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Analysis of passenger queues at airport terminals
Research in Transportation Business & Management 1 (2011) 144–149
Contents lists available at ScienceDirect
Research in Transportation Business & Management
Analysis of passenger queues at airport terminals Raik Stolletz University of Mannheim, Business School, Chair of Production Management, Schloss, 68131 Mannheim, Germany
a r t i c l e
i n f o
Article history: Received 19 January 2011 Received in revised form 22 June 2011 Accepted 29 June 2011 Available online 27 July 2011 Keywords: Passenger queues Check-in Economies of scale
a b s t r a c t This paper presents an approach for analyzing passenger queueing processes at several stages of the service chain in airport terminals with a focus on passenger check-in systems. The queueing process at the check-in is driven by the arrival process of the passengers and the service process, including the desired technical and personnel resources. The key features of these systems are the time-dependent process of arrivals, generally distributed processing times, and systems with multiple servers. Time dependent service quality measures are analyzed, which are related to managerial decisions about the required number of check-in operators. Different analytical methods are reviewed to derive service quality measures for such dynamic and stochastic systems. Due to the dynamics in the demand and in the provided capacity, the use of standard queueing models is rather limited. A stationary backlog carryover (SBC) approach is developed to approximate the relevant performance measures. Numerical experiments demonstrate the reliability of these different approaches. Based on economies of scale, positive effects for common check-in operations can be shown. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Airline passengers queue up at several steps during the passenger processing at the departure airport, the transfer airport, and the arrival airport. The check-in is the first step in the service process chain at passenger airports, followed by the security check. If the passengers do not have a direct flight, they change the plane at one or more transfer airports. Additional security checks may be mandatory. Depending on their nationality, passengers have to pass through the immigration office at the first airport they enter a country or at their final destination airport. Operational decisions for these queueing processes involve the determination of the number of check-in operators (or check-in counters) and the number of offered security stations over the course of a day. The service offered to the passengers is measured by the mean queue length, the mean waiting time, or the service levels that a certain percentage of passengers wait less than a given threshold (see Graham (2003)). This paper analyzes the queueing process at the check-in in greater detail. Nevertheless, the proposed approach for the performance analysis is applicable as well to any of the queues described above. The check-in service could consist of assigning seats to the passenger, printing out the boarding pass, and handling the baggage drop-off. The distribution of the processing time depends on many factors, for example staff experience and passenger characteristics (see Brunetta, Righi, & Andreatta (1999)). Many airlines provide self-service check-in (SSCI) to reduce the amount of time a check-in agent needs to handle one passenger. If passengers are traveling with cabin luggage only and
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are using the self-service facilities, there is no further need for queueing up at the check-in counter. Some airlines require that passengers print out their own boarding pass before they are allowed to queue at the check-in desk. Compared to the conventional check-in, SSCI reduces the demand for check-in operations on the one hand and may reduce the average processing time on the other hand. This results in lower overall operator requirements for SSCI compared to the conventional check-in. Fig. 1 shows the flow chart of the check-in process. The check-in service could be provided for single or multiple flights of an airline (common check-in). Some airlines have their own ground handling staff to offer such services, whereas other airlines hire a ground handling company, especially when the number of flights from an airport is small. Based on the distribution of the arrival rate of passengers over time and target service measures, the requirement of check-in operators has to be set for each time interval. Two kinds of setting the timedependent requirements for check-in agents could be distinguished: based on contracts and based on service level agreements. 1. In the first case, the number of operators needed is predetermined by contracts between the airline and the ground handler for each flight. In the case of a check-in for a single flight, the contract is based on the planned time of departure. The check-in starts at a specified amount of time before the take-off time and closes, for example, 30 min prior to the departure. During this time interval the number of agents can be time-dependent and depends additionally on • the number of passengers, • the airports of departure and destination, and • the time of the day.
R. Stolletz / Research in Transportation Business & Management 1 (2011) 144–149
145
Passenger arrives
Airline provides SSCI?
no
yes
SSCI is mandatory
no
Passenger picks SSCI
yes
no
yes
Passenger picks seat and prints boarding pass
yes
Cabin luggage only? no
Baggage drop-off counter?
no
yes Passenger queues up in baggage drop-off line
Passenger queues up in check-in line
Passenger proceeds to security check Fig. 1. Flow chart of the check-in process.
The planned profile of open check-in counters is usually experience based and not justified according to any goal on the waiting time at the check-in. On the one hand, a high number of operators results in short passenger waiting times and in a low utilization of the operators. On the other hand, if the number of operators is too low, the utilization is high and the customers have to queue up.
number of possible profiles, a fast performance analysis is necessary to evaluate the performance measures for these profiles. This is a prerequisite to analyze the tradeoff between the utilization of the operators and the service quality offered to the passengers. To this end, the check-in system needs to be analyzed as a time-dependent and stochastic queueing system. The main contribution of this paper is twofold:
2. In the second planning situation the requirements are based on service level agreements between the ground handler and the airline. In this case the ground handler has more freedom to assign the operators to different check-in stations, based on some bounds on the performance measures mentioned above.
• An approximation method for analyzing the performance measures for time-dependent and stochastic check-in systems is developed. Numerical examples show that the proposed approach overcomes traditional approximation methods. • The analysis of the tradeoffs between the utilization of check-in operators and the service provided to the passengers shows the influence of the distribution of processing times and economies of scale via common check-in operations.
