A simulation-based heuristic that promotes business profit while increasing the perceived quality of service industries

International Journal of Production Economics 211 (2019) 60–70

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International Journal of Production Economics journal homepage: www.elsevier.com/locate/ijpe

A simulation-based heuristic that promotes business profit while increasing the perceived quality of service industries

T

Federico Trigosa,∗, Alan R. Vazquezb, Leopoldo Eduardo Cárdenas-Barrónc a

Tecnológico de Monterrey, EGADE Business School, San Pedro Garza García, NL, Mexico KU Leuven, Department of Biosystems, Leuven, Belgium c Department of Industrial and Systems Engineering, School of Engineering and Sciences, Tecnológico de Monterrey, E. Garza Sada 2501 Sur, Monterrey C.P. 64849, NL, Mexico b

ARTICLE INFO

ABSTRACT

Keywords: Heuristics OR in banking Productivity and competitiveness Queueing Simulation

This paper tackles the problem of minimising the direct server costs of complex multiple-server queueing systems while improving the quality perceived by customers. The problem is solved by finding the server roster, defined as the allocation of active individual servers in each time interval of the working day. As no closed solution is available at this time, it was obtained by simulation modelling. The complex queueing system was solved by a simulation-based heuristic that includes the dynamic arrivals, entity reneging and/or balking, multi-servicing, and the individual service-time distributions of servers. The distributions of the random variables of single or multiple-branch businesses were identified by sampling procedures for multiple company-branch deployment. The beneficial financial impact of the proposed solution is confirmed in numerical real-life banking scenarios. The quality perceived by customers (evaluated through the expected queueing time) was also improved. Finally, additional insights that might improve the system performance are highlighted.

1. Introduction Customers usually dislike waiting in long queues because they perceive a loss of valuable time that could be spent on more productive activities. Although the waiting experience is exhausting and tedious to most people, it is a necessary part of several real-life situations. Long queues are not only boring, but irritating and stressful. From a business perspective, waiting lines represent a cost to both customers and service providers. Thus, reducing these costs would boost the business profit. It is worth mentioning that long waiting times and long queue lengths sometimes deter customers (a customer arriving at the system may balk at the queue length and walk away) and/or trigger reneging behaviour (when a customer joining the queue later abandons it due to the long waiting time). Whereas long/slow queues provide a negative experience for customers in any service, short/fast lines promote a memorable positive experience. A major influencer of the queue length and waiting time is the service capacity; that is, the number of available servers per time interval. The server capacity directly affects the company costs, because increasing the server capacity increases the direct labor cost. Physical banking is a representative example in which service capacity is important.

∗

Regular banks have a complex queue system that cannot be mathematically analysed in a closed form due to its dynamic nature. Instead, this paper proposes a simulation-based heuristic that minimises the server cost (measured by the number of total working-server hours per day) while controlling the expected queueing time. The findings are applied to a real banking operation. Although waiting time is not regulated by an international body, some comparisons can be made. The average wait time at bank branches in the United Kingdom is 2.5 min, with a maximum of 12 min (Fearnley, 2011). At the United States, fried chicken fast-food chain Raising Cane (Benshosan, 2017), drive-thru customers wait (on average) 2 min and 48 s for their order. In a study of 24 European countries (Antena 3, 2010), the average waiting time was 4 min in Spain and 10 min in the remaining 23 countries. According to the Latin America Federation of Banks (Glen de Tobón et al., 2009), some Brazilian states define a maximum waiting time of 15 min on regular bank working days, and 30 min on days before or after holidays; an Argentinian province law stipulates a maximum waiting time of 30 min. The contributions of this work are threefold. First, it proposes a simulation-based heuristic for solving the roster problem in complex queue systems. Second, it offers practical insights for determining the input random variables at single or multiple company branches. Third,

Corresponding author. E-mail addresses: [email protected] (F. Trigos), [email protected] (A.R. Vazquez), [email protected] (L.E. Cárdenas-Barrón).

https://doi.org/10.1016/j.ijpe.2019.01.009 Received 6 April 2018; Received in revised form 20 September 2018; Accepted 7 January 2019 Available online 25 January 2019 0925-5273/ © 2019 Elsevier B.V. All rights reserved.

