Queueing systems with scheduled arrivals, i.e., appointment systems, are typical for frontal service systems,

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1 MANAGEMENT SCIENCE Vol. 54, No. 3, March 28, pp in ein inform doi /mnc INFORMS Scheduling Arrival to Queue: A Single-Server Model with No-Show INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(. Additional information, including right and permiion policie, i available at Refael Hain, Sharon Mendel School of Mathematical Science, Tel Aviv Univerity, Tel Aviv 69978, Irael {hain@pot.tau.ac.il, haronmendel@hotmail.com} Queueing ytem with cheduled arrival, i.e., appointment ytem, are typical for frontal ervice ytem, e.g., health clinic. An apect of cutomer behavior that influence the overall efficiency of uch ytem i the phenomenon of no-how. The conequence of no-how cannot be underetimated; e.g., Britih urvey reveal that in the United Kingdom alone more than 12 million general practitioner (GP appointment are mied every year, coting the Britih health ervice an etimated 25 million annually. In thi tudy we anwer the following key quetion: How hould the chedule be computed when there are no-how? I it ufficiently accurate to ue a chedule deigned for the ame expected number of cutomer without no-how? How important i it to invet in effort that reduce no-how i.e., given that we apply a chedule that take no-how into conideration, i the exitence of no-how till cotly to the erver and cutomer? Key word: appointment cheduling; health care; no-how; queue; ingle erver Hitory: Accepted by Michael Fu, tochatic model and imulation; received October 6, 26. Thi paper wa with the author 4 1 month for 4 reviion. Publihed online in Article in Advance January 25, Introduction Queueing ytem with cheduled arrival are known a appointment ytem. A better-deigned appointment ytem can reduce waiting time for cutomer and increae the utilization of expenive peronnel and other reource. An important apect of cutomer behavior that influence the overall efficiency of uch ytem i the phenomenon of no-how. Appointment ytem of all kind experience diappointment incidence on a regular bai. Empirical tudie of no-how cited by Cayirli and Veral (23 indicate that in many health clinic the volume of no-how trongly affect the ytem performance. Some medical webite (Medical New Today 24 quote figure of up to about 4% of no-how. A Britih urvey publihed in 23 (BBC New 23 reveal that in the United Kingdom alone more than 12 million general practitioner (GP (family doctor appointment are mied every year, coting the Britih health ervice an etimated 25 million annually. A urvey performed by KPMG Conulting upport thee figure (New Health Network 22. A urvey publihed in 26 (Developing Patient Partnerhip 26, Guardian Unlimited 26 reveal that thee figure have not decreaed over the year, and over 11 million GP appointment, and jut over 5 million practice nure appointment, are mied every year jut in the United Kingdom. Source in the American Air Force are alo found to complain about no-how (ee American Air Force 26. The above i jut a drop in the ocean of practical evidence that nohow are a crucial factor in the efficient performance of appointment ytem, and pecifically in the performance of national health ervice (NHS all over the world. Neverthele, depite the extenive work done on appointment ytem, the iue of no-how ha been tudied very little. The firt to preent an extenive work dealing with appointment cheduling of outpatient ervice wa Bailey (1952. A literature review on appointment policie can be found in the work of Mondchein and Weintraub (23. Cayirli and Veral (23 provide a comprehenive urvey of reearch on appointment cheduling in outpatient ervice. They claify work by the reearch methodology ued: analytical, imulation-baed, and cae tudy. A more general bibliography of queue in health care wa provided by Preater (21. A a bae point for our tudy, we ue an analytical model for cheduling arrival to queue where all cutomer how up (Pegden and Roenhine 199. We alo refer to Stein and Cote (1994, who baed their tudy on Pegden and Roenhine (199. They ue the bae model to tudy larger ytem and alo to tudy and compare the reult obtained when adding a contraint of equally paced cheduled appointment. The firt to addre appointment ytem with nohow in an analytic approach wa Mercer, who tudie the nonequilibrium ditribution of the queue length and alo give the reult for the equilibrium 565