In addition to bounds on performance measures given by the airlines, there could also be some limitations of queue lengths due to security reasons. The managerial decisions are concerned with defining operator profiles with respect to given bounds on the performance measures in both planning situations. Due to the large
The paper is organized as follows: Section 2 presents a literature review. The considered model of the check-in process is described in
146
R. Stolletz / Research in Transportation Business & Management 1 (2011) 144–149
Section 3. The approximation method is developed in Section 4. The numerical study in Section 5 compares the proposed approach to existing ones. In doing so, the influence of the distribution of the processing time is studied and economies of scale of a common checkin are shown. Section 6 discusses the results and concludes with suggestions for further research. 2. Literature review Operational research is widely used to support decisions in planning and operating aviation activities, see the overview in Barnhart, Belobaba, and Odoni (2003). In analyzing the requirements for check-in operations, the most challenging is to take into account the time-dependent arrivals of passengers. Therefore the review concentrates on a discussion of these distributions and gives an overview on approaches for the performance approximation of timedependent and stochastic systems. The arrival rate of the passengers of a single flight depends on the scheduled departure time and on the flight itself. The distribution of the arrival rate over time (also called passenger earliness of arrival profile, see Barros and Tomber (2007)) depends for example, on the destination, on the flight category (business or charter), or on the time of the day. Thus for example, the mean arrival time in the morning is much shorter than in the afternoon (see Robertson, Shrader, Pendergraft, Johnson, and Silbert (2002)). Common for all the different distributions is that they are strongly dynamic and cannot be replaced by average values. The other main characteristic of the arrival process is the stochasticity. Fig. 2 shows an example for a passenger earliness of arrival profile (Robertson et al., 2002). For longhaul flights passengers usually arrive earlier and this distribution is shifted to the left. The earliness of arrival profile also depends on the kind of passenger. For example, 71% of business travelers at Frankfurt Airport do not arrive earlier than 2 h before the scheduled departure time, while only 45% of leisure travelers arrive within 2 h before departure (Media Frankfurt GmbH, Werbemonitor 2010). Several general approaches for the performance approximation of time-dependent queueing systems are discussed in literature (see the overviews in Ingolfsson, Akhmetshina, Budge, Li, and Wu (2007) and Stolletz (2008a)). In general, ordinary differential equations can be solved (Kleinrock, 1975) or the randomization approach could be applied (Grassmann, 1977) for Marcovian systems. Both approaches are not suitable for queues with generally distributed arrival and service processes. Similarly, fluid approximations (Mandelbaum &
20% 18%
% passengers arrived
16% 14%
Massey, 1995) and infinite server approximations (Ingolfsson et al., 2007) are not able to handle generally distributed service and interarrival times. Fluid approximations are often called cumulative diagrams in airport literature. Tosic (1992) analyzed different queueing models for airport terminals and used cumulative diagrams to model the check-in facilities. Janic (2003) modeled the check-in process using cumulative diagrams, too. Barros and Tomber (2007) analyze the passenger and baggage screening with cumulative diagrams. Brunetta et al. (1999) argued that stationary stochastic queueing models could not be applied because of the importance of the dynamic effects and therefore apply cumulative diagrams to analyze different processing facilities. In general, approximations with stationary approaches are often used to approximate the performance of time-dependent service systems, see for example the Stationary Independent Period by Period (SIPP) approach (Green, Kolesar, & Soares, 2001) and the Stationary Backlog Carryover (SBC) approach (Stolletz, 2008a). Numerical experiments show that the SBC approach outperforms the SIPP approach for the M(t)/M/c(t) queue (Stolletz, 2008a). The performance of single server M(t)/G(t)/1 systems with generally distributed processing times is well approximated by the SBC approach (Stolletz, 2008b). Therefore this paper extends the SBC approach to the analysis of M(t)/G/c(t) loss-waiting queues with time dependent inter-arrival times, generally distributed processing times, and time-dependent numbers of servers.
3. Description of the check-in model This paper focuses on a basic queueing model for a check-in system with a single queue and a time-dependent number c(t) of check-in counters, as depicted in Fig. 3. This M(t)/G/c(t) system assumes an inhomogeneous Poisson arrival process with instantaneous arrival rates λ(t). The service time for a passenger corresponds to a generally distributed random variable with rate μ and the squared coefficient of variation CV 2. The number of waiting positions is assumed to be infinite. Waiting customers are served according to the First-Come-First-Serve (FCFS) policy. Time-dependent performance measures of interest are the expected utilization, the expected queue length, and the expected waiting time. Many airlines provide separate check-in for economy class, business class, and first class. If there is a strict separation, a passenger of a certain class could be only served by the dedicated group of operators. The model described above applies for each class because they do not influence each other. Some check-in queues are organized in a way that check-in agents of a higher class (for example business class) are allowed to serve the next waiting customer of a lower class (economy class), if no other high-class passenger is waiting. In this case the models cannot be analyzed independently. Such models with heterogeneous customers, heterogeneous operators, and priorities are called X-type or N-type models. They are analyzed for other service operations in steady-state, for example call centers (see Stolletz and
12% 10%
Check-in Counter
8%
1 1
6% Queue
4%
Arriving passenger
λt
2% 0%
150 140 130 120 110 100 90 80 70 60 50 40 30 20 10
∞
2 1
0
Time before departure [min] Fig. 2. Passenger earliness of arrival profile. Source: Robertson et al. (2002).
ct Fig. 3. Basic queueing model for the check-in.