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it provides practical recommendations for improving the system performance. The remainder of the paper is organised as follows. Section 2 reviews the relevant literature, and Section 3 describes the proposed methodology, including the conceptual model (the particularities of our generalised queuing system for banking), the simulation model, and the simulation-based heuristic. This section also proposes two sampling plans. Section 4 demonstrates the sampling plan and simulation heuristic in a real numerical example. Section 5 ends the paper with conclusions and suggestions for further research.

Jouini et al. (2009) investigated multi-class call centre models in the presence and absence of reneging. Their method estimates the delays to clients on arrival. Ernst et al. (2004) presented a collection of references for roster design. They acknowledged the highly constrained, complex nature of the related optimisation problem, which usually depends on particular industry requirements. Simulation is a major engineering tool for analysing and solving complex systems that defy an analytical closed solution. Fu (1994, 2015) comprehensively reviewed the simulation techniques for optimising stochastic discrete-event queuing systems. General simulation algorithms and their applications are exhaustively reviewed in Amaran et al. (2014). Van Dijk (2000) demonstrated three applications of a hybrid simulated queueing approach. Nagy et al. (2013) simulated an IP multimedia subsystem network using single servers and a FIFO queueing discipline, and Šeda et al. (2011) simulated the various characteristics of a queueing system. More complex systems have also been modelled in simulation studies. For example, Munoz and Ruspini (2013) simulated the changing number of servers, arrival rates, and service rates in fuzzy queueing systems. Gustafsson (2003) modelled queueing systems in a Poisson simulation that captures the dynamic, stochastic nature of real arrivals. Similarly, Yang and Liu (2012) quantified the dynamics of a general queueing system in stochastic simulation algorithms of transfer function models. More applications of queueing simulations can be found in Day et al. (2010), Mayor and Silvester (1996), and Wu (2009). Simulation-based optimisation remains challenging due to the complexity of the models, and the large number of combinations of the levels of the variables involved. Arena is a powerful simulation software that models diverse real-life situations. It was used by Li et al. (2009), who simulated the production planning of a dedicated remanufacturing system operating on a prioritised stochastic batch-arrival mechanism, accounting for factors that impact on the profit. By integrating the Supply Chain Operations Reference (SCOR) model and the a discrete event simulation tool through Arena, Persson and Araldi (2009) altered the input parameters of a supply chain, and analysed the dynamic effects with a view to understanding the behaviour of the entire supply chain. In a subsequent paper, Persson (2011) presented an alternative version of the SCOR model and tested it in a company. Another application of Arena is reported in Melouk et al. (2013), who optimised the decisions of a steel manufacturing company in a simulation model. The literature review showed that no closed mathematical formulation currently exists for solving the roster problem under reneging, balking, dynamic arrivals, servers with individual service-time distributions, and multi-servicing of customers. Therefore, this paper introduces an effective simulation-based heuristic that accounts for the complexity of real queuing systems.

