min L j. (6.4) j:q j > (For example, L = n for the Erlang distribution with n phases and L = for the hyperexponential distribution.) Then f (L ) T () >. Moreover, if L 2, then f (i) T () = for i L 2. 8

23 Now Theorem 6.3 and formula (6.39) enable us to apply Theorem 6.2 to phase-type distributions Delayed distribution of patience Assume that, up to a fixed time c >, customers do not abandon. For example, customers could be listening to a recorded announcement. Such situations inspire us to consider delayed distributions of patience, which can be represented by c + τ, where τ represents (im)patience as before. The case of deterministic patience is important as well. As examples, one can consider overflowing, or Internet applications, where the waiting of jobs in queue is usually bounded. Lemma 6.3 (Building blocks) Assume that the density of patience time vanishes over the interval [, c], for some c >. (That means that all customers are willing to wait at least c.) Assume that the density of patience time is positive at c: g c >. For β (both negative and positive) consider the staffing level n = λ µ + β λ µ + o( λ). For β = let n = λ + a, < a <. (6.4) µ a. If β > J = nµ λ e c(nµ λ) λ β µ h( ˆβ c ) g c + o e c(nµ λ), (6.42) λ ˆβ c = β µ g c. (6.43) If β = and a If β = and a = If β < J µa ( e µac ). (6.44) J c. (6.45) J ec(λ nµ) λ β µ + h( ˆβ c ). (6.46) g c VRU. Customers that do not get service within a deterministic target time are sent to another call center or to the 9

24 b. If β > If β = and a If β = and a = If β < J J = λ β 2 µ + o. (6.47) λ J µ 2 a 2 ( e µac ) ce µac. (6.48) µa J c2 2. (6.49) cec(λ nµ) λ β µ + h( ˆβ c ). (6.5) g c Remark 6.4 In the case β =, performance measures are very sensitive to the remaining term n λ/µ. Therefore, in (6.4) this term is asymptotically small in comparison to o( λ) in the other cases. Theorem 6.4 (Performance measures) Under the assumptions of Lemma 6.3, the performance measures of the M/M/n+G system with delayed patience distribution are approximated by: a. Delay probability. If β >, the asymptotic delay probability coincides with the Erlang-C approximation [9]: PW > [ + β ]. (6.5) h( β) If β = and a If β = and a = If β < PW = = π n 2 a e µac + o n. (6.52) PW = = π n 2 µc + o n. (6.53) PW = e c(λ nµ) h( β) µ β µ + h( ˆβ c) g c. (6.54) b. Probability to abandon. If β > PAb W > e c(nµ λ) β µ β2 µ λ h( ˆβ c ). (6.55) g c 2

25 If β = and a If β = and a = If β < PAb W > = n PAb W > = n ae µac e µac + o. (6.56) n µc + o. (6.57) n PAb W > = β + o n. (6.58) n (See Remark 6.2 on page 7.) c. Average offered waiting time. If β > (Erlang-C approximation). E[V V > ] = n βµ + o n (6.59) If β = and a If β = and a = If β < E[V V > ] µa ce µac. (6.6) e µac E[V V > ] c 2. (6.6) E[V ] E[V V > ] c. (6.62) d. Average waiting time. E[W ] E[V ]; E[W W > ] E[V V > ]. (6.63) Remark 6.5 Formulae (6.6)-(6.63) imply that, for β, average wait (both offered and actual) converges to positive constants. That distinguishes the case of delayed distributions from Theorems 6. and 6.2, where E[W ] converged to zero. The important case of deterministic patience times gives rise to similar statements: Theorem 6.5 (Deterministic patience) Assume that patience time is deterministic and equal to c >. a. Delay probability. If β > : PW > [ + β ]. (6.64) h( β) 2

26 If β = and a If β = and a = If β < PW = = π n 2 a e µac + o n. (6.65) PW = = π n 2 µc + o n. (6.66) PW = e c(λ nµ) β h( β). (6.67) b. Probability to abandon. If β > PAb W > e c(nµ λ) λ β µ. (6.68) If β = and a If β = and a = If β < PAb W > = n PAb W > = n ae µac e µac + o. (6.69) n µc + o. (6.7) n PAb W > = β + o n. (6.7) n c. Average offered waiting time. If β > E[V V > ] = n βµ + o n. (6.72) If β = and a If β = and a = If β < E[V V > ] µa ce µac. (6.73) e µac E[V V > ] c 2. (6.74) E[V ] E[V V > ] c. (6.75) d. Average waiting time. E[W ] E[V ]; E[W W > ] E[V V > ]. (6.76) 22

