In this e-companion, we provide the proofs of propositions, lemmas, and corollaries appearing

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1 Appedices I this e-compaio, we provide the proofs of propositios, lemmas, ad corollaries appearig i the paper. We also provide full detail for calculatig the PGF of the CWTP s, i.e. Ez W (i, for Model 1. EC.1. Proof of Propositio 1 From (1, we have Ez X t+1 R(zEz (X t + +D t ( R(z Ez (X t + +D t, X t + Ez (X t + +D t, X t < R(z ( (q + pz Ez X t, X t + E Ez D t, X t < X t, X t < R(z ( (q + pz Ez Xt, X t + E(q + pz X t, X t < ( 1 R(z (q + pz z j P(X t j + + (q + pz j P(X t j. (EC.1 Takig the limit of both sides as t yields X(z R(z ((q + pz X 2 (z + X 1 (q + pz. Replacig X 2 (z with (X(z X 1 (z/z i above ad solvig it for X(z completes the proof. EC.2. Proof of Lemma 1 The lemma ca be proved usig Rouche s theorem (see, e.g. Titchmarsh 1939 but as explaied i Ada et al. (2006 that would require the radius of covergece for A(z ad cosequetly R(z to be larger tha oe, which is ot the case for some heavy-tailed distributios such as discrete Pareto distributio (e.g. Johso et al To iclude such distributios, we use Theorem (3.2 give i Ada et al. (2006. By this theorem, we eed to show (i A(0 > 0, (ii A(z is differetiable at at z 1, ad (iii A (1 <. For (i, ote that A(0 r 0 q, which is positive sice r 0 > 0 ad q > 0 by assumptio. Coditio (ii holds for referral distributios R with fiite mea. Fially, A (1 p + µ R < by the stability coditio. ec1

2 ec2 e-compaio to Izady: Appoitmet Capacity Plaig i Specialty Cliics EC.3. Proof of Corollary 1 For (5, settig µ X lim z 1 d X(z ad simplifyig yield the result. For (6, dz E(z E z R+D R(z ( Ez D, X < + Ez D, X where D is the limitig distributio of D t as t. This gives E(z R(z (X 1 (q + pz + (q + pz P(X. EC.4. Aalysis of Coditioal Waitig Time Periods (CWTP s We start by explaiig how CWTP s are calculated. Cosider the example depicted i Figure EC.1 where system capacity is assumed to be two patiets per uit time. Suppose patiet six is radomly tagged ad has bee referred durig slot J i steady state. The umber of referrals before (after the tagged patiet i slot J is deoted by B (B ad is equal to 2 (1 i the example. W (i the equals F (i for i 1, 2,..., where F (i represets the umber of patiets i frot of the tagged customer i the begiig of the ith CWTP before the service begis, ad x is the largest iteger less tha or equal to x. I the example, W (1 3 1 ad W ( Figure EC.1 A Simple Illustratio for Waitig Time Calculatios. (1 (1 (2 (2 W F / 1 W F / H (1 B +DJ1 7 H (2 0 (1 6 W (1 (1 5 N H ( R T ( 1 D (1 2 j T j 7 9 j 0 F (1 (XJ- + +B3 4 6 F ( M N 1 ( 2 (1 (1 R S 3 (1 ( 2 T M (1 F (1 - W ( B2 B J T (1 J+1 J+2 T (2 J Hece i order to calculate W (i, oe eeds to fid F (i. Clearly, F (1 (X J + +B ad is equal to 3 i the example. To obtai F (i, i 2, we first defie T (i as the idex of the first iterval i the