2 Hain and Mendel: Scheduling Arrival to Queue: A Single-Server Model with No-Show 566 Management Science 54(3, pp , 28 INFORMS INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(. Additional information, including right and permiion policie, i available at ditribution (ee Mercer 196, More recent are the work of Hutzchenreuter (25 and of Kaandorp and Koole (26. The former compare the performance of a election of appointment-cheduling rule under D + noie/m/1 model, uing imulation for olving intance of them. Kaandorp and Koole (26 define a local earch cheduling algorithm and prove that it converge to the optimal chedule with repect to a weighted um of the average expected waiting time of cutomer, idle time of erver, and tardine in chedule. The algorithm i flexible and can incorporate no-how. Thi paper analyze the cheduling of finite appointment ytem with no-how by providing a mathematical formulation of uch a model with a ingle erver and analyzing the effect of no-how on the performance of the ytem. We aim to anwer the following key quetion: How hould the (optimal chedule be computed when there are no-how? How i the efficiency of the ytem affected if the cheduled arrival are retricted to be equally paced? How important i it to reduce no-how, i.e., given that we apply a olution that take no-how into conideration, i the exitence of no-how till cotly to the erver and cutomer? Our reult indicate that no-how greatly affect the chedule and hould be taken into account when deigning an appointment ytem. We tudy two type of appointment ytem: In 2, time interval between cheduled arrival are not necearily equal. A cutomer can be cheduled to arrive at any time after, or even at the ame time, a the cutomer cheduled to arrive before her. In 3, we tudy ytem with equal time interval between cheduled arrival. We tudy each type of ytem eparately, and alo compare the reult obtained for each type. Further reult and detail can be found in Mendel ( General Model We aume that each cutomer ha a probability of p of howing up, and if he how up he doe o exactly at hi cheduled arrival time. The objective i to determine a chedule for a fixed number of cutomer, minimizing the um of expected cutomer waiting cot and expected erver availability cot. There i a ingle erver to provide a Markovian ervice who remain available until the lat cutomer leave the ytem. The erver ha no prior knowledge of which cutomer will how up; hence, he ha to remain available at leat until the time the lat cutomer i cheduled to arrive. Cutomer are erved in the order of their cheduled appointment. Denote the cheduled arrival of the kth cutomer by t k, then if t i = t j, i<j, and both cutomer i and j how up, cutomer i i erved before cutomer j. We aume that ervice time follow an exponential ditribution and are identically ditributed. For appointment ytem in health ervice thi identical ditribution aumption i often reaonable becaue the extent of the ailment i not known at the time the appointment i made. A poible extenion for further reearch may be multiple cla model, where appointment are drawn from different ditribution. Such an extenion hould involve a different olution method, becaue there i a equencing problem involved Notation n: Number of cutomer to be cheduled. c w : Cutomer waiting cot per unit of time. c : Server availability cot per unit of time. : Relative cot meaure = c / c + c w. 1/ : Mean ervice time (ervice exponentially ditributed. x i : Time interval between the cheduled arrival time of the ith and the i + 1 t cutomer. x: A vector of interval between cheduled arrival x = x 1 x 2 x n 1. t i : Time of the ith cheduled arrival. t 1 = and t i = i 1 j=1 x j, i = 2 n. wi : Expected waiting time in the queue of the ith cutomer if he how up. w1 =. p: Probability of a cutomer to how up. N t i : Number of cutomer in the ytem jut prior to the time of the ith cheduled arrival. S n p /M/1: A queueing ytem with n cheduled independent cutomer, each howing up with probability p according to a pecified chedule S n p, and a ingle Markovian erver Objective Function The objective i to determine a vector x minimizing the um of the expected cutomer waiting cot and erver availability cot: 1 x = c w p = c w p n w i +c n w i +c ( n 1 x i +E erver time after t n [ n 1 x i +w n + p ] (1 In thi expreion we ue the fact that the expected erver time after t n i the um of the time he ha to erve cutomer who howed up before t n and are till in the ytem at t n, which i the amount of time the lat cutomer would have waited if he had howed up, and the expected ervice time of the lat cutomer if he how up. The objective function can be implified by omitting the contant c p/. We alo divide by c + c w p to obtain that the objective to be minimized i n 1 n 1 x = 1 w i + x i + w n (2 i=2

