Managing Service Systems with an Offline Waiting Option and Customer Abandonment: Companion Note

Managing Service Systems with an Offline Waiting Option and Customer Abandonment: Companion Note Vasiliki Kostami Amy R. Ward September 25, 28 The Amusement Park Ride Setting An amusement park ride departs at deterministically spaced intervals, and can carry only a certain number of customers. Hence it is desirable to extend our analysis to include situations in which customers are served in batches at set time intervals. The parameter regime we have considered, in which the arrival and service rates are large, is applicable to most popular amusement park rides, because generally hundreds or thousands of customers arrive to the ride, and board the ride, each hour. Recall that we are considering a system in which the arrival rate becomes large, and the service rate is θ for some θ R. We superscript any process or quantity associated with the system having arrival rate and service rate µ = θ by. The numbering of all the equations in this Companion Note begins after the number of the last equation, 38, in the Technical Appendix. The required modification to the model is the service process definition. In a slight abuse of notation, we use SI and SO to denote the cumulative number of customers that have boarded the ride from the inline and offline queues, even though the service processes are no longer renewal. The reader is to understand that in this Companion Note, SI and S O refer to the processes defined below. Let l 2/3 1, and assume service occurs only at discrete time points l, 2l, 3l,..., which represent ride departure times. At each discrete time il, the number of customers that can enter into service board the ride is n 1/3 1/6 θ. Then, the service rate is µ = n /l = θ customers per hour. The service process is defined recursively as follows. At time, no customers have boarded the ride, so that S I S O. 1

Suppose that at discrete time il, there are Q I customers in the inline queue and Q O customers in the offline queue. Then, the number of customers served from each queue that board the ride is and B Oil Hence B I il { [ 1 ] αn + min αn + Q O, QI αn Q I Q I αn Q I < αn { [ αn ] 1 αn + min + Q I, QO 1 αn Q O 1 αn Q O Q O < 1 αn. S I il S I i 1l + B I il S Oil S Oi 1l + B Oil. No customers board the ride in between the discrete time points l, 2l, 3l,..., and so for any t >, S I t = S I t l l and S Ot = S O t l. l The evolution equations for the queue-length process are very similar to 3 and 4 in Section 2 Q I t Q Ot A t i=1 A t i=1 1{w I W t i w OW t i } S I t 39 1{w I W t i > w OW t i } S Ot N γq Osds. 4 The difference is that now, because the service process counts the cumulative number of customers that have boarded the ride and not the number of customers that have departed after riding, the processes Q I and Q O track only the customers waiting to ride and do not include the customers riding or in service. Hence the wait time estimates W I t Q I t µα and W Ot O t µ1 α do not include any customers currently on the ride. This is reasonable because at time t the time remaining until the next ride departs is t/l l t, which becomes negligible as 2

increases. Finally, note that the bound on O in 1 is now Q Ot O t Q Ot + N [ t sup s t W γq O s ] + Osds N γq Osds, 41 where WO t represents the actual time a customer arriving to the offline queue at time t must wait to board the ride. We expect that the discrete review system behaves similarly to the continuous time system. The following proposition shows that Theorems 1 and 2, and hence also Corollary 1, remain valid for the discrete review system. Proposition 1 Theorems 1 and 2 remain valid for the model defined through 39-4. The process Q appearing in Theorems 1 and 2 again solves the stochastic equation 11 but the infinitesimal variance of the Brownian motion X is σ 2 A. We end the body of this Companion Note by showing how to apply Proposition 1 to one popular roller coaster ride at Six Flags Magic Mountain, Tatsu. Suppose the arrival rate has been estimated. To use the approximation, we must determine the parameter θ. Tatsu has capacity for approximately 16 people to ride every hour, and so θ = 16, or θ = 16. The approximation is not very sensitive to l, because Proposition 1 remains valid for any review period of size f for 1/2 < f < 1. The key is that the time between ride departures is roughly on the order of seconds if the number of people that can ride every hour is around one or two thousand which is true for most roller coasters. Proof of Proposition 1 For the proof of Proposition 1 we will need the following lemma whose proof we defer at the end of this Companion Note. This final Lemma states that Lemmas 1-3 remain valid in the modified model in the Amusement Park Ride Setting, in which customers are served in batches at set time intervals. In this setting, as is true for the processes Q I and Q O, the workload processes PI and P O, and the actual waiting time processes W I and W O, refer to the customers waiting to board the ride and do not include the customers currently riding. Furthermore, τ t 1 γq Osds. 3