2. Literature review One of the most important quality elements in the service sector is the queue waiting time before customers are served by an agent (which may or may not be human). Nordgren (1999) established that customer waiting time is the most detrimental time usage of service companies. For instance, a customer who has waited a long time can give the company a bad reputation. The process is apparently simple: customers arrive at the system, wait in line until an agent (one of many working in the system) is available to serve them, and finally leave the system. However, difficulties arise when the servicing time of each server follows an individual random distribution. Moreover, the complexity increases when the time between arrivals varies throughout the working day. Additional sources of complexity include: customer balking, customer reneging, and customer multi-servicing (where a customer service is interrupted by lack of a service element, such as paperwork). In the last case, the customer temporarily leaves the system to obtain the needed element, and returns to the queue after some finite random time. The returning customer may or may not be prioritised over other customers. This type of queueing system has been modelled by several approaches. For instance, Sharma and Sharma (1996) modelled batches of queueing customers with different arrival rates depending on the system capacity. Claeys et al. (2010) also adopted a batch-service queueing model with dynamic arrivals of batches. Hur and Paik (1999) developed an optimal N-policy for a single server queue with Markovian arrival rates and a general distribution of service times. In their model, the arrival rates depend on the server's status. However, all of these contributions adopt a closed-form mathematical model, which relies on strong assumptions. Therefore, they cannot fully capture the above-mentioned complexity of queueing dynamics. Slegers et al. (2009) developed a computational procedure that allocates processors (servers) offering different types of services on a computing grid with different arrival rates. They averaged the queue sizes obtained in four heuristics and compared the result to the optimal solution. Service companies handling different arrival rates at different hours of the day face a similar problem. Heuristics that search for the optimal solution taking into consideration servers with different service distributions are also required. Kim and Kim (2015) studied a queueing system in an emergency care centre (ECC) that bases the waiting times on the patients' conditions. They used a hybrid priority model using the first-in-first-out (FIFO) discipline for non-priority patients and a Jackson Network for priority patients. They assumed exponential service times for various service types. The arrival distributions were assumed as Poisson distributions with different parameters for different classes of patients. However, in a more realistic situation, the arrival times might follow other distributions. Moreover, the performances of ECC personnel can alter throughout the day, and the service time distributions can differ between the night time and early daylight hours. Koole and Mandelbaum (2002) emphasised the practical aspects of queueing systems in call centres, offering definitions of the system and queueing models. However, they did not provide a sampling plan for adequately collecting the data of call centres. To apply efficient methodologies, practitioners need a standard sampling plan for queueing at call centres and other systems.

3. Methodology This section describes the classical steps of a simulation study, and is organised as follows. Subsections 3.1 and 3.2 describe the conceptual and simulation models, respectively. Section 3.3 suggests practical sampling procedures for obtaining the input data of the simulation model, and subsection 3.4 describes the simulation model validation procedure. Finally, subsection 3.5 introduces the heuristic that intends to minimise the server working hours while limiting the expected queueing time throughout the working day. The notations used in this paper are given in Table 1. 3.1. Conceptual model Improved customer service is a main goal of multiple-branch banking operations serving customers in the base-of-the-pyramid sector (Prahalad, 2006). In our case, improving the customer service means 61

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important element of perceived quality. In the present study, the queueing time over the working day is limited by a constraint. Banking service catalogues may include several hundreds of potential banking transactions (deposits, payments and others). The time between customers arrivals throughout the working day is a dynamic random variable. To identify the distribution of the customer-arrival intervals, the working day is divided into several time intervals. The bank branch has a limited space for cashier facilities, which sets an upper limit on the maximum number of servers in any time interval. To maximise the efficiency, we enforce a single FIFO queue discipline rule (Gross and Harris, 2008). Some customers lack certain information or paperwork (customer-related issues) when being served, which interrupts the service. Such customers momentarily leave the server (sometimes with a ticket that prioritises them when they rejoin the queue), to acquire the item or items needed to resume the service. This situation is called multi-servicing. The customer-service model must find the appropriate service capacity (number of servers) and allocate the individual servers to the available service stations (teller facilities) during each time interval. Simultaneously, it should minimise the total service cost (number of paid server hours) during each day without exceeding the maximum limit of the expected customer queueing time (defined by management and influenced by international benchmarks).

Table 1 Notation. Parameter

Description

α µs

Significance level. Service capacity of the s th server (customer/time). Mean service time of the s th server. Maximum number of active service stations in the system over all working periods. The 1 half-width confidence interval of the mean queueing time. Maximum number of services that a customer can receive. Probability that a customer needs a second service.