3 Hain and Mendel: Scheduling Arrival to Queue: A Single-Server Model with No-Show Management Science 54(3, pp , 28 INFORMS 567 INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(. Additional information, including right and permiion policie, i available at where = c / c + c w p i the relative erver availability cot, or obtained by uing the relative cot meaure = / + p 1. To evaluate the effect of no-how only on the variable cot, i.e., iolating the expected ervice time from the objective becaue it i inevitable and moreover a contant that doe not depend on the chedule, we define x a the expected total cot of waiting in the ytem. It conit of the um of the expected cot of cutomer waiting in the queue and the idle time of the erver. x can be obtained from the objective function (1 by x = 1 x E total ervice cot = 1 x c pn/. Rewriting and implifying the equation, we obtain x = x p n 1 ( Recurive Expreion for wi In thi ection we develop a general expreion for wi a a function of x. The expected waiting time in the queue for a cutomer who how up depend upon the number of cutomer he encounter upon arrival: wi i 1 j=1 j/ Pr N t i = j. The probability that there are j cutomer in the ytem at t i depend upon whether or not the i 1 th cutomer how up. For 1 j<iand i 2, i j 1 Pr N t i = j = p Pr N t i 1 = j + k 1 k= Pr k departure between t i 1 and t i i j p Pr N t i 1 = j + k k= Pr k departure between t i 1 and t i Becaue the ervice i Markovian, the probability of k departure between the i 1 t and the ith cheduled arrival (auming there are at leat k cutomer in the ytem at t i 1 i the probability of exactly k event in a Poion proce with the rate of. Thu, Pr N t i = j i j 1 = p Pr N t i 1 = j + k 1 x i 1 k k! k= i j p Pr N t i 1 = j + k x i 1 k k! i j 1 = k=1 ( p xi 1 k k= Pr N t i 1 = j + k 1 x i 1 k 1 k 1! + 1 p + p Pr N t i 1 = j 1 1 j<i i 2 Similarly, the probability that the ytem i empty jut prior to t i for i 2i Pr N t i = i 1 = p Pr N t i 1 = k 1 k=1 Pr time between t i 1 and t i uffice for at leat k departure i p Pr N t i 1 = k k= Pr time between t i 1 and t i uffice for at leat k departure i 1( x = p Pr N t i 1 = k 1 i 1 l l! k=1 l=k i 2( x + 1 p Pr N t i 1 = k i 1 l l! k= i 2 = Pr N t i 1 = k k= l=k ( k 1 x 1 i 1 l p x i 1 k l! k! l= 2.4. Solution Method A cloed-form olution for thi optimization model exit only for n = 2 cutomer, a cae of limited practical interet; for detail and reult ee Mendel (26. For larger ytem we mut obtain a olution numerically. The objective function ued by Pegden and Roenhine (199 i believed to be convex depite the appearance of nonconvex term, although no one ha been able to prove the convexity. Each member in the um that form the no-how model objective function (2 i a linear combination of a member of Pegden and Roenhine objective function; hence, (2 i convex if and only if Pegden and Roenhine objective function i convex. Baed on thi aumption, we obtain what i believed to be an optimal olution by applying equential quadratic programming (SQP; ee Mendel (26 for detail.. Solution Analyi.1. Schedule. A p decreae, appointment are cheduled cloer together. From a certain value, ome cutomer at the beginning of the chedule are cheduled to arrive at the ame time. The interval width increae for the firt few cutomer, then tay almot contant until it decreae for the lat few cutomer. With no-how, thi phenomenon expand, and a p decreae and the relative erver availability cot increae, not only are the firt few cutomer cheduled to arrive together, but o are the lat few cutomer. The latter phenomenon occur for relatively

4 Hain and Mendel: Scheduling Arrival to Queue: A Single-Server Model with No-Show 568 Management Science 54(3, pp , 28 INFORMS Figure 1 Interarrival Time S 1 7 /M/1 Figure 3 Expected Waiting Time S 5 9 /M/ INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(. Additional information, including right and permiion policie, i available at x i =.1 =.3 =.5 =.7 = i low value of p. For example, in a ytem with 1 cutomer, we begin to oberve thi pattern only for p 3 and 9. Figure 1 and 2 preent the pacing between cheduled arrival for variou value of and 1 cutomer. The interval number are given on the horizontal axi and the pacing between cheduled arrival are given on the vertical axi. Point i x i on a certain graph in thee figure i the value found for x i in the relevant model, i.e., the cheduled interarrival time between cutomer i and cutomer i + 1 for a ytem of S 1 p /M/1, with p and a tated in the figure. The intuitive explanation for thee pacing i imilar to that given by Stein and Cote (1994. The lat few cutomer are cheduled to arrive cloer, or even together, to avoid the erver being idle while only a few cutomer remain to arrive. The cheduling of the firt few cutomer to arrive cloe or even together fit what i known