27 6..4 Patience with balking Lemma 6.4 (Building blocks) Consider the QED operational regime n = λ µ + β λ µ + o( λ), λ. Assume that the patience-time distribution has an atom at zero. encountered, customers abandon immediately with probability PBlk >, or In other words, if wait is Ḡ() = PBlk. Assume, in addition, that the survival function Ḡ is differential at the origin: Ḡ () = g. (Here g is the right-side derivative of the patience-time distribution function at the origin.) Then a. b. J = λ PBlk + (nµ λ) g λ 2 PBlk 3 + o λ 2. (6.77) J = n 2 µ 2 PBlk 2 + o n 2. (6.78) Theorem 6.6 (Performance measures) Under the assumptions of Lemma 6.4, the performance measures of the M/M/n+G queueing system in the QED regime can be approximated by: a. Probability to encounter queue decreases at rate Delay probability decreases at rate n : PV > h( β) n PBlk + o n. (6.79) n : PW > ( PBlk) h( β) + o n. (6.8) n PBlk b. Conditional probability to abandon PAb V > converges to the balking probability: PAb V > = PBlk + n g µ PBlk + o. (6.8) n Conditional probability to abandon PAb W > decreases at rate n : PAb W > = n g µ PBlk ( PBlk) + o. (6.82) n 23

28 The unconditional probability to abandon decreases at rate n : PAb = h( β) + o n. (6.83) n c. Conditional average offered wait E[V V > ] decreases at rate n : E[V V > ] = n The average offered wait decreases at rate n 3/2 : E[V ] = n 3/2 h( β) µ PBlk 2 + o n 3/2 µ PBlk + o. (6.84) n. (6.85) d. Conditional average waiting time E[W W > ] decreases at rate n : E[W W > ] = n µ PBlk + o. (6.86) n The average wait E[W ] decreases at rate n 3/2 : E[W ] = ( PBlk) h( β) n3/2 µ PBlk 2 + o n 3/ Patience with scaled balking. (6.87) Lemma 6.5 (Building blocks) Assume that the patience distribution depends on the system size n. Specifically, let the balking probability P n Blk = p b, for some p b >. Assume n that the derivative of the survival function Ḡn at the origin is independent of the system size: Ḡ n() = g. Then a. where J = n µg h( ˆβ) + o n, (6.88) ˆβ = µ (β + p b ). (6.89) g b. J = [ ˆβ ] ( nµg h( ˆβ) + o. (6.9) n) 24

29 Theorem 6.7 (Performance measures) Under the assumptions of Lemma 6.5, the performance measures of the M/M/n+G queueing system in the QED regime can be approximated by: a. The probability of delay and positive offered wait converge to a constant that depends on β, p b and g µ : [ PV > PW > where ˆβ is defined by formula (6.89). + b. Conditional probabilities to abandon decrease at rate g µ n : h( ˆβ) ], (6.9) h( β) PAb V > = [ ] g n µ h( ˆβ) β + o n. (6.92) PAb W > = g n µ [ h( ˆβ) ˆβ ] + o ( n ). (6.93) The unconditional probability to abandon PAb also decreases at rate n and can be approximated by the product of (6.92) and (6.9). c. Conditional average offered wait E[V V > ] decreases at rate n : E[V V > ] = [ h( n g µ ˆβ) ˆβ ] + o n. (6.94) The average offered wait E[V ] also decreases at rate n and can be approximated by the product of (6.94) and (6.9). d. The average waiting time is equivalent to the average offered wait: E[W ] E[V ]; E[W W > ] E[V V > ] (6.95) e. The ratio between the probability to abandon of delayed customers and average wait of delayed customers converges to the value of the patience density at the origin: 6.2 Numerical experiments PAb W > E[W W > ] g. (6.96) In the main paper we analyzed the quality of QED approximations for service grade β =. Here we perform experiments with several other service grades. Example (Figure 2): β =.5. The approximations for the first two distributions are excellent again. The slopes of the two corresponding curves in the first plot remain the same as 25