R. Stolletz / Research in Transportation Business & Management 1 (2011) 144–149
Helber (2004)). The SBC approach could be extended to X-type or N-type models. However, in order to show the characteristics of timedependent and stochastic check-in systems the analysis is restricted to models with homogeneous operators and homogeneous passengers.
approximation results directly in the backlog rates bi and the expected utilizations E[Ui] for each period i with bi = ˜λi ⋅Pi ðBÞ = ˜λi ·
4. Performance analysis approach
c
(i) The approximated utilization E[Ui] for period i is analyzed using a stationary M/G/ci/ci loss model, which yields an artificial blocking probability Pi(B). A backlog bi is generated from these (artificially) blocked customers. The backlog rate bi is given by ˜ i ·Pi ðBÞ: bi = λ
ð1Þ
This number of artificially blocked customers is carried over as additional arrivals in the subsequent period i + 1. For example, ˜ i + 1 of period i + 1 is built by the artificial arrival rate λ + 1
= λi
+ 1
+ bi = λi
+ 1
˜λ = μ ci i
ci !∑ki = 0
In the SBC-approach the time horizon is divided into small periods i of equal length l with constant arrival rates λi for each period i. This approach was developed for M(t)/M(t)/c(t) and M(t)/G/1 queueing systems with an infinite waiting room and Poisson arrivals, see Stolletz (2008a) and Stolletz (2008b). The basic idea of this approximation procedure is divided into three steps for each period i: (i) the approximation of the utilization, (ii) the derivation of a modified arrival rate, and (iii) the approximation of waiting-based performance measures using a stationary M/G/c-approximation. In the first step each period is analyzed with a loss queueing model using ˜ i . The loss models of consecutive periods are an artificial arrival rate λ connected via an artificial backlog, which equals the (artificial) blocking rate of the preceding period. To derive the artificial arrival rate of the subsequent period, this backlog rate is added to the original arrival rate, see Fig. 4. This step results in an approximation of the utilization. Based on this utilization, a modified arrival rate is derived in the second step such that a stationary waiting system would reach the same expected utilization. In the third step, waiting based performance measures are approximated using this modified arrival rate and a stationary approximation. To approximate an M(t)/G/c(t) queueing model for the check-in the SBC approach is extended in the following way. Starting with the first period the following steps are applied for each period i:
˜i λ
147
+ ˜λi ·Pi ðBÞ
ð2Þ
˜ 1 = λ1 . Applying Erlang's loss formula and is initialized with λ for Pi(B) (see for example Gross and Harris (1998)), the SBC
E½Ui =
˜ λi = μ k!
k and
˜λ ð1−P ðBÞÞ ˜λ −b λ + bi−1 −bi i i i = i = i : ci μ ci μ ci μ
ð3Þ
ð4Þ
(ii) For the approximation of waiting-based performance measures a stationary M/G/ci/∞ waiting model is used. In order to obtain approximations of the performance measures of interest, the modified arrival rate λiMAR serves as an input to the stationary queueing model. This modified arrival rate λiMAR is set such that the approximated utilization E[Ui] of step (i) is achieved. In the case of an M(t)/G/c(t) queue this results in MAR
λi
= ci ·μ·E½Ui :
ð5Þ
(iii) An approximation for the performance measures for a stationary M/G/ci queue is applied with the modified arrival rate λiMAR. This approach can be combined with any stationary approximation. According to Bolch, Greiner, de Meer, and Trivedi (1998, p. 233) the approximation of Cosmetatos (1976) gives very good results for different squared coefficients of variation of the processing times CV 2. The approximation of the expected waiting time E[WM/G/ci] is based on the expected waiting time E[WM/M/ci] of the M/M/ci queue and the expected waiting time E[WM/D/ci] of the M/D/ci queue. Exact derivations of performance measures are known for both queue types, see for example Bolch et al. (1998). The approximation of Cosmetatos (1976) leads to h i h i h i 2 2 E WM = G = ci ≈CV E WM = M = ci + 1−CV E WM = D = ci :
ð6Þ
Using Little's law the expected queue length is given by h i h i MAR E QM = G = ci = λi E WM = G = ci :
ð7Þ
Algorithm 1 shows the whole procedure. The performance measures of interest for time t are approximated by the values of the respective period. Numerical experiments in Stolletz (2008a) indicate that the average processing time is a good period length, i.e., l = μ − 1 is used. Algorithm 1. SBC-approach for the M(t)/G/c(t) queue
Fig. 4. Idea of the backlog carry over of the SBC approximation (step (i)).
1: Input: λi, μ, ci, CV2 2: Initialize: i = 0 ; b0 = 0 3: for i = 1 to I do ˜ i = λi + bi−1 4: λ 5: procedure M = G = ci = ci ˜λi ; μ; ci 6: return Pi(B), E[Ui] 7: end procedure 8: bi = ˜λi ·Pi ðBÞ 9: λiMAR = ci · μ · E[Ui] 10: procedure M/M/ci/∞(λiMAR, μ, ci) 11: return E[WM/M/ci] 12: end procedure 13: procedure M/D/ci/∞(λiMAR, μ, ci) 14: return E[WM/D/ci] 15: end procedure 16: E[WM/G/ci] = CV2E[WM/M/ci] + (1 − CV2)E[WM/D/ci] 17: E[QM/G/ci] = λiMARE[WM/G/ci] 18: end for
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250
Table 1 Arrival rates of the base case.
1
0–30 0.5
30–60 1.3
60–90 1.9
60–120 1.5
SBC SIPP Sim
0.9
200
E[W] (in sec.)
Time period Arrival rate
0.8
SBC (SCV 1) SBC (SCV 0) Sim (SCV 1) Sim (SCV 0)
150
100
0.7 50
E [U]
0.6 0.5
0
0.4
0
1000
2000
3000
4000
5000
6000
7000
time [sec.]
0.3 0.2
Fig. 7. Approximated expected waiting time for the base case with different CV2.
0.1 0
0
1000
2000
3000
4000
5000
6000
7000
time [sec.] Fig. 5. Approximated expected utilization for the base case.