a Bailey Law (Bailey 1952, which Figure 2 x i Interarrival Time S 1 3 /M/1 =.1 =.3.5 =.5 =.7 = i w i recommend cheduling the firt cutomer together to later reduce the erver idle time..2. Cutomer Expected Waiting Time. Our objective function i a combination of expected cutomer waiting time and erver availability time. However, if the expected waiting time are too high, the ytem management may conider reevaluating the relative cot, o greater conideration would be given to waiting cot. Figure 3 and 4 preent expected waiting time of cutomer who how up, for variou relative erver availability cot. The variance of the expected waiting time of cutomer who how up i quite high, and the expected waiting time of a cutomer who how up increae a hi cheduled place in line increae, i.e., w i+1 >w i i = 1 n 1. Thi could be explained by the queue dicipline. A more cutomer are cheduled to arrive together, the more they wait if more than one of them arrive. We alo note that the Figure 4 w i Expected Waiting Time S 5 7 /M/ w 5 w 4 w 3 w 2 w 5 w 4 w 3 w 2

5 Hain and Mendel: Scheduling Arrival to Queue: A Single-Server Model with No-Show Management Science 54(3, pp , 28 INFORMS 569 INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(. Additional information, including right and permiion policie, i available at expected waiting time for cutomer who how up increae with p, i.e., if fewer cutomer are expected to arrive, the cutomer who how up are le likely to wait due to ervice of earlier cutomer. Neverthele, thee cutomer are waiting much longer than they would if they were erved by a ytem that erve the ame expected number of cutomer and i deigned and operate a an appointment ytem where all cutomer how up..3. Quantifying the Impact of No-Show on Cutomer Expected Waiting Time. Comparing the expected waiting time for cutomer who how up for a S n p < 1 /M/1 model to the S n 1 /M/1 model where np = n, we find that if all cutomer who do not how up would have notified in advance that they were not going to how up, a chedule could have been deigned with maller expected waiting time for thoe who how up. Figure 5 and 6 each preent a comparion between the expected waiting time for cutomer who how up in a S n p < 1 /M/1 model and the ame expected waiting time of the equivalent S n 1 /M/1 model where np = n. In thee graph the relative erver availability cot,, i given on the horizontal axi, and the expected waiting time meaure of cutomer who how up are given on the vertical axi. The waiting time of the model with nohow are drawn in olid line, wherea the one of the model where all cutomer how up are drawn in dahed line. We note that a p decreae, the impact of the no-how increae. If a large portion of cutomer do not how up, the expected waiting time of the cutomer who do how up i much higher than the expected waiting time they would have had if the chedule wa originally deigned only for them. The influence of the no-how phenomenon i greater Figure 5 Show waiting time Waiting Meaure Comparion S /M/1 and S 5 1 /M/1 w max 1. w max w mean.5 w mean Figure 6 Show waiting time Waiting Meaure Comparion S 1 5 /M/1 and S 5 1 /M/1 w max w max w mean.5 w mean for maller relative erver availability cot, meaning that the impact of no-how i more evere when the importance given to cutomer waiting time, to minimize them, i higher..4. Objective Function Value. For high, the minimum value found for the objective function for p = 1, where all cutomer how up, i higher than for ome other howing up probabilitie. Auming the relative cot i given and a ytem operating cot are all repreented in (2, the operating cot can be brought to a lower expected value by manipulating a pecific p portion of the cutomer to how up..5. Operational Cot per Cutomer. Two other meaure of the ytem that are of much interet concern operational cot per cutomer. Thee meaure are defined with repect to x and x by dividing them by np. x /np i a meaure for ytem waiting cot per cutomer, wherea x /np alo take into account the ervice cot. Figure 7 and 8 preent the value of thee meaure a a function of p for variou value of. For both meaure, the maximum value they obtain i not necearily when all cutomer how up. When comparing the behavior of the total cot with thoe of the cot per cutomer, we find that their pick are not obtained at the ame howing-up probability, and moreover, the pick of the operational meaure graph are obtained for a lower p. Thi may imply that cutomer who how up pay more if many of the cheduled cutomer do not how up..6. Cot ofno-show. To evaluate the cot of no-how, we compare the cot of ytem where all cutomer how up to ytem with no-how with the ame expected number of cutomer who how up.