1/µs F HW

¯q (t , St ) /2, W

k pms2

pms j

pR R st St = {s1, …, st } FSCSS T TBA TBE t Lq

Wq

¯ q* W

Probability that a customer needs an extra service after j 1 services Probability of a non-balking customer being classified as reneging customer. Number of replications in the simulation model. Number of active server stations in the interval t 1 to t (t T ). Set of active service stations during time intervals 1 to t. Fastest Server to the Closest Service Station (server allocation rule). Number of time periods in the system. Time between customer arrivals. Time between customer exits. A particular time period in the system. Expected queue length (customers). Expected queue waiting time.

3.2. Simulation model

Upper limit of the average queue waiting time throughout the working day.

The proposed simulation model of the complex queue system was inspired by Kelton et al. (2014) and is shown in Fig. 1. The model possesses the following properties:

controlling the expected queueing time throughout the working day. In a good business strategy, the customer queueing time must approach world benchmarks. The problem consists of finding the server-roster that minimises the server hours while controlling the expected queueing time of customers (a managerial decision) throughout the working day. For practical purposes, all bank branches owned by the same bank have independent and random input variables, so the present study considers an arbitrary bank branch. In many cases, bank customers classified at the base of the pyramid (who unfamiliar with electronic banking) can allocate only a few minutes to banking, sometimes taking time out of work. Therefore, reducing the mean queueing time is particularly beneficial to customers. The expected customer queueing time is considered as an

(i) The system maximum service capacity is F stations at any given time. (ii) The dynamic distributions of times between the arrival of customers are arranged into time intervals throughout the working day. (iii) Each customer checks the current length of the queue and compares it against their perception of a long queue (modelled by a random distribution). If the queue length at arrival exceeds the perception, the customer balks and departs from the queue; otherwise, the customer joins the queue. Joining customers are split into two categories: normal customers (who complete the

Fig. 1. Flow diagram of the simulation model. 62

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(iv)

(v)

(vi)

(vii)

(viii)

service before departing), and renege customers. Renege customers perceive a long wait time (selected from an abandon-time distribution). If their current queue time exceeds their extended perception, they may leave the system (renege). A non-balking customer becomes renege customer with probability pR . For simulation purposes, each reneging customer is cloned into a logical customer. The original and logical customers are modelled at the top and bottom of Fig. 1, respectively. There is a one-to-one initial relation between the two customer classes. Each logical reneging customers waits for a random time (selected from the abandon-time distribution). After this simulated time period, the model requests the queue position of the corresponding original customer. If the customer remains in the queue, the program eliminates the physical and logical customer in the system (simulating reneging). If the physical customer is no longer in the queue, the physical customer has been already served, and the logical customer is deleted from the system. The non-reneging customers (and some reneging ones, as mentioned above) wait in the queue until they arrive at a service station. If two or more servers are available, the customer approaches the closest one (closest to the end of the queue). The available servers (note that particular servers are available at particular time intervals during the working day) are allocated to the service stations, following a server allocation rule (a decision variable). Once the customer has reached the server, a random service time is generated depending on the individual server at the service station. When the service is finished, whether the customer requires a second service (i.e. satisfies the multi-servicing condition) is decided by a Bernoulli random variable with probability pms2 . If the customer requires an extra service, he/she returns to the system (after a randomly generated time interval). The customer rejoins the general queue with or without a higher priority, depending, on the model policy. The numbers of extra services needed by multi-service customers are generated from Bernoulli random variables with known probabilities pms2 , …, pmsk where pmsj is the probability of requiring the j th extra service after finishing the j 1th service. The system defines pmsk +1 = 0 , and the number of multi-services cannot exceed k. Finally, the customer finishes the last service and leaves the system.