3 e-compaio to Izady: Appoitmet Capacity Plaig i Specialty Cliics ec3 ith CWTP, so T (1 J + 1 ad T (i T (i 1 + W (i for i 2. I the example, for istace, T (1 J + 1 ad T (2 J + 3. We further defie M (i (N (i as the umber of patiets i frot of (behid the tagged customer at the ed of ith CWTP, ad S (i as the umber of patiets i frot of the tagged patiet at the ed of ith CWTP who miss their appoitmets ad subsequetly reschedule for i 1. Clearly, S (i Biomial ( M (i, p. By assumptio, followig missig her (i 1th appoitmet, the tagged customer will be placed i the backlog i the begiig of the followig iterval, i.e. iterval T (i+1, behid ew referrals as well as all S (i 1 patiets. We thus have F (i (M (i 1 + N (i R T (i 1 + S(i 1 for i 2. For the example, F (2 ( We further have M (i F (i W (i, which by W (i defiitio is o-egative, ad N (i H (i + W (i 1 (R T (i +j +D T (i +j for i 1, where H (i represets the umber of patiets behid the tagged patiet i the begiig of the ith CWTP. For the example, M ( , N ( Note that H (1 B +D J ad H (i Biomial ( mi{n (i 1, (M (i 1 + 1}, p for i 2, where the mi operator gives the patiets behid the tagged patiet that are due to be served at the same time as her. Also, sice durig each coditioal waitig time period the size of the backlog is certaily larger tha, D T (i +j Biomial (, p for i 1 ad 0 j < W (i. For the example, H (1 1 ad H (2 0. A summary of otatios used i this sectio ad their correspodig equatios ad defiitios are preseted i Table EC.1. Usig the otatios ad formulas give i Table EC.1, Theorems EC.1, EC.2, ad EC.3 provide the ecessary equatios for calculatig the PGF of W (i for all values of i. Theorem EC.1. For {z C : z 1}, where W (i (z E z (i W K (i (z, y E z F (i y H(i, 1 m0 P(F (i m, if z 0, z 1 1 z 1 1/ m0 K(i (z 1.εm, 1 ε m z 1/ ε m 1, otherwise, (EC.2

4 ec4 e-compaio to Izady: Appoitmet Capacity Plaig i Specialty Cliics Table EC.1 A Summary of Notatios for Waitig Time Calculatios i Model 1. Notatio Equatio Defiitio B - referrals before the tagged patiet B - referrals after the tagged patiet W (i F (i { ith coditioal waitig time (X F (i J + + B for i 1 umber of patiets i frot of the (M (i 1 + N (i R T (i 1 + S (i 1 for i 2 tagged patiet i the begiig of the ith CWTP T (i { J + 1 for i 1 T (i 1 + W (i for i 2 idex of the first iterval i the ith CWTP M (i F (i W (i umber of patiets i frot of the tagged patiet at the ed of ith CWTP N (i N (i H (i + W (i 1 (R T (i +j + D T +j umber of patiets behid the tagged (i patiet at the ed of ith CWTP S (i - umber of o-shows i frot of the tagged patiet at the ed of ith CWTP that rejoi the queue H (1 B + D J umber of patiets behid the tagged patiet i the begiig of the 1st CWTP H (i - umber of patiets behid the tagged patiet i the begiig of the ith CWTP for i 2 z 1/ z 1/ e iarg(z/, where z is the absolute value of z, i is the imagiary uit, ad Arg(z represets the pricipal value of the argumet of z, i.e. it is a mappig i the iterval ( π, π, ε m e i2πm/ for m 0, 1,...,, i.e. ε m s are complex roots of z 1, ad P(F (i m ca be obtaied by umerically ivertig the PGF K (i (z, 1. Proof For z 0, W (i F (i implies that W (i (0 P(W (i 0 1 m0 P(F (i m. For z 0, the proof is more complicated. Let k (i (j, k deote the mass fuctio of K (i (z, y, i.e. k (i (j, k P(F (i j, H (i k. Ivokig the relatios W (i F (i, M (i F (i W (i, ad N (i H (i + W (i 1 A T (i +j where, for a give i 1, A T (i +j R T (i +j + D T (i +j radom radom variable A with A(z R(z(q + pz for 0 j < W (i, we have P (i (z, x, y E z W (i x M (i y (i N t0 E t0 z t x t E x F (i y H(i, W (i t A(y t t0 t+ 1 jt z t x t A(y t x j y k k (i (j, k k0 z t x F (i t y H(i + t 1 A T (i +j, W (i t are i.i.d as the

5 e-compaio to Izady: Appoitmet Capacity Plaig i Specialty Cliics ec5 1 z t x j y k A(y t k (i (t + j, k. t0 k0 I order to relate P (i (z, x, y to K (i (z, y, we use the siftig property of the Kroecker delta fuctio δ(x (Wag ( t+j l z t z 1 δ(j l, (t, j N 0, 1, (EC.3 l0 to arrive at 1 1 P (i (z, x, y k (i (t + j, ku(z, y t+j l x l y k δ(j l, t0 k0 l0 where u(z, y (za(y 1. Note that we have also chaged the power of x from j to l as both sums are over the same iterval. Usig the followig relatio betwee the Kroecker delta fuctio ad the complex roots of uity we get δ(j l P (i (z, x, y t0 k0 l m0 ε t+j l m, (t, j, l N 3, (EC.4 1 k (i (t + j, ku(z, y t+j l x l y k m0 1 εt+j l m 1 1 ( k (i (t + j, k (u(z, yε m t+j y k x u(z, yε m m0 t0 k0 l ( l K (i x (u(z, yε m, y. u(z, yε m m0 Workig out the secod sum yields l0 l P (i (z, x, y K (i (x, y + u(z, y x u(z, y 1 u(z, y x 1 u(z, y 1 m0 0 m 1 m k ε m K (i (u(z, yε m, y, if x/u(z, y ε k for some 0 k 1, u(z, yε m x ε m K (i (u(z, yε m, y, otherwise. u(z, yε m x (EC.5 Settig x y 1 i above completes the proof.