Lemma 4 Lemmas 1-3 also hold when the system evolution equations are specified through 39-4. Note that in this setting the processes TI, T O, and I no longer appear in the system evolution equations, and so Lemma 1 is modified to state that as, Q, PI, PO, τ,,,, a.s., u.o.c.. 42 We must show the following. i For any T >, sup t T w I α ii As, Q, Ĩ Q, Ĩ. iii As, Proof of i sup WI t W I t and t T Q I t w O 1 α Q O t, in probability, as. sup WO t W O t, in probability. t T Modify the definitions of Ũ 1 and Ũ 2 in the proof of Theorem 1i so that Ũ1 t, s = w O Ã t s 1 α Ã { θ + w I w O α 1 α w O w I + w } O 1 t s 1 α l Ũ2 t, s = w I Ã t s α Ã { θ + w O w I 1 α α w I w O + w } I 1 t s. α l With ξ and ξ defined exactly as in the proof of Theorem 1i, observe that when w I α Q I ξ > w O 1 α Q O ξ, because the inline offline queue does not become empty during [ξ, ξ ], the offline inline queue may become empty, and service occurs in discrete time intervals S I t S I ξ S Ot S O ξ t ξ αn l t ξ 1 αn. l 4

Then, substitution of the above bounds into the equivalent of 2 in the proof of Theorem 1i in this setting specifically, replace S I T I t S I T I ξ with S I t S I ξ and SO T O t S O T O ξ with S O t S O ξ shows that for large enough w I α Q I t w O 1 α Q Ot ɛ } {Ũ 2 + 1 + max 1 t, ξ, Ũ 2 t, ξ + w O 1 t N γq 1 α Osds. Since l 1 = 1/3 as, the remainder of the proof proceeds exactly as the proof of Theorem 1 i, noting that by Lemma 4 as. 1 γq Osds a.s., u.o.c., Proof of ii We first obtain a useful equivalent representation for the process Q O t = Q I t + Q O t. Define so that ɛ t γ 1 αw I Q s Q αw O + 1 αw Os ds, I Q t = A t SI t SOt 1 αw I γ Q sds αw O + 1 αw I +ɛ t N γq Osds + γq Osds. Next we define recursively the process that tracks the cumulative number of empty ride seats I I il I i 1l + [ n Q il ] +. The process I does not increase in between the discrete time points l, 2l, 3l,..., and so for any t >, t I t = I l. l 5

Then, SI t + SOt = = and so It follows that t/l i=1 t/l i=1 BI il + BOil n 1{Q il n } + Q il 1{Q il < n }, S I t + S Ot + I t = t n. l where Q t = X t + ɛ t X t A t 1 αw I γ Q sds + I t 43 αw O + 1 αw I t n N γq l Osds + γq Osds. Furthermore, because the ride will depart with empty seats only when no customers are waiting, Q il I il I i 1l =. 44 i=1 By Lemma 4, there exists a convergent subsequence Q i, I i Q, Ĩ i i as i. We show that the limit Q, Ĩ satisfies Q, Ĩ = φ κ, ψ κ X, 45 where X is a Brownian motion with drift θ and variance σa 2. We first show C1 in Definition 1 is satisfied. On the subsequence i, because X i t i = Ã i t Ñ i τ i t + t i t 1 l i the functional central limit theorem shows X i i X. l i t t l i + θt l i t, 6

Also on the subsequence i, as in the proof of part ii of Theorem 1 which requires the tighness of Q established in Lemma 4, ɛ i i, as i. We conclude from 43 that the limit Q, Ĩ satisfies Qt = Xt 1 αw I γ Qsds + Ĩt. 46 αw O + 1 αw I For C2, note that on the subsequence i, the definition of the Reimann-Stieltjes integral implies that when we take limits as i on both sides of the equality in 44, QtdĨt =. Since furthermore I i = and I i is non-decreasing, it follows that Ĩ = and Ĩ is nondecreasing. We conclude that 45 is valid. From the representation 18, this is equivalent to the stochastic equation for Q in 11. Since the subsequence i was arbitrary, this part of the proof is complete. Proof of iii The argument is very similar to the proof of Theorem 2 in the Technical Appendix Kostami and Ward 28b, and so is omitted. The exception is that the upper-bound on O in 41 replaces the upper-bound on O in 1. Proof of Lemma 4 We divide the proof of Lemma 4 into three parts, with each part re-proving Lemmas 1, 2, and 3 for the modified model in the Amusement Park Ride Setting, in which customers are served in batches at set time intervals. Proof of 42 Lemma 1 equivalent We require defining the following two comparison systems. Comparison system 1 is the model in the Amusement Park Ride Setting without abandonments. In particular, the inline and offline queue-length processes, Q B,I and Q B,O, satisfy equations 39 and 4 with 7