5. Numerical study To show the reliability of the SBC approach the results are compared to the SIPP approach and to the fluid approach. The timedependent expected utilization and queue length are compared to simulation results with 20,000 replications. In the base case check-in for a single flight with c = 2 agents and exponentially distributed processing times of μ − 1 = 1 min is assumed. The arrival rates per minute are given on a 30-minute basis with the values given in Table 1. Figs. 5 and 6 show the expected utilization and the expected queue length for the SBC approach, the SIPP approach, and the simulation. Both measures are well approximated by the SBC approach. Due to the underlying assumption of stationarity in the SIPP approach, the measures do not change within a 30-minute period. The steady-state will be reached at the end of such an interval, but not at its beginning. In the first minutes the steady-state is not reached and the SIPP approach significantly overestimates the utilization and the queue λ length. Due to the assumption of ρ = i <1 the fluid approach ci μ h i λ results in an expected utilization of E Uifluid = i , which equals the ci μ steady-state result of the SIPP approach. The queue length in the fluid 18
approach is approximated with zero in each period and therefore not shown in the figures. The influences of the distribution of the processing time are manyfold. Obviously, a reduction of the mean processing time, for example due to self-service facilities, reduces the utilization, the queue length, and the mean waiting time. Furthermore, a reduction of variation has a significant influence on the performance measures. Fig. 7 shows the expected waiting time for the base case with a squared coefficient of variation of 1 (exponentially distributed processing times) and 0 (deterministic processing times). Similar results could be shown for the expected queue length. To show economies of scale related to common check-in operations, a system with c = 10 (instead of c = 2) operators and with arrival rates that are five times higher as in the base case is analyzed. Figs. 8 and 9 show the expected queue length and the expected waiting time, which have the same shape due to Little's law. Compared to Figs. 6 and 7, the queue length for the larger system increases moderately, whereas the waiting time is reduced significantly in all periods. Another advantage of a common check-in for flights with different times of departures allows to smooth the demand and hence to smooth operator requirements. This results in better solutions in the following step of shift scheduling, see Stolletz (2010). To summarize the numerical study, it shows that the conventional approaches described in the literature are not able to approximate the performance of a time-dependent and stochastic check-in system. Its utilization is much better approximated using the SBC-approach. Especially the queue length and the waiting time are underestimated
16
SBC SIPP Sim
16
SBC SIPP Sim
14
14
12
12
E [Q]
E [Q]
10 10 8
8 6
6 4
4
2
2
0
0
1000
2000
3000
4000
5000
6000
time [sec.] Fig. 6. Approximated expected queue length for the base case.
7000
0
0
1000
2000
3000
4000
5000
6000
7000
time [sec.] Fig. 8. Approximated expected queue length for the common check-in.
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100
of the stochasticity inside the system and allows to create a smoother demand via adding up different earliness of arrival profiles. Further research is needed to integrate performance analysis into an optimization procedure for such systems. Another direction of further research is the analysis of check-in systems with self-service facilities, heterogeneous passenger groups, and heterogeneous groups of operators. While the check-in is only one part of the complex service network at an airport, the operations of succeeding steps should be integrated. For example, the influence of temporary shortfalls in the capacity of the baggage handling system should be analyzed.
SBC SIPP Sim
90 80
E[W] (in sec.)
149
70 60 50 40 30 20 10 0
References 0
1000
2000
3000
4000
5000
6000
7000
time [sec.] Fig. 9. Approximated expected waiting time for the common check-in.
with the widely used cumulative diagrams (fluid approximation). Based on the analysis of the variability of the service time and the study of economies of scale two major managerial implications can be stated. First, a reduction of the variability shows a significant improvement of the performance measures. Second, economies of scale can be gained by common check-in operations. To do so, running a larger system with the same utilization reduces the waiting time significantly on the one hand. On the other hand, common check-in systems are able to achieve a higher utilization of operators with a good quality of service. 6. Conclusion The service quality at check-in stations is mainly driven by the number of available check-in counters, the dynamic arrival rate of passengers, and the distribution of the processing time. With respect to the planning situations described in Section 1, the SBC approach yields a good approximation of performance measures and hence, it results in better operator requirement profiles in both planning situations. The developed SBC approach for the performance analysis is computationally tractable, i.e., it can be used to evaluate many different operator profiles within seconds. It outperforms the SIPP approach and the commonly used cumulative diagrams. Numerical experiments show that besides the mean processing time, the squared coefficient of variation has great impact on waiting-time based performance measures. Economies of scale can be achieved through the implementation of a common check-in. This reduces the influence
Barnhart, C., Belobaba, P., & Odoni, A. R. (2003). Applications of operations research in the air transport industry. Transportation Science, 37(4), 368–391. Barros, d. A. G., & Tomber, D. D. (2007). Quantitative analysis of passenger and baggage security screening ar airports. Journal of Advanced Transportation, 41(2), 171–193. Bolch, G., Greiner, S., de Meer, H., & Trivedi, K. S. (1998). Queueing networks and Markov chains: Modeling and performance evaluation with computer science applications. New York: John Wiley & Sons, Inc. et al. Brunetta, L., Righi, L., & Andreatta, G. (1999). An operations research model for the evaluation of an airport terminal: SLAM (simple landside aggregate model). Journal of Air Transport Management, 5, 161–175. Cosmetatos, G. (1976). Approximate equilibrium results for the multi-server queue (m/g/r). Operational Research Quarterly, 27(3), 615–620. Graham, A. (2003). Managing airports — An international perspective (2. ed). Elsevier. Grassmann, W. K. (1977). Transient solutions in Markovian queueing systems. Computers and Operations Research, 4(1), 47–53. Green, L. V., Kolesar, P. J., & Soares, J. (2001). Improving the SIPP approach for staffing service systems that have cyclic demands. Operations Research, 49(4), 549–564. Gross, D., & Harris, C. M. (1998). Fundamentals of queueing theory et al. (3 ed). New York: John Wiley & Sons. Ingolfsson, A., Akhmetshina, E., Budge, S., Li, Y., & Wu, X. (2007). A survey and experimental comparison of service level approximation methods for nonstationary M(t)/M/s(t) queueing systems INFORMS. Journal of Computing, 19(2), 201–214. Janic, M. (2003). Assessment and management of quality of service at an airport passenger terminal. Transportation Planning and Technology, 26(3), 239–263. Kleinrock, L. (1975). Queueing systems volume I: Theory. New York: John Wiley & Sons, Inc. Mandelbaum, A., & Massey, W. A. (1995). Strong approximations for time-dependent queues. Mathematics of Operations Research, 20(1), 33–64. Robertson, C., Shrader, S., Pendergraft, D., Johnson, L., & Silbert, K. (2002). The role of modeling demand in process re-engineering. In E. Yücesan, C. -H. Chen, J. Snowdown, & J. Charnes (Eds.), Proceedings of the 2002 Winter Simulation Conference (pp. 1454–1458). Stolletz, R. (2008). Approximation of the non-stationary M(t)/M(t)/c(t)-queue using stationary queuing models: The stationary backlog-carryover approach. European Journal of Operational Research, 190(2), 478–493. Stolletz, R. (2008). Non-stationary delay analysis of runway systems. OR Spectrum, 30(1), 191–213. Stolletz, R. (2010). Operational workforce planning for check-in counters at airports. Transportation Research Part E, 46, 414–425. Stolletz, R., & Helber, S. (2004). Performance analysis of an inbound call center with skills-based routing. OR Spectrum, 26, 331–352. Tosic, V. (1992). A review of airport passenger terminal operations analysis and modelling. Transportation Research Part A: Policy and Practice, 1, 3–26.