recorded, and the customer is re-recorded when returning to the service. P4 : If applicable, record the identification number of a reneging customer, the time at reneging, and the queue position just before reneging. The service stations are numbered from right to left, with service station one being closest to the end of the queue (see Fig. 2). Using the data base generated by this manual system, the distributions required by the simulation model in 3.2 can be identified in goodness-of-fit tests. 3.3.2. Automated sampling plan Although the sampling plan described in the previous subsection is effectively applicable to a single branch of a company, manual sampling is economically infeasible for companies with many independent bank branches. Let us state some aspects of physical banking: 1: Servers run their services from a computer terminal that records the starting and finishing times of the service, along with the customer and server identification numbers. Thus, the service times of individual servers can be recorded automatically. Furthermore, if the service requires several automated sub-services, the specific runtime of each sub-service can be recorded. The automated subservice times do not usually sum to the total service time, because human interaction is required between each sub-service. Fig. 3 shows the elements comprising the server service time. The customer is first welcomed by the server (human interaction time). Next, a sub-service (such as an electronic deposit) is conducted. The customer is then asked whether another transaction is required, which involves more human interaction. If so, another electronic transaction is made. After several repeats of this process, the total service time has an electronic time component and a human contact element as shown in Fig. 3. 2: According to Burke's theorem (Gross and Harris, 2008), the times between customer exists and customer entries have identical distributions in a multi-server, single queueing system with Poisson arrival times and exponentially distributed service times. Burke's theorem is unidirectional. Although the present study explores a terminating system (one that never reaches steady state), the distribution of the times between customer exists and customer entries might be equal. This hypothesis can be tested statistically on a caseby-case basis. If true for an individual instance, the times between exits can be recorded by the system at the end of each service, and the distribution of the times between arrivals can be obtained by goodness-of-fit tests without requiring manual sampling from the distribution of the times between exits. 3: If the length/time of the queue is sufficiently short, the balking and reneging behaviours become negligible. 4: If the customer is adequately informed in advance, the multiservicing becomes negligible.

3.3. Data gathering This subsection explains the data collection process. Once the data are collected, the statistical distributions of the independent variables (such as the time between arrivals and service times) are determined in goodness-of-fit statistical tests (Banks et al., 2010). After describing the classical manual sampling used in many analyses, this subsection develops an automated sampling plan (some assumptions apply) that can be implemented in several company branches. The automated plan lowers the costs and is easier to implement than manual sampling plans.

If a particular case satisfies items 1 through 4, using the records of every service (customer identification number, arrival time, server identification number, start and finish times of the service, etc.), all input data distributions required by the simulation model can be obtained in an automated fashion. Moreover, if the system records the start and finish of every sub-service (deposits, payments, and other transactions), the distributions of the total human, and non-human (electronic) interaction times of the service can be deduced, providing useful information for service improvements. Hence, if the service meets the conditions in items 1 through 4, the random input distributions of the simulation model in 3.2 can be automatically obtained, enabling deployment in multiple company branches.

3.3.1. Manual sampling plan The random variable distributions in the model were identified at four check points (see Fig. 2).

P1: Assign a unique customer identification number, and records the arrival time of the customer at the system. If the customer balks, records queue length at balking. P2: Record the customer identification number and the time at which the customer leaves the queue to be served. If more than one server is available at this time, the customer is guided to the nearest available server. P3: Record the customer identification number at the service completion time. If the customer requires an extra service, this fact is 63

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Fig. 2. Layout of manual sampling.

3.4. Model validation

replications) obtained from the simulation model. Also define St = {s1, …, st } as the set of active service stations in each time interval (from 1 to t). Note that st F , t T . The set From St , which contains information on server availability and the enforceable server-allocation rules, dictates the allocation of individual servers to the service stations (i.e. the roster). The heuristic finds the least number of servers that maintain the expected queueing time within the specified limit throughout the working day, and allocates them accordingly. The heuristic logic first notes that the expected queueing time in the first period is independent those of the remaining periods. The procedure then begins by mini¯ q* within mising the number of servers and allocating them to retain W the confidence interval of the expected queueing time in the simulation (thus satisfying the perceived service quality defined earlier). As the expected queueing time is within the imposed limit during the previous t 1 periods, the heuristic processes the t th period only. The solution is obtained after running the procedure from t = 1 to T. The heuristic is described as follows: BSBH: Backward Simulation Based Heuristic