6 Hain and Mendel: Scheduling Arrival to Queue: A Single-Server Model with No-Show 57 Management Science 54(3, pp , 28 INFORMS Figure 7 x per Showing Cutomer S 1 p /M/1 = 7 Figure 9 Cot Comparion S /M/1 and S 5 1 /M/ Φ INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(. Additional information, including right and permiion policie, i available at Φ(x/np p Figure 9 and 1 each preent a comparion between the cot of a S n p < 1 /M/1 model and the cot of the equivalent S n 1 /M/1 model where np = n. In thee graph the relative erver availability cot i given on the horizontal axi, noted by, and the cot are given on the vertical axi. The cot of the model with no-how are drawn in olid line, wherea the cot of the model where all cutomer how up are drawn in dahed line. We note that a the probability of cutomer not howing up increae, the cot of no-how increae. The influence of the phenomenon on the cot i greater on a maller relative erver availability cot. Thee finding conform with the finding detailed in.3 for the expected waiting time. Neverthele, the impact of no-how on the expected waiting time, meaured in percentage, i not a high. Thu, the no-how alo increae the erver idle time. Figure 8 Ω(x/np x per Showing Cutomer S 1 p /M/1 = 3 Cot Ω The Equally Spaced Model Following the work of Stein and Cote (1994, we alo conider the cae where the cutomer are cheduled to arrive at the ytem at equally paced time, i.e., x 1 = x 2 = = x n 1. We add to their model the attribute that each cutomer how up with a probability p 1. A tated by Stein and Cote, the equally paced model i of interet becaue it provide a realitic retriction to the cheduling problem, becaue in mot appointment ytem the appointment are cheduled uing a fixed interval between cheduled arrival Solution Method The equation repreenting our model remain the ame, with the exception that x i = x eq i. Thu, baed on (2 our objective i to minimize Figure 1 Cot Ω n 1 eq x eq = 1 w i + n 1 x eq + w n (4 i=2 Φ Cot Comparion S 1 5 /M/1 and S 5 1 /M/1 Φ Ω Φ Ω p

7 Hain and Mendel: Scheduling Arrival to Queue: A Single-Server Model with No-Show Management Science 54(3, pp , 28 INFORMS 571 INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(. Additional information, including right and permiion policie, i available at The olution method are imilar to thoe ued in the unretricted model. Even though we are now looking for the value of a ingle variable x eq that optimize the objective function (4 (a oppoed to a vector of value, there i till no cloed-form olution. Hence, a in the unretricted model, we obtain the olution numerically for the implified cae of = 1 by uing SQP method Analyi ofthe Equally Spaced Solution A found by Stein and Cote for the baic model, we alo find that for a ytem with no-how, the effect of adding a contraint to force equally paced interval doe not materially change the value of the objective function for any combination of p and, and determine an interval that i approximately the average of the equivalent unretricted model. Alo, the expected waiting time in the equally paced model behave in a imilar manner to the expected waiting time in the unretricted model, and have imilar value Relation Between the Equal Spacing Solution and the Showing-Up Probability. From a practical point of view, it i of interet to tudy the relation between the equal pacing olution and the howingup probability. Auming a given erver availability relative cot, the equal pacing olution depend upon the howing-up probability. We find that for a wide range of relative cot value there eem to be an almot linear relation between the equal pacing olution and the howing-up probability. Thi can be noted in Figure 11 and 12, which preent, for a given number of cutomer, the equal pacing olution for given relative cot a a function of the howing-up probabilitie No-Show Impact on the Equal Spacing Solution. A comparion