30 in Figure 6 of the main paper:.25 and 2/3, respectively. Note that, in contrast to that figure, the difference between exact values and approximations does not decrease monotonically in λ. That is due to approximation of the QED staffing level in formula (4.35) from the main paper by the nearest integer value. Figure 2: Service grade β =.5, performance measures and approximations probability to abandon Probability to abandon vs. average waiting time exact U(,4) approx U(,4) exact expmix approx expmix exact Erlang approx Erlang exact delexp approx delexp average waiting time, sec Average waiting time vs. arrival rate exact U(,4) approx U(,4) exact expmix approx expmix exact Erlang approx Erlang exact delexp approx delexp.2. 2 P(Ab W>) 5 5 average waiting time, sec Probability to abandon given delay vs. arrival rate exact U(,4) approx U(,4) exact expmix approx expmix exact Erlang approx Erlang exact delexp approx delexp arrival rate probability of wait arrival rate Delay probability vs. arrival rate exact U(,4) approx U(,4) exact expmix approx expmix exact Erlang approx Erlang exact delexp approx delexp arrival rate The delayed exponential distribution also demonstrates very good fit: the average wait and the delay probability are very close to the Erlang-C approximation, and the probability to abandon decreases exponentially. However, quality of approximations for the Erlang distribution is not so good (in fact, the worst one among all special cases considered in this subsection). Approximations for the aver- 26

31 age wait and the delay probability coincide with Erlang-C formulae (and, therefore, with the approximation for delayed exponential). The fit of PW > is not bad at all. However, the fit of E[W ] is less good and the fit of PAb W > is the worst of all. The reason seems to be unstableness of approximation (6.3) for small positive service grades β. Figure 3: Service grade β =, performance measures and approximations probability to abandon Probability to abandon vs. average waiting time exact U(,4) approx U(,4) exact expmix approx expmix exact Erlang approx Erlang exact delexp approx delexp average waiting time, sec Average waiting time vs. arrival rate exact U(,4) approx U(,4) exact expmix approx expmix exact Erlang approx Erlang exact delexp approx delexp.2.5 P(Ab W>) average waiting time, sec Probability to abandon given delay vs. arrival rate exact U(,4) approx U(,4) exact expmix approx expmix exact Erlang approx Erlang exact delexp approx delexp probability of wait arrival rate Delay probability vs. arrival rate exact U(,4) approx U(,4) exact expmix approx expmix exact Erlang approx Erlang exact delexp approx delexp arrival rate arrival rate Example 2 (Figure 3): β =. Now the approximations for the Erlang distribution are much better than in Figure 2. In particular, the fit of PAb W > graph is reasonable for small values of λ and good for large values. (Recall from formula (6.3) that conditional probability 27

32 to abandon decreases at rate /n.) In the delayed exponential case, the probability to abandon is negligible for all values of λ. Figure 4: Service grade β =.5, performance measures and approximations Probability to abandon vs. average waiting time Average waiting time vs. arrival rate probability to abandon exact U(,4) approx U(,4) exact expmix approx expmix exact Erlang approx Erlang exact delexp approx delexp average waiting time, sec exact U(,4) approx U(,4) exact expmix approx expmix exact Erlang approx Erlang exact delexp approx delexp.5 P(Ab W>) average waiting time, sec Probability to abandon given delay vs. arrival rate exact U(,4) approx U(,4) exact expmix approx expmix exact Erlang approx Erlang exact delexp approx delexp arrival rate probability of wait arrival rate.9.8 Delay probability vs. arrival rate exact U(,4).7 approx U(,4) exact expmix approx expmix.6 exact Erlang approx Erlang exact delexp approx delexp arrival rate Example 3 (Figure 4): β =.5. The fit for the first two distributions (g > ) is fine. The approximation for PAb W > coincides for the last two distributions with g =. (See Remark 6.2.) The average wait decreases at rate n /4 for the Erlang patience and converges to delay time in the delayed exponential case. Finally, in the last two cases delay probability converges to one exponentially (but with very different rates). 28