Once the simulation model is built, its output variables must be compared with the real system parameters obtained from real data. These evaluations usually involve hypothesis comparison tests of the means of the variables and parameters, as suggested by Banks et al. (2010). 3.5. Simulation based heuristic After deeming the simulation model adequate at a particular instance by common validation techniques (Banks et al., 2010), the target limit of the expected queueing time throughout the working day (which ¯ q* . The roster is usually dictated by management) was defined as W problem can then be solved. Before proceeding, we introduce the following notations. Let R be the number of replications (defined at the validation step) in each simulation run, T + be the number of time intervals in the working day, and st be the number of active server stations in the interval t 1 ¯ q (t , St ) and HW /2, W¯q (t , St ) be the accumulated to t (with t T ). Let W average queueing time and the 1 half-width confidence interval of the expected queueing time, respectively (computed from 0 to t, with R

1: Set S0 = { } and define a server allocation rule. 2: For t = 1 to T

Fig. 3. Elements of the total service time. 64

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3: st = F , St = St 1 st . ¯ q (t , St ) HW /2, W¯q (t , St ) W ¯ q*} and (st 1) 4: While {(W st = st 1, St = St 1 st . 5: 6: End While 7: st = st + 1, St = St 1 st . 8: If st > F stop, the problem is infeasible at period t. 9: End For

intervals of the servers are listed in the first, second and third columns, respectively. The fourth column gives the number of minutes worked by the servers in each time interval, and their service averages, standard deviations and distributions are displayed in the fifth, sixth and seventh columns, respectively. Note that 56.37 service hours were paid on this working day in the real system (last row of Table 3). As stated above, the total service time includes both electronic transaction time and human interaction time. The seven servers occupied at most five service stations. There were more servers than stations because the servers require breaks throughout the 12-h working day. Table 4 shows the distributions of the electronic times of the service (excluding the human interactions). The electronic times of all tellers followed exponential distributions, possibly because the staff were adequately trained to use the banking software. Such adequate training would standardise the operation. However, the human interaction part of the service was not necessarily standardised, suggesting that this part could be improved by additional training. Since the BSBH requires a server allocation rule, let us consider allocating the fastest available server to the service station nearest the end of the queue, the second-fastest available server to the service station second nearest the end of the queue, repeating the allocation until all required-available servers are deployed at a given time. This server allocation rule is referred to as Faster Server to the Closest Service Station (FSCSS). Table 5 shows the same information as Table 3, but with the servers sorted from fastest to slowest (in terms of average total service time) in order to easily apply FSCSS server allocation rule. Although no customers balked in this real-life example, 2.5% of the customers reneged ( pR = 0.025). The reneging waiting time followed a random distribution of 40 + Weibull(150,0.66) seconds. If the potential reneging customer is still in the queue at this time, the customer will leave the system. The probability of exiting the system after a single service was 0.9431. Therefore, customers who required a second service (with probability pms2 = 1 0.9431) received a card/ticket identifying them as multi-service customers. The time for which multi-service customers remained outside the system followed a triangular distribution of (45,60,90 ) seconds; subsequently, returning customers joined the general queue with a higher priority than single-service customers. No customer required three or more services (k = 2). All multi-service information was obtained on a second effort.