Contents lists available at ScienceDirect
Research in Transportation Business & Management
Analysis of passenger queues at airport terminals Raik Stolletz University of Mannheim, Business School, Chair of Production Management, Schloss, 68131 Mannheim, Germany
a r t i c l e
i n f o
Article history: Received 19 January 2011 Received in revised form 22 June 2011 Accepted 29 June 2011 Available online 27 July 2011 Keywords: Passenger queues Check-in Economies of scale
a b s t r a c t This paper presents an approach for analyzing passenger queueing processes at several stages of the service chain in airport terminals with a focus on passenger check-in systems. The queueing process at the check-in is driven by the arrival process of the passengers and the service process, including the desired technical and personnel resources. The key features of these systems are the time-dependent process of arrivals, generally distributed processing times, and systems with multiple servers. Time dependent service quality measures are analyzed, which are related to managerial decisions about the required number of check-in operators. Different analytical methods are reviewed to derive service quality measures for such dynamic and stochastic systems. Due to the dynamics in the demand and in the provided capacity, the use of standard queueing models is rather limited. A stationary backlog carryover (SBC) approach is developed to approximate the relevant performance measures. Numerical experiments demonstrate the reliability of these different approaches. Based on economies of scale, positive effects for common check-in operations can be shown. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Airline passengers queue up at several steps during the passenger processing at the departure airport, the transfer airport, and the arrival airport. The check-in is the first step in the service process chain at passenger airports, followed by the security check. If the passengers do not have a direct flight, they change the plane at one or more transfer airports. Additional security checks may be mandatory. Depending on their nationality, passengers have to pass through the immigration office at the first airport they enter a country or at their final destination airport. Operational decisions for these queueing processes involve the determination of the number of check-in operators (or check-in counters) and the number of offered security stations over the course of a day. The service offered to the passengers is measured by the mean queue length, the mean waiting time, or the service levels that a certain percentage of passengers wait less than a given threshold (see Graham (2003)). This paper analyzes the queueing process at the check-in in greater detail. Nevertheless, the proposed approach for the performance analysis is applicable as well to any of the queues described above. The check-in service could consist of assigning seats to the passenger, printing out the boarding pass, and handling the baggage drop-off. The distribution of the processing time depends on many factors, for example staff experience and passenger characteristics (see Brunetta, Righi, & Andreatta (1999)). Many airlines provide self-service check-in (SSCI) to reduce the amount of time a check-in agent needs to handle one passenger. If passengers are traveling with cabin luggage only and
E-mail address: [email protected]. 2210-5395/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.rtbm.2011.06.012
are using the self-service facilities, there is no further need for queueing up at the check-in counter. Some airlines require that passengers print out their own boarding pass before they are allowed to queue at the check-in desk. Compared to the conventional check-in, SSCI reduces the demand for check-in operations on the one hand and may reduce the average processing time on the other hand. This results in lower overall operator requirements for SSCI compared to the conventional check-in. Fig. 1 shows the flow chart of the check-in process. The check-in service could be provided for single or multiple flights of an airline (common check-in). Some airlines have their own ground handling staff to offer such services, whereas other airlines hire a ground handling company, especially when the number of flights from an airport is small. Based on the distribution of the arrival rate of passengers over time and target service measures, the requirement of check-in operators has to be set for each time interval. Two kinds of setting the timedependent requirements for check-in agents could be distinguished: based on contracts and based on service level agreements. 1. In the first case, the number of operators needed is predetermined by contracts between the airline and the ground handler for each flight. In the case of a check-in for a single flight, the contract is based on the planned time of departure. The check-in starts at a specified amount of time before the take-off time and closes, for example, 30 min prior to the departure. During this time interval the number of agents can be time-dependent and depends additionally on • the number of passengers, • the airports of departure and destination, and • the time of the day.
R. Stolletz / Research in Transportation Business & Management 1 (2011) 144–149
145
Passenger arrives
Airline provides SSCI?
no
yes
SSCI is mandatory
no
Passenger picks SSCI
yes
no
yes
Passenger picks seat and prints boarding pass
yes
Cabin luggage only? no
Baggage drop-off counter?
no
yes Passenger queues up in baggage drop-off line
Passenger queues up in check-in line
Passenger proceeds to security check Fig. 1. Flow chart of the check-in process.