33 Figure 5: Service grade β =, performance measures and approximations probability to abandon Probability to abandon vs. average waiting time exact U(,4) approx U(,4) exact expmix approx expmix exact Erlang approx Erlang exact delexp approx delexp average waiting time, sec Average waiting time vs. arrival rate exact U(,4) approx U(,4) exact expmix approx expmix exact Erlang approx Erlang exact delexp approx delexp..5 P(Ab W>) average waiting time, sec Probability to abandon given delay vs. arrival rate exact U(,4) approx U(,4) exact expmix approx expmix exact Erlang approx Erlang exact delexp approx delexp probability of wait arrival rate Delay probability vs. arrival rate exact U(,4) approx U(,4) exact expmix approx expmix exact Erlang approx Erlang exact delexp approx delexp arrival rate arrival rate Example 4 (Figure 5): β =. Here we encounter two interesting phenomena. First, the conditional probabilities to abandon start to be very similar for the four distributions and close to β/ n. (Recall formula (4.6) from the main paper and take into account that h( ˆβ) is small for large negative β.) Another interesting phenomenon is observed in the last plot: PW > curve for exponential mixture is relatively far from the uniform one. To explain it, note that for large negative β PW = gµ + g µ h( ˆβ) h( β) h( ˆβ) h( β) g µ h( ˆβ) h( β), 29

34 recall that the normal hazard-rate h( ) decreases rapidly for large negative ˆβ, and that the absolute value of ˆβ is larger for the uniform distribution. (Recall definition (4.3) from the main paper.) Conclusions. Overall, the QED approximations are very good even for moderate staffing levels. Below (Subsections 7.3 and 8.2) we compare them with the quality-driven and the efficiencydriven approximations observing that, in most cases, the QED approximations are preferable. In the main case (g > ), the linear PAb / E[W ] relation is confirmed for all values of the service grade. For relatively large positive β we observe convergence to the Erlang-C asymptotic formulae for the average wait and the delay probability. For relatively large negative β the probability to abandon converges to β/ n for all distributions in consideration. (Recall Remark 6.2 after Theorem 6.2.) 6.3 Proofs of the QED results Proof of Lemma 6.. We provide a detailed proof of b and prove a new asymptotic statement d that was not presented in the main paper. b. In the QED regime, J = x exph λ (x)dx = x x exp [ ] λ(ḡ(u) ) β λµ du dx. Straightforward calculations imply that J A = x exp xβ λµ λg x 2 dx = β 2πµ β 2 [ ( )] µ µ exp Φ β 2 λg λg g 2g g [ = ˆβ ] ( nµg h( ˆβ) + o. (6.97) n) Asymptotic equivalence between J and J A is demonstrated via the Laplace argument, using inequality (.7) from the main paper. Approximation for δ integrals is proved very similarly to a. Consider the second part of the argument, exponential bounds for the δ integrals above: ( x exph λ (x)dx x exp αλ x δ ) δ 2 2 β λµ dx, δ (α was defined by formula (.9) in the main paper) = exp αλ δ 2 δ 2 β λµ δ 3 xe αλx dx δ

35 = exp αλ δ 2 δ [ δ 2 β λµ αλ e αλδ + ] (αλ) 2 e αλδ The tail part of the J A integral, x exp xβ λµ λg x 2 dx, 2 δ can be treated as a special case of J (linear survival function near the origin). d. First, calculate the integral J 2A = x 2 exp β λµx λg x 2 β 2 µ dx = exp 2 2g (changing variables) = exp β 2 µ 2g ( β µ y β ) µ 2 exp g g λ λ = o(e νλ ), ν >. (6.98) λg (x + β µ x 2 g λ exp 2 λg y 2 dy. 2 ) 2 dx, and, after exact calculations, [ ( )] 2π = (λg ) 3/2 + β2 µ µ β 2 µ Φ β exp g g 2g = [ ] ˆβ2 + (nµg ) 3/2 ˆβ + o h( ˆβ) n 3/2. (λg ) 3/2 β µ g The last equality follows from the definition of ˆβ and λ nµ (λ, n ). Finally, in the same way as in Parts a and b of Lemma. from the main paper, we can validate the approximation of by J 2 = J 2A = x x 2 exp λ Ḡ(u)du nµx dx x 2 exp β λµx λg x 2 dx. 2 Proof of Theorem 6.. d. Here an alternative approach to the proof in the main paper is presented. We must prove that E λ [W ] lim λ E λ [V ] =, where the performance measures are indexed by the arrival rate in the QED regime. Recall that W = min(v, τ), where V and τ are independent. 3