γ =. Under the same arrival sequence, on a sample path basis, Q t Q B,It + Q B,Ot, for all t. Comparison system 2 is a conventional single-server queue with no abandonments and deterministic, non-batched service. In particular, the queue-length process evolution equation is Q Ct A t µ 1{Q Cs > }ds. Under the same arrival sequence, on a sample path basis Q B,Iil + Q B,Oil = Q Cil for every i =, 1, 2,.... We conclude that on a sample path basis Q t Q C t t l + A t A l. 47 l l It is well known that Q C / a.s., u.o.c., as. Furthermore, A t A t l l = A t A t t l + t l, l l and, as, A a.s., u.o.c. and t t/l l. Hence Q a.s., u.o.c., as. It then follows that τ a.s., u.o.c., as. Proof that the sequence { Q, Ĩ } is tight in D Lemma 2 equivalent It is sufficient to verify that the sequence { Q } is tight in D. Tightness of the sequence { Q, Ĩ } then follows from equation 43, because the functional central limit theorem shows the sequence {X / } is tight, and tightness of the sequence {ɛ / } can be established very similarly to Lemma 2. Let T >. We verify conditions 16.17 and 16.18 in Theorem 16.8 in Billingsley. B16.17 We must show that for η > arbitrarily small, there exists an a and a large enough 8

such that P sup t T Q t a < η,. 48 It is well-known that for Q C defined as in the first part of this proof, Q C / weakly converges to a reflected Brownian motion with drift θ and variance σ 2 A. Furthermore, A t A t l l t = à t à l + t t l. l l The functional central limit theorem implies that à weakly converges to a Brownian motion, and the definition of l implies t t/l l as. Therefore, the condition 48 follows from the bound in 47. B16.18 It is sufficient to show that for γ > and η > arbitrarily small, there exists a δ small enough and a large enough such that P sup sup t T δ v,s [t,t+δ] Q s Q v γ < η,. 49 Without loss of generality, assume s < v. We require an upper and a lower bound on the process Q v Q s. For the upper bound, define a comparison system Q Ct A s + t A s µ By similar reasoning as in the previous paragraph, Q v Q s Q C For the lower bound, since v s l s 1{Q Cζ > }dζ. l + A v A v l l. 5 Q v Q s v v s s = A v A l + A l A s + A l A l l l l l v s SI v + SI s SOv + SOs N γq Oζdζ + N γq Oζdζ, 9

and at most n customers are served every l time units, Q v Q s v/l l 1 i= s/l l v N A i + 1l A il n 2n 51 s γq Oζdζ + N γq Oζdζ. Noting that Q O t Q t for all t, 1 A i + 1l A il n = à i + 1l à il 2/3 θ, and 1 v s N γq Oζdζ N γq Oζdζ = Ñ τ v Ñ τ s v s γ Q Oζdζ, it follows from 5 and 51 that where M U Q C M L v s l Q v Q s maxmu, ML, 52 l + à v l l à s l +γv s sup t T Q t v à v à l v + v l l l l + Ñ τ v + Ñ τ s v s + 2/3 θ l l + 2 n. l l The condition 49 follows because every term on the right-hand side of 52 becomes arbitrarily small with high probability as δ converges to, for v s < δ and large enough. In particular, v s/l l v s as, and so, since Q C weakly converges to a continuous limit process a reflected Brownian motion with initial position, it follows that Q C v s/l l can be made arbitrarily small with high probability as δ becomes small. Furthermore, à conveges to a continuous lmiit process and so the terms à v à v l l and à v l l à s l l 1

become arbitrarily small with high probability as δ becomes small. The constant terms all converge to, and, because we have shown the convergence in 42 in Lemma 4, which implies 15 remains valid in this setting, Ñ τ weakly converges to. Finally, because we have already shown condition B16.17 is satisfied, the term γv s sup t T Q t becomes arbitrarily small with high probability as δ becomes small. Proof of 16 and 17 Lemma 3 equivalent As in the proof of Lemma 3, it is sufficient to show that as P I αw O 1 αw I + αw O Q and P O 1 αw I 1 αw I + αw O Q. For the inline queue, note that the number of batches required to serve all customers in the inline queue exceeds Q I t/n and is less than Q I t/n. Since each batch requires l time units to process and so Q l I t Q P n I t l I t, n PI t Q l I t l. n Since l as and by parts i and ii of this Proposition the weak convergence in Corollary 1 remains valid, l Q I n = l n Q I αw O 1 αw I + αw O Q. We conclude as. P I αw O 1 αw I + αw O Q Since whenever the number of customers in the offline queue exceeds 1 αn at a discrete review time point, at least 1 αn customers are served, Q O t 1 αn + 1 l exceeds the amount of time required for all customers in the offline queue that do not abandon to be served. Hence the number of customers in the offline queue that eventually do abandon 11

must be less than or equal to Therefore, A t N «Q t+ O t 1 αn +1 l N γq Osds. γq Osds Q l O t A t Q P n Ot l O t. n It follows from the observation that Q O t 1 αn + 1 l = Q O t 1 αµ + l as that 1 A as by identical argument as that in the proof of Lemma 3. As in the preceding paragraph, we conclude as. P O 1 αw I 1 αw I + αw O Q 12