6 ec6 e-compaio to Izady: Appoitmet Capacity Plaig i Specialty Cliics It is evidet from Theorem EC.1 that i order to obtai W (i (z we eed to have K (i (z, y. The followig theorem shows how we ca recursively calculate K (i usig L (i 1, where L (i (x, y E x M (i y N(i ad is obtaied through Theorem EC.3. Theorem EC.2. For {(z, y C 2 : z 1, y 1}, K (1 (z, y E z F (1 y H(1 (X 1 (η + η x R(y r 0 yµ R, if z 0 (( η z (X(z X1 (z + X 1 (η R(z R(y, otherwise, µ R (z y (EC.6 ad K (i (z, y ( 1 1 j r 0 q j η k l (i 1 (j, k, if z 0 k0 (η 1 R(z (L (i 1 (z θ 1 1 j z η, z ( z θ j 1 1 j z k l (i 1 (j, k + θ j η k l (i 1 (j, k otherwise, η for i 2, where η q + py, θ q + pz, k0 l (i (j, k is the probability mass fuctio of L (i (x, y, i.e. l (i (j, k P(M (i j, N (i k. It ca be obtaied by umerically ivertig the two-dimesioal PGF L (i (x, y usig the discrete (fast Fourier trasform method or Taylor series expasio method. k0 (EC.7 Proof To prove (EC.6, ote that sice F (1 (X J + + B ad H (1 B + D J, K (1 (z, y E z F (1 y H(1 E z (X J + y D J Ez B y B E z (X J + y D J R(z R(y µ R (z y, (EC.8 where the first equality is due to the idepedece of the system backlog i the begiig of each iterval ad the umber of referrals durig that iterval, ad the secod equality has bee obtaied

7 e-compaio to Izady: Appoitmet Capacity Plaig i Specialty Cliics ec7 by takig ito accout that a arbitrary customer is more likely to arrive i a iterval with more arrivals (see e.g. Brueel ad Kim (1993. We the have E z (X j + y D J Ez X J y D J, X J + Ey D J, X j < (q + py Ez X J, X z J + E(q + py X J, X J < (q + py (X(z X z 1 (z + X 1 (q + py. (EC.9 Substitutig (EC.9 i (EC.8, ad applyig the L Hopitals rule times for the case of z 0, yields the result. For i 2, sice F (i (M (i 1 + N (i R T (i 1 + S(i 1 with S (i Biomial ( M (i, p ad H (i Biomial ( mi(n (i 1, (M (i 1 + 1, p, we have K (i (z, y E z F (i y H(i E z (M (i 1 +N (i R T (i 1 +S (i 1 y H(i R(zE E z (M (i 1 +N (i S (i 1 y H(i (M (i 1, N (i 1 R(zE (q + pz M (i 1 (q + py mi(n(i 1, (M (i 1 +1 E z (M (i 1 +N (i (M (i 1 N (i 1. Coditioig the above o whether or ot M (i 1 + N (i 1 > 1 yields K (i (z, y R(zE (q + pz M (i 1 (q + py M (i 1 1 z M (i 1 +N (i 1 +1, M (i 1 + N (i 1 > 1 + R(zE (q + pz M (i 1 (q + py N(i 1, M (i 1 + N (i 1 1 ( 1 ( q + py R(z E z q + pz M (i 1 z N (i 1, M (i 1 + N (i 1 > 1 z q + py + R(zE (q + pz M (i 1 (q + py N(i 1, M (i 1 + N (i 1 1. Expadig the expected values i the above equatio, takig ote of the fact that 0 M (i 1, 0 N (i <, ad that the first term i the right had side of the first equality above is equal to 0 for z 0, completes the proof.