36 It can be derived from the proof of Lemma 6. (Part b) that, δ >, E λ [V ; V > δ] lim λ E λ [V ] = δ xv λ (x)dx xv λ (x)dx. (6.99) (Specifically, formula (6.98) shows that an exponential bound is available for δ.) Now, E λ [V ; V > τ] lim λ E λ [V ] ( Eλ [V ; V > τ; τ > δ] = lim λ E λ [V ] E λ [V ; τ < δ] lim λ E λ [V ] = P τ < δ. + E ) λ[v ; V > τ; τ < δ] E λ [V ] (6.) The first term of (6.) converges to zero due to (6.99). The last equality follows from the independence between V and τ. The probability P τ < δ can be made arbitrarily small, since τ has no mass at zero. Hence, Now, E λ [V ; V > τ] E λ [V ; V τ] lim = and lim =. (6.) λ E λ [V ] λ E λ [V ] E λ [W ] = E λ [min(v ; τ)] = E λ [V ; V τ] + E λ [τ; τ < V ] E λ [V ]. The second statement of d follows from PW > PV >. f. Use formula (3.7) and the QED asymptotics for J and J : E[V Ab] = (λ nµ)j + J (λ nµ)j + βµ nj + J βµ nj + Now we shall prove formula (6.3). Note that (β/g ) ( ˆβ/h( ˆβ)) + /( µg h( ˆβ)) n ˆβ/h( ˆβ) [ ] n g µ h( ˆβ) ˆβ. ˆβ E[W Ab] = E[τ τ < V ] = E[τ; τ < V ] PAb = = v(x) ( x tdg(t)) dx, PAb E[τ; τ < x]v(x)dx PAb where v(x) is the density of the offered wait (recall formula (2.3)). Then x tdg(t) = xg(x) 32 x G(t)dt.

37 Note that ɛ > δ > such that for x [, δ], and Then, for x [, δ], (g ɛ)x 2 xg(x) (g + ɛ)x 2 (g ɛ)x 2 2 x 2 2 (g 3ɛ) x x G(t)dt (g + ɛ)x 2. 2 tdg(t) x2 2 (g + 3ɛ). Since x tdg(t) is bounded by x2, we can construct an exponential bound for δ v(x) ( x tdg(t)) dx in the spirit of Lemma 6., part b. Then, based on the Laplace method, we deduce that ( x ) v(x) tdg(t) dx g x 2 v(x)dx, 2 and E[W Ab] g 2PAb λπ n J 2 g 2PAb λj 2 E + λj. (6.2) From part d of Lemma 6. we observe that the numerator of (6.2) is equal to λg J 2 = For the denominator of (6.2) [ ] + ˆβ2 ˆβ + o n. nµg h( ˆβ) 2PAb (E + λj) = 2( + (λ nµ)j) 2( βµ nj) 2 Dividing the numerator of (6.2) by the denominator: E[V Ab] = n 2 g µ + ˆβ 2 βh( ˆβ) h( ˆβ) ˆβ + o n = [ n 2 g µ h( ˆβ) ˆβ ˆβ ] + o n. ( ˆβ ) h( ˆβ) Finally, we prove inequalities (6.5). The right one is a consequence of (5.2) and (5.5). The left inequality is equivalent to or 2h(x) > x + h(x) x 2h 2 (x) 3xh(x) + x 2 >,. 33

38 which follows from convexity of h and formula (5.3). g. From formula (2.3) for the density of the virtual offered wait it follows that x [ ] V P E[S] > t exp λ(ḡ(u) ) β λµ du dx µ t/ λµ V > =. (6.3) λ J Then using the Laplace method we show that the last expression is equivalent to exp β λµx λg x 2 β 2 µ dx exp t/ λµ 2 2g t + β µ exp λg y 2 dy 2 µλ g λ = J J (using the asymptotic expression for J, taken from Lemma. of the main paper, Part a) ( ) g Φ ˆβ + µ t Φ( ˆβ). λ Now, in order to complete the proof, we need to substitute µ by n and the virtual offered wait V by the waiting time W in the left-hand part of (6.3). The validity of the first substitution can be verified using λ nµ. For the second substitution we must prove PW > PV > and P W E[S] > t n V P E[S] > t. n Both relations directly follow from W = min(v, τ) and V p (n ) (see part d). h. Conditional probability to abandon: g t/ λµ t/ λµ P Ab xv(x)dx v(x)dx V E[S] > t n g = t/ λµ t/ λµ Calculating the numerator of (6.4), we get λ exp g t 2 2µ βt t/ λµ v(x)( Ḡ(x))dx t/ λµ v(x)dx x exp β λµx λg x 2 dx 2 exp β λµx λg x 2. (6.4) dx 2 ˆβ ( g Φ ˆβ + µ t ) Φ( ˆβ). 34