The planned profile of open check-in counters is usually experience based and not justified according to any goal on the waiting time at the check-in. On the one hand, a high number of operators results in short passenger waiting times and in a low utilization of the operators. On the other hand, if the number of operators is too low, the utilization is high and the customers have to queue up.
number of possible profiles, a fast performance analysis is necessary to evaluate the performance measures for these profiles. This is a prerequisite to analyze the tradeoff between the utilization of the operators and the service quality offered to the passengers. To this end, the check-in system needs to be analyzed as a time-dependent and stochastic queueing system. The main contribution of this paper is twofold:
2. In the second planning situation the requirements are based on service level agreements between the ground handler and the airline. In this case the ground handler has more freedom to assign the operators to different check-in stations, based on some bounds on the performance measures mentioned above.
• An approximation method for analyzing the performance measures for time-dependent and stochastic check-in systems is developed. Numerical examples show that the proposed approach overcomes traditional approximation methods. • The analysis of the tradeoffs between the utilization of check-in operators and the service provided to the passengers shows the influence of the distribution of processing times and economies of scale via common check-in operations.
In addition to bounds on performance measures given by the airlines, there could also be some limitations of queue lengths due to security reasons. The managerial decisions are concerned with defining operator profiles with respect to given bounds on the performance measures in both planning situations. Due to the large
The paper is organized as follows: Section 2 presents a literature review. The considered model of the check-in process is described in
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Section 3. The approximation method is developed in Section 4. The numerical study in Section 5 compares the proposed approach to existing ones. In doing so, the influence of the distribution of the processing time is studied and economies of scale of a common checkin are shown. Section 6 discusses the results and concludes with suggestions for further research. 2. Literature review Operational research is widely used to support decisions in planning and operating aviation activities, see the overview in Barnhart, Belobaba, and Odoni (2003). In analyzing the requirements for check-in operations, the most challenging is to take into account the time-dependent arrivals of passengers. Therefore the review concentrates on a discussion of these distributions and gives an overview on approaches for the performance approximation of timedependent and stochastic systems. The arrival rate of the passengers of a single flight depends on the scheduled departure time and on the flight itself. The distribution of the arrival rate over time (also called passenger earliness of arrival profile, see Barros and Tomber (2007)) depends for example, on the destination, on the flight category (business or charter), or on the time of the day. Thus for example, the mean arrival time in the morning is much shorter than in the afternoon (see Robertson, Shrader, Pendergraft, Johnson, and Silbert (2002)). Common for all the different distributions is that they are strongly dynamic and cannot be replaced by average values. The other main characteristic of the arrival process is the stochasticity. Fig. 2 shows an example for a passenger earliness of arrival profile (Robertson et al., 2002). For longhaul flights passengers usually arrive earlier and this distribution is shifted to the left. The earliness of arrival profile also depends on the kind of passenger. For example, 71% of business travelers at Frankfurt Airport do not arrive earlier than 2 h before the scheduled departure time, while only 45% of leisure travelers arrive within 2 h before departure (Media Frankfurt GmbH, Werbemonitor 2010). Several general approaches for the performance approximation of time-dependent queueing systems are discussed in literature (see the overviews in Ingolfsson, Akhmetshina, Budge, Li, and Wu (2007) and Stolletz (2008a)). In general, ordinary differential equations can be solved (Kleinrock, 1975) or the randomization approach could be applied (Grassmann, 1977) for Marcovian systems. Both approaches are not suitable for queues with generally distributed arrival and service processes. Similarly, fluid approximations (Mandelbaum &
20% 18%
% passengers arrived
16% 14%
Massey, 1995) and infinite server approximations (Ingolfsson et al., 2007) are not able to handle generally distributed service and interarrival times. Fluid approximations are often called cumulative diagrams in airport literature. Tosic (1992) analyzed different queueing models for airport terminals and used cumulative diagrams to model the check-in facilities. Janic (2003) modeled the check-in process using cumulative diagrams, too. Barros and Tomber (2007) analyze the passenger and baggage screening with cumulative diagrams. Brunetta et al. (1999) argued that stationary stochastic queueing models could not be applied because of the importance of the dynamic effects and therefore apply cumulative diagrams to analyze different processing facilities. In general, approximations with stationary approaches are often used to approximate the performance of time-dependent service systems, see for example the Stationary Independent Period by Period (SIPP) approach (Green, Kolesar, & Soares, 2001) and the Stationary Backlog Carryover (SBC) approach (Stolletz, 2008a). Numerical experiments show that the SBC approach outperforms the SIPP approach for the M(t)/M/c(t) queue (Stolletz, 2008a). The performance of single server M(t)/G(t)/1 systems with generally distributed processing times is well approximated by the SBC approach (Stolletz, 2008b). Therefore this paper extends the SBC approach to the analysis of M(t)/G/c(t) loss-waiting queues with time dependent inter-arrival times, generally distributed processing times, and time-dependent numbers of servers.