8 ec8 e-compaio to Izady: Appoitmet Capacity Plaig i Specialty Cliics Theorem EC.3. For i 1, ad {(x, y C 2 : x 1, y 1}, L (i (x, y E x M (i y (i N K (i (x, y + u(1, y x u(1, y 1 u(1, y x u(1, y 1 1 m0 0 m 1 m k ε m K (i (u(1, yε m, y, if x/u(1, y ε k for some 0 k 1, u(1, yε m x ε m K (i (u(1, yε m, y, otherwise, u(1, yε m x (EC.10 where u(z, y (za(y 1. Proof Settig z 1 i (EC.5 yields the result. The recursive structure of the equatios required for calculatig W (i (z is depicted i Figure EC.2. For example, for calculatig W (2, oe eeds to calculate K (2, which depeds o L (1, which i tur requires K (1. Figure EC.2 The Recursive Structure for Calculatig W (i (z. W(z W (1 (z W (2 (z W (3 (z K (1 (z,y K (2 (z,y K (3 (z,y L (1 (x,y L (2 (x,y L (3 (x,y EC.5. Proof of Propositio 2 From (8, we have Ez X t+1 Ez (X t N t + +D t +R t R(zE E z (X t N t + +D t N t N 1 R(zE (q + pz N z j P(X t j + N + (q + pz j P(X t j.

9 e-compaio to Izady: Appoitmet Capacity Plaig i Specialty Cliics ec9 where the last equality is due to (EC.1. Takig the limit of both sides as t yields N 1 X(z R(zE (q + pz N z j x j+n + (q + pz j x j ( R(z (q + pz c X 2,c (z c + R(z z ( c0 X(z X 1,c (q + pz c c0 z c (q + pz c c + c0 z X 1,c (q + pz c c0 z c (q + pz c X 1,c (z c, where the third equality is obtaied by replacig X 2,c (z with (X(z X 1,c (z/z c. Solvig the c0 above for X(z completes the proof. EC.6. Proof of Lemma 2 By theorem (3.2 i Ada et al. (2006, we eed to show (i A(0 > 0, (ii A(z is differetiable at at z 1, ad (iii A (1 <. For (i, ote that A(0 r 0 q, which is positive sice r 0, q, ad are positive by assumptio. Coditio (ii holds for referral distributios with fiite mea. For (iii, A (1 µ R + µ N q < by the stability coditio. EC.7. Proof of Corollary 2 For (11, the proof is similar to that of Corollary 1. For (12, E(z E z R+D R(zE E z D N R(zE N 1 (q + pz i x i + (q + pz N x i i0 in where D is the limitig distributio of D t as t. This gives E(z R(z X 1,c (q + pz + (q + pz c (1 P(X < c c. c0 EC.8. Proof of Propositio 3 From (13, we have φ ij P(X t+1 j X t i P ( mi{(i + + D t + R t, k} j X t i 0, j < i, P ((i + + D t + R t j X t i, i j < k, (EC.11 P ((i + + R t + D t k X t i, j k.

10 ec10 e-compaio to Izady: Appoitmet Capacity Plaig i Specialty Cliics For i j < k, we the have P ( (i + + D t + R t j X t i P(R t + D t j (i + X t i, R t lr l l0 P(D t j (i + l X t ir l l0 l0 ( mi{, i} (p(i + j (i + l ( mi{,i} j+(i + +lrl q(i j (i + + l j (i + l(j i + ( mi{, i} (p(i + j (i + l ( i j+lrl q(i j (i + +, l ad for j k, P ( (i + + D t + R t k X t i P ( D t + R t k (i + X t i which completes the proof. Refereces P ( D t k (i + l X t i r l l0 mi{i,} l0 mk (i + l ( mi{i, } m ( q((i 1 + k (i + mi{i,} lk i (p(i + r l mi{i,} mk (i + l m ( mi{i,} mrl q(i + ( mi{i, } m ( m p((i 1+, q((i 1 + Ada, I. J. B. F., J. S. H. va Leeuwaarde, E. M. M. Wiads O the applicatio of rouch s theorem i queueig theory. Oper. Res. Lett. 34( Brueel, H., B. G. Kim Discrete-Time Models for Commuicatio Systems Icludig ATM. Kluwer, Bosto. Johso, N. L., S. Kotz, A. W. Kemp Uivariate Discrete Distributios. Joh Wiley & Sos, New York. Titchmarsh, E.C The Theory of Fuctios. Oxford Uiversity Press, New York. Wag, R Itroductio to Orthogoal Trasforms: With Applicatios i Data Processig ad Aalysis. Cambridge Uiversity Press.