39 The denominator of (6.4) is equal to λg ( ) g Φ ˆβ + µ t Φ( ˆβ). Dividing the numerator by the denominator, we get (6.7). Proof of Lemma 6.2. a. β >. Here and in the proofs below we denote o( ) deviation terms in the staffing rules (6.2) and (6.2) by f(λ). Apply Lemma 4. with k = 2 ; l = ; k 2 = ; l 2 = k + 2; m = ; (6.5) (condition k l > k 2 l 2 = is valid for k > ) to derive that J A = exp β λµx f(λ)µx λg kx k+2 dx (6.6) (k + 2)! exp β λµx f(λ)µx dx = nµ λ λg k (β λµ) k+3 + o exp β λµx λg kx k+2 ( (k + 2)! dx + o ( ) λ (k+)/2. λ (k+)/2 (We use that nµ λ = β λµ + f(λ)µ.) Now note that x J = exp λ Ḡ(u)du λx β λµx f(λ)µx dx. (6.7) Under the assumptions of Lemma 6.2, ɛ > δ > such that, for u [, δ], ) (g k + ɛ)u k+ (k + )! Ḡ(u) (g k ɛ)u k+ (k + )!. (6.8) From Lemma 4.3 (m =, n =, k = /2, l = ), there exists ν > such that δ exp β λµx f(λ)µx dx = o e λν. (6.9) Formulae (6.8) and (6.9) enable us to apply the Laplace method in order to show that J from (6.7) can be approximated by J A from (6.6). We use a similar reasoning in order to derive (6.23). (Lemma 4. is applied with m = and other parameters taken from (6.5).) Specifically, J A = x exp β λµx f(λ)µx λg kx k+2 dx (6.) (k + 2)! 35

40 = x exp β λµx f(λ)µx dx = and, then substitute instead of (6.). (nµ λ) 2 J = b. β =. Using Lemma 4.2 with we get and taking m =, = = (k + 3) λg k (β λµ) k+4 + o x exp β λµx λg kx k+2 ( (k + 2)! dx + o ( λ (k+2)/2 ) (λ ) x x exp λ Ḡ(u)du λx β λµx f(λ)µx dx k =, l = k + 2, k 2 = k + 2, l 2 =, m =, J A k + 2 J A k + 2 = exp f(λ)µx λg kx k+2 dx (k + 2)! ( (k + 2)! λg k ) /(k+2) Γ k o λ (k+)/2 = x exp f(λ)µx λg kx k+2 dx (k + 2)! ( (k + 2)! λg k ) 2/(k+2) 2 Γ k o λ (k+)/2 Then we use (6.8), the Laplace method and Lemma 4.3 in order to substitute x J = exp λ Ḡ(u)du λx f(λ)µx dx, and J = x x exp λ Ḡ(u)du λx f(λ)µx dx, λ (k+)/2 ), (6.). (6.2) into (6.) and (6.2), respectively. Note that Lemma 4.3 cannot be applied immediately to get and δ δ δ exp f(λ)µx λg kx k+2 (k + 2)! dx = o (e ) λν x exp λ Ḡ(u)du λx f(λ)µx dx = o (e ) λν ( f(λ) can be positive). However, this problem can be easily solved. For example, exp f(λ)µx λg kx k+2 dx exp (k + 2)! 2 λg kx k+2 dx (k + 2)! 36 δ