3. Description of the check-in model This paper focuses on a basic queueing model for a check-in system with a single queue and a time-dependent number c(t) of check-in counters, as depicted in Fig. 3. This M(t)/G/c(t) system assumes an inhomogeneous Poisson arrival process with instantaneous arrival rates λ(t). The service time for a passenger corresponds to a generally distributed random variable with rate μ and the squared coefficient of variation CV 2. The number of waiting positions is assumed to be infinite. Waiting customers are served according to the First-Come-First-Serve (FCFS) policy. Time-dependent performance measures of interest are the expected utilization, the expected queue length, and the expected waiting time. Many airlines provide separate check-in for economy class, business class, and first class. If there is a strict separation, a passenger of a certain class could be only served by the dedicated group of operators. The model described above applies for each class because they do not influence each other. Some check-in queues are organized in a way that check-in agents of a higher class (for example business class) are allowed to serve the next waiting customer of a lower class (economy class), if no other high-class passenger is waiting. In this case the models cannot be analyzed independently. Such models with heterogeneous customers, heterogeneous operators, and priorities are called X-type or N-type models. They are analyzed for other service operations in steady-state, for example call centers (see Stolletz and
12% 10%
Check-in Counter
8%
1 1
6% Queue
4%
Arriving passenger
λt
2% 0%
150 140 130 120 110 100 90 80 70 60 50 40 30 20 10
∞
2 1
0
Time before departure [min] Fig. 2. Passenger earliness of arrival profile. Source: Robertson et al. (2002).
ct Fig. 3. Basic queueing model for the check-in.
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Helber (2004)). The SBC approach could be extended to X-type or N-type models. However, in order to show the characteristics of timedependent and stochastic check-in systems the analysis is restricted to models with homogeneous operators and homogeneous passengers.
approximation results directly in the backlog rates bi and the expected utilizations E[Ui] for each period i with bi = ˜λi ⋅Pi ðBÞ = ˜λi ·
4. Performance analysis approach
c
(i) The approximated utilization E[Ui] for period i is analyzed using a stationary M/G/ci/ci loss model, which yields an artificial blocking probability Pi(B). A backlog bi is generated from these (artificially) blocked customers. The backlog rate bi is given by ˜ i ·Pi ðBÞ: bi = λ
ð1Þ
This number of artificially blocked customers is carried over as additional arrivals in the subsequent period i + 1. For example, ˜ i + 1 of period i + 1 is built by the artificial arrival rate λ + 1
= λi
+ 1
+ bi = λi
+ 1
˜λ = μ ci i
ci !∑ki = 0
In the SBC-approach the time horizon is divided into small periods i of equal length l with constant arrival rates λi for each period i. This approach was developed for M(t)/M(t)/c(t) and M(t)/G/1 queueing systems with an infinite waiting room and Poisson arrivals, see Stolletz (2008a) and Stolletz (2008b). The basic idea of this approximation procedure is divided into three steps for each period i: (i) the approximation of the utilization, (ii) the derivation of a modified arrival rate, and (iii) the approximation of waiting-based performance measures using a stationary M/G/c-approximation. In the first step each period is analyzed with a loss queueing model using ˜ i . The loss models of consecutive periods are an artificial arrival rate λ connected via an artificial backlog, which equals the (artificial) blocking rate of the preceding period. To derive the artificial arrival rate of the subsequent period, this backlog rate is added to the original arrival rate, see Fig. 4. This step results in an approximation of the utilization. Based on this utilization, a modified arrival rate is derived in the second step such that a stationary waiting system would reach the same expected utilization. In the third step, waiting based performance measures are approximated using this modified arrival rate and a stationary approximation. To approximate an M(t)/G/c(t) queueing model for the check-in the SBC approach is extended in the following way. Starting with the first period the following steps are applied for each period i:
˜i λ
147
+ ˜λi ·Pi ðBÞ
ð2Þ
˜ 1 = λ1 . Applying Erlang's loss formula and is initialized with λ for Pi(B) (see for example Gross and Harris (1998)), the SBC
E½Ui =
˜ λi = μ k!
k and
˜λ ð1−P ðBÞÞ ˜λ −b λ + bi−1 −bi i i i = i = i : ci μ ci μ ci μ
ð3Þ
ð4Þ
(ii) For the approximation of waiting-based performance measures a stationary M/G/ci/∞ waiting model is used. In order to obtain approximations of the performance measures of interest, the modified arrival rate λiMAR serves as an input to the stationary queueing model. This modified arrival rate λiMAR is set such that the approximated utilization E[Ui] of step (i) is achieved. In the case of an M(t)/G/c(t) queue this results in MAR
λi
= ci ·μ·E½Ui :
ð5Þ
(iii) An approximation for the performance measures for a stationary M/G/ci queue is applied with the modified arrival rate λiMAR. This approach can be combined with any stationary approximation. According to Bolch, Greiner, de Meer, and Trivedi (1998, p. 233) the approximation of Cosmetatos (1976) gives very good results for different squared coefficients of variation of the processing times CV 2. The approximation of the expected waiting time E[WM/G/ci] is based on the expected waiting time E[WM/M/ci] of the M/M/ci queue and the expected waiting time E[WM/D/ci] of the M/D/ci queue. Exact derivations of performance measures are known for both queue types, see for example Bolch et al. (1998). The approximation of Cosmetatos (1976) leads to h i h i h i 2 2 E WM = G = ci ≈CV E WM = M = ci + 1−CV E WM = D = ci :
ð6Þ
Using Little's law the expected queue length is given by h i h i MAR E QM = G = ci = λi E WM = G = ci :
ð7Þ
Algorithm 1 shows the whole procedure. The performance measures of interest for time t are approximated by the values of the respective period. Numerical experiments in Stolletz (2008a) indicate that the average processing time is a good period length, i.e., l = μ − 1 is used. Algorithm 1. SBC-approach for the M(t)/G/c(t) queue
Fig. 4. Idea of the backlog carry over of the SBC approximation (step (i)).