between the equal pacing olution of ytem where all cutomer how up Figure 11 x eq Equal Spacing Solution S 5 p /M/1 =.1 =.3 =.5 =.7 = p Figure 12 x eq Equal Spacing Solution S 1 p /M/1 =.1 =.3 =.5 =.7 = p to ytem with no-how with the ame expected number of howing-up cutomer, reveal that the pacing with no-how i maller, and it decreae a the howing probability p decreae. Figure 13 and 14 compare the equal pacing olution of S n p < 1 /M/1 model and the correponding S n 1 /M/1 model where np = n. The relative erver availability cot,, i given on the horizontal axi, and the pacing i given on the vertical axi. The pacing with no-how are drawn in olid line, wherea thoe of the model where all cutomer how up are in dahed line. If cutomer would have notified in advance that they were not going to how up, a more paciou chedule could have been deigned for thoe who do how up. Thi align with the impact of no-how on the expected waiting time and cot, a detailed in.3 and.6. Due to expected no-how, the chedule i more condened, with longer expected waiting time leading to Figure 13 x eq np = 8 Comparion x eq p =.8 p =

8 Hain and Mendel: Scheduling Arrival to Queue: A Single-Server Model with No-Show 572 Management Science 54(3, pp , 28 INFORMS INFORMS hold copyright to thi article and ditributed thi copy a a courtey to the author(. Additional information, including right and permiion policie, i available at Figure 14 x eq np = 3 Comparion x eq p =.3 p =.375 p =.6 p = higher cot than there would have been if the ytem wa deigned only for the cutomer who do how up eventually. We alo note that the impact of nohow on the pacing i le for extreme relative ervice availability cot, i.e., the impact when i very mall or very large i not a notable a it i for intermediate value of. In thee extreme cae the chedule i highly influenced by the relative cot; hence, the impact of no-how i not a ignificant. Acknowledgment Thi reearch wa upported by the Irael Science Foundation (Grant 237/2. Reference American Air Force. 26. No-how for appointment wate time, money. Retrieved July 1, 27, mil/new/tory.ap?toryid= Bailey, N A tudy of queue and appointment ytem in hopital outpatient department with pecial reference to waiting time. J. Roy. Statit. Soc BBC New. 23. GP back no how fine. Retrieved July 1, 27, Cayirli, T., E. Veral. 23. Outpatient cheduling in health care: A review of literature. Production Oper. Management Developing Patient Partnerhip. 26. Mied appointment increaing depite ame day appointment. Retrieved July 1, 27, Guardian Unlimited. 26. Unwelcome break. Retrieved July 1, 27, Hutzchenreuter, A. 25. Queueing model for outpatient appointment cheduling. Thei, Faculty for Mathematic and Economic, Ulm Univerity, Germany. Kaandorp, G., G. Koole. 26. Optimal outpatient appointment cheduling. Working paper, Department of Mathematic, Vrije Univerity, Amterdam. Medical New Today. 24. Why 42% of clinic appointment are no-how: An iue of repect. Retrieved July 1, 27, newid= Mendel, S. 26. Scheduling arrival to queue: A model with no-how. M.Sc. thei, Department of Statitic and Operation Reearch, Tel Aviv Univerity. Retrieved July 1, 27, Mercer, A A queueing problem in which the arrival time of the cutomer are cheduled. J. Roy. Statit. Soc., Ser. B Mercer, A Queue with cheduled arrival: A correction, implification and extenion. J. Roy. Statit. Soc., Ser. B Mondchein, S. V., G. Y. Weintraub. 23. Appointment policie in ervice operation: A critical analyi of the economic framework. Production Oper. Management New Health Network. 22. What patient really think of the NHS. Retrieved July 1, 27, template3.ap?id=41. Pegden, C. D., M. Roenhine Scheduling arrival to queue. Comput. Oper. Re Preater, J. 21. A bibliography of queue in health and medicine. Keele Mathematic Reearch Report 1-1, Keele Univerity, Staffordhire, UK. Stein, W. E., M. J. Cote Scheduling arrival to queue. Comput. Oper. Re