41 for λ large enough. c. β <. As in part a, we approximate J by J A = exp β λµx f(λ)µx λg kx k+2 dx, (6.3) (k + 2)! and, then, apply the Laplace method to show that J J A. However, since β is a positive number, the integrand increases near zero, which requires additional work that involves somewhat cumbersome calculations. Define x = ( ) /(k+) [ β λµ f(λ)µ] (k + )!, λg k to be equal to the point where the integrand of (6.3) reaches a maximum (note that x converges to zero at rate λ /(2k+2) ). Performing the variable change y = x x, we get Note that x exp J A = exp[ β λµ f(λ)µ] x β λµy f(λ)µy λg k(y + x ) k+2 (k + 2)! x exp β λµydy = β λµ expβ λµx. dy. (6.4) Since β is negative, the integral above decreases at rate exp λk/(2k+2) and we can change the λ integral limits in (6.4) to. Now we expand (y+x ) k+2 from (6.4). The free term (x ) k+2 is taken out of the integral and the (k + 2)y(x ) k+ term is cancelled by [ β λµ f(λ)µ] y. We must show now that the quadratic term in the expansion dominates the others. In other words, J A exp k + k + 2 [ ] (k + )! /(k+) (λ nµ) (k+2)/(k+) = exp g k k + k + 2 [ ] (k + )! /(k+) (λ nµ) (k+2)/(k+) λg k 2πk! (λg k ) /(2k+2) ((k + )!(λ nµ)) k/(2k+2). exp λg k 2k! (x ) k y 2 dy We shall prove that the quadratic term in the integral (6.4) dominates the cubic term, an argument that can be repeated for the terms with larger degrees of y using Remark 4.2. Ignoring cumbersome constants, we must show that exp λ (k+2)/(2k+2) y 2 λ (k+3)/(2k+2) y 3 dy 37

42 exp λ (k+2)/(2k+2) y 2 dy = 2πλ (k+2)/(4k+4). The equivalence above follows from Lemma 4. with (Note that condition (4.) prevails for k >.) k = k + 2 2k + 2, l = 2, k 2 = k + 3 2k + 2, l 2 = 3, Formula (6.27) for J is proved via the approximation J A x exp β λµx f(λ)µx λg kx k+2 dx x J A, (k + 2)! where the second equivalence is obtained via the change of variables y = x x. Proof of Theorem 6.2. a. Delay probability. β >. Recall the asymptotic expression for E: E = λ µ h( β), which does not depend on the patience distribution G. Hence, PW > = λj E + λj λ β µ λ h( β) µ + λ β µ = [ + β ]. h( β) β =. From Part b of Lemma 6.2 and taking into account that E PW = = E E + λj E λj π λ k/(2k+4) 2µ π n k/(2k+4) 2 k + 2 Γ ( k+2 [ ) k + 2 Γ ( k+2 g k µ k+ (k + 2)! λ µ π 2 : [ ) ] k+2. g k (k + 2)! ] k+2 β <. Use that PW = E λj µλj h( β). b. Probability of delayed customers to abandon. β >. PAb V > = + (λ nµ)j λj 38 λg k (β λµ) k+2 λ β µ

43 β =. PAb V > = (β λµ + f(λ)µ)j λj g k β k+ (λµ) (k+)/2 n (k+)/2 g k (βµ) k+. λj n (k+)/(k+2) k + 2 Γ ( k+2 [ ) g k µ k+ (k + 2)! ] k+2. β <. PAb V > = (β λµ + f(λ)µ)j λj µ β λ β. n c. Average offered waiting time. β >. E[V V > ] = J J β λµ βµ n. β =. β <. E[V V > ] = J J n /(2k+2) E[V V > ] = J J x n /(2k+2) [ ] β(k + )! /(k+). g k [ ] β(k + )! /(k+). g k d. Average waiting time. Since the survival function Ḡ is strictly decreasing near the origin, the proof of Part d of Theorem 6. can be duplicated. Proof of Theorem 6.3. We start with some definitions. Consider the underlying Markov process of the phase-type distribution F T. Let S denote the set of states that correspond to the positive values of the initial distribution: i S iff q i >. Then let S be the set of states that can be reached by one jump from some initial state. Formally, j S iff j S and there exists i S such that R ij >. Finally, we define recursively the set S k which comprises states that can be reached by k jumps: j S k iff j S,..., S k and there exists i S k such that R ij >. According to the definition (6.4) of L, the absorbing state S L. We shall number the states of the underlying Markov process in the following way: first, the states from S, then the states that belong to S,..., S L, etc. A relevant part of the generator 39