1: Input: λi, μ, ci, CV2 2: Initialize: i = 0 ; b0 = 0 3: for i = 1 to I do ˜ i = λi + bi−1 4: λ 5: procedure M = G = ci = ci ˜λi ; μ; ci 6: return Pi(B), E[Ui] 7: end procedure 8: bi = ˜λi ·Pi ðBÞ 9: λiMAR = ci · μ · E[Ui] 10: procedure M/M/ci/∞(λiMAR, μ, ci) 11: return E[WM/M/ci] 12: end procedure 13: procedure M/D/ci/∞(λiMAR, μ, ci) 14: return E[WM/D/ci] 15: end procedure 16: E[WM/G/ci] = CV2E[WM/M/ci] + (1 − CV2)E[WM/D/ci] 17: E[QM/G/ci] = λiMARE[WM/G/ci] 18: end for
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250
Table 1 Arrival rates of the base case.
1
0–30 0.5
30–60 1.3
60–90 1.9
60–120 1.5
SBC SIPP Sim
0.9
200
E[W] (in sec.)
Time period Arrival rate
0.8
SBC (SCV 1) SBC (SCV 0) Sim (SCV 1) Sim (SCV 0)
150
100
0.7 50
E [U]
0.6 0.5
0
0.4
0
1000
2000
3000
4000
5000
6000
7000
time [sec.]
0.3 0.2
Fig. 7. Approximated expected waiting time for the base case with different CV2.
0.1 0
0
1000
2000
3000
4000
5000
6000
7000
time [sec.] Fig. 5. Approximated expected utilization for the base case.
5. Numerical study To show the reliability of the SBC approach the results are compared to the SIPP approach and to the fluid approach. The timedependent expected utilization and queue length are compared to simulation results with 20,000 replications. In the base case check-in for a single flight with c = 2 agents and exponentially distributed processing times of μ − 1 = 1 min is assumed. The arrival rates per minute are given on a 30-minute basis with the values given in Table 1. Figs. 5 and 6 show the expected utilization and the expected queue length for the SBC approach, the SIPP approach, and the simulation. Both measures are well approximated by the SBC approach. Due to the underlying assumption of stationarity in the SIPP approach, the measures do not change within a 30-minute period. The steady-state will be reached at the end of such an interval, but not at its beginning. In the first minutes the steady-state is not reached and the SIPP approach significantly overestimates the utilization and the queue λ length. Due to the assumption of ρ = i <1 the fluid approach ci μ h i λ results in an expected utilization of E Uifluid = i , which equals the ci μ steady-state result of the SIPP approach. The queue length in the fluid 18
approach is approximated with zero in each period and therefore not shown in the figures. The influences of the distribution of the processing time are manyfold. Obviously, a reduction of the mean processing time, for example due to self-service facilities, reduces the utilization, the queue length, and the mean waiting time. Furthermore, a reduction of variation has a significant influence on the performance measures. Fig. 7 shows the expected waiting time for the base case with a squared coefficient of variation of 1 (exponentially distributed processing times) and 0 (deterministic processing times). Similar results could be shown for the expected queue length. To show economies of scale related to common check-in operations, a system with c = 10 (instead of c = 2) operators and with arrival rates that are five times higher as in the base case is analyzed. Figs. 8 and 9 show the expected queue length and the expected waiting time, which have the same shape due to Little's law. Compared to Figs. 6 and 7, the queue length for the larger system increases moderately, whereas the waiting time is reduced significantly in all periods. Another advantage of a common check-in for flights with different times of departures allows to smooth the demand and hence to smooth operator requirements. This results in better solutions in the following step of shift scheduling, see Stolletz (2010). To summarize the numerical study, it shows that the conventional approaches described in the literature are not able to approximate the performance of a time-dependent and stochastic check-in system. Its utilization is much better approximated using the SBC-approach. Especially the queue length and the waiting time are underestimated
16
SBC SIPP Sim
16
SBC SIPP Sim
14
14
12
12
E [Q]
E [Q]
10 10 8
8 6
6 4
4
2
2
0
0
1000
2000
3000
4000
5000
6000
time [sec.] Fig. 6. Approximated expected queue length for the base case.
7000
0
0
1000
2000
3000
4000
5000
6000
7000
time [sec.] Fig. 8. Approximated expected queue length for the common check-in.
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of the stochasticity inside the system and allows to create a smoother demand via adding up different earliness of arrival profiles. Further research is needed to integrate performance analysis into an optimization procedure for such systems. Another direction of further research is the analysis of check-in systems with self-service facilities, heterogeneous passenger groups, and heterogeneous groups of operators. While the check-in is only one part of the complex service network at an airport, the operations of succeeding steps should be integrated. For example, the influence of temporary shortfalls in the capacity of the baggage handling system should be analyzed.
SBC SIPP Sim
90 80
E[W] (in sec.)
149
70 60 50 40 30 20 10 0
References 0
1000
2000
3000
4000
5000
6000
7000
time [sec.] Fig. 9. Approximated expected waiting time for the common check-in.
with the widely used cumulative diagrams (fluid approximation). Based on the analysis of the variability of the service time and the study of economies of scale two major managerial implications can be stated. First, a reduction of the variability shows a significant improvement of the performance measures. Second, economies of scale can be gained by common check-in operations. To do so, running a larger system with the same utilization reduces the waiting time significantly on the one hand. On the other hand, common check-in systems are able to achieve a higher utilization of operators with a good quality of service. 6. Conclusion The service quality at check-in stations is mainly driven by the number of available check-in counters, the dynamic arrival rate of passengers, and the distribution of the processing time. With respect to the planning situations described in Section 1, the SBC approach yields a good approximation of performance measures and hence, it results in better operator requirement profiles in both planning situations. The developed SBC approach for the performance analysis is computationally tractable, i.e., it can be used to evaluate many different operator profiles within seconds. It outperforms the SIPP approach and the commonly used cumulative diagrams. Numerical experiments show that besides the mean processing time, the squared coefficient of variation has great impact on waiting-time based performance measures. Economies of scale can be achieved through the implementation of a common check-in. This reduces the influence
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