The heuristic allocates the roster in each period. Step 1 defines the empty set S0 . Once the roster has been designed from time 0 to t 1, (by the active-service set St 1 and the server allocation rule). The heuristic is then ready to process period t (step 2). The search begins at full capacity (F servers) in Step 3, and is decremented in Step 5 until one server remains, or until the confidence interval of the accumulated ¯ q* (the while condition) at st . At the beginning queueing time exceeds W of step 7, the capacity st does not satisfy the queueing time constraint (or st = 0 ), if so the capacity st + 1 is the last capacity satisfying the queueing time constraint (unless st = F also violates the time constraint). If the current iteration solution is infeasible (st = F , i.e. the system cannot satisfy the expected wait time even after deploying F servers in period t), the heuristic terminates at step 8. In a feasible system, the set ST = {s1, …, sT } defines the required server capacity throughout the working day. The final roster is solved by applying the server-allocation rule to ST . 4. Numerical demonstration of the proposed heuristic Based on Fearnley (2011); Benshosan (2017); Antena 3 (2010); Glen de Tobón et al. (2009), the top management of a particular bank branch ¯ q* = 300 s). The demand an expected queueing time of five minutes (W bank branch has six physical service stations (F = 6), and opens from 9:00 to 21:00 h. Using the manual sampling plan in subsection 3.3.1, the input file located at:

https://www.dropbox.com/sh/39fx5fohl5nkkil/ AADwKcjmIJltmF3dVyZvPTbPa?dl = 0

(1)

was obtained. No multi-services are explicitly recorded in this file. Hereafter, all statistical tests are conducted at the = 0.05 significance level and the random-distribution notation follows Kelton et al. (2014). Tables 2–5 were obtained by goodness-of-fit tests as explained in subsection 3.3. Table 2 shows the distributions of the times between arrivals (in seconds) of customers (dynamic arrivals). Note that some customers arrive before 9:00 a.m. The first column of Table 2 lists the time periods over which the intervals between the arrival times were averaged (second column). The distributions are listed in the third column. Table 3 tabulates the service times and availability of the seven servers. The server identification numbers, stations and active time

4.1. Model construction The simulation model in subsection 3.2 can be encoded in any commercial general-purpose simulation software. Fig. 4 shows a general view of the model layout. Here, the model was encoded in Arena Simulation software. The model is divided into five blocks. Block I encompasses the dynamic arrival process of the system, including the balking behaviour when the physical queue grows large according to customers expectations. Block II processes the unique physical queue, and block III builds a complementary virtual queue to handle the reneging process. Block IV constructs the simultaneous service locations from elementary blocks. Block V runs the multi-service system in several trials based on the related stochastic process.

Table 2 Times between customer arrivals in each time interval throughout the working day (numerical example). Time interval

Average (seconds)

Distribution (seconds)

8:37 to 9:00 9:00 to 10:00 10:00 to 11:00 11:00 to 12:00 12:00 to 13:00 13:00 to 14:00 14:00 to 15:00 15:00 to 16:00 16:00 to 17:00 17:00 to 18:00 18:00 to 19:00 19:00 to 20:00 20:00 to 21:00

87.40 31.00 32.80 31.18 29.80 28.70 25.70 31.10 31.80 33.00 29.40 29.40 35.30

2 + EXPO(85.4) 3 + EXPO(28) 3 + EXPO(29.8) 2 + 138 BETA(0.673, 2.51) 2 + EXPO(27.8) 2 + EXPO(26.7) 3 + EXPO(22.7) 0.999 + EXPO(30.1) 0.999 + EXPO(30.8) 0.999 + EXPO(32) 0.999 + EXPO(28.4) 0.999 + 177 BETA(0.365, 1.91) 0.999 + EXPO(34.3)

4.2. Model validation and sensitivity analysis For validation purposes, the simulation model was compared with the real system using the statistical techniques proposed by Banks et al. (2010). The validation is summarised in the first two rows of Table 6. The half-width confidence interval of the expected queueing time (determined from R=220 simulation replicates) was less than one minute. As the simulated confidence limits of both the expected queueing time and expected queue length contained the real system parameters, the hypothesis that the model correctly simulates the real system cannot be rejected. Next, a sensitivity analysis of the simulation model was carried out 65

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Table 3 Service-time distributions and server availability throughout the working day (numerical example). Teller

Service

Time

station 1

2 1 3 4 4 1 5 2 2 3

2 3 4 5 6 7

13:03–15:44 16:00–18:51 8:37–15:28 18:01–21:13 8:50–18:00 8:51–21:11 15:56–20:49 15:55–21:04 8:37–12:59 15:20–21:13

Work

Average

Std dev

Distribution

(minutes)

(seconds)

(seconds)

(seconds)

161 171 411 192 550 740 293 309 262 293 3382 56.37 h

174

11.91637529

32 + EXPO(142)

106.3

59.06293254

4 + ERLA(34.1, 3)

107.84 126.648 153.9 172.6 119.8

61.6892211 82.18417366 9.994998749 110.7329219 61.66100875

26 24 54 16 13

1 2 3 4 5 6 7

Average

Std dev

Distribution

(minutes)

(seconds)

(seconds)

81.6 97.1 108.1 98.9 86.4 94.7 98.8

58.6 72.1 85.1 75.90 61.4 70.7 76.8

23 25 23 23 25 24 22

+ + + + + + +

GAMM(46.5, 1.76) GAMM(65.8, 1.56) EXPO(99.9) ERLA(78.3, 2) ERLA(35.6, 3)

simulated Scenarios compared with the original system yielded significantly different results, opening the door for improving the system. Hence, to reduce the expected queueing time and expected queue length, companies could invest in the following measures: server training to improve the service speed (shorten the mean service time) and the service distribution, providing customer information to decrease the multi servicing demands, allocating servers to service stations based on their mean service time (following the FSCSS allocation rule), and controlling human interaction in the service. Among these measures, allocating servers by their average service times might inspire them to increase their efficiency and earn a payroll bonus. Therefore, server allocation rule FSCSS is enforced in the subsequent analysis.

Table 4 Electronic service times and distributions on individual tellers (numerical example). Teller

+ + + + +

EXPO(58.6) EXPO(72.1) EXPO(85.1) EXPO(75.9) EXPO(61.4) EXPO(70.7) EXPO(76.8)

4.3. Simulation results

under the “Other things equal” criterion. In Scenario (a), all servers were set to operate at the speed of the fastest available server (Server 2; see Row 3 of Table 6). In Scenario (b), all customers were assumed knowledgeable enough to avoid multi-servicing (Table 6 Row 4). In Scenario (c), there were no reneging customers (Table 6 Row 5); notice that both parameters were worsened in this scenario because all arriving customers were serviced. In Scenario (d), all servers were allocated by the FSCSS rule (Table 6, Row 6), and in Scenario (e), all servers consumed electronic service time only (i.e. did not interact with the customers (Table 6, Row 7). Although this scenario is not physically possible, it is investigated for comparative purposes. Finally, Scenario (f) considers that all servers spend only fifteen seconds on human interaction, and the remainder of their service on electronic time (Table 6, Row 8). The significant influence of human interaction suggest that efforts to standardise this part of the service would improve the system efficiency. Notice that in all scenarios except Scenario (c), the confidence limits of the expected queueing time and expected queue length excluded the real system parameters, informing that those

The first of the analysed models considered sorted servers and availability as in Table 5. Table 7 shows the first set of results from the BSBH. The first row shows the solution during the first working hour (9:00 through 10:00). The expected queueing time of the customers served by the six servers was within 13.9157 ± 1768 s with 95% confidence. As the confidence interval of the expected queueing time was within the 300-s limit, the number of servers was decremented until the confidence interval exceeded the imposed 300-s limit (at three servers). Thus, the roster during the first working hour was set to S1 = {s1 = 4} . In the second row of Table 7, the number of servers was set to four during the first hour (9:00 through 10:00), and six during the second hour (10:00 through 11:00). The confidence interval was 68.1321 ± 7.191 s. The BSBH decremented the number of servers during the second hour al the way to two, when the confidence interval of the expected queueing time exceeded the 300 s constraint, namely, S2 = (s1 = 4, s2 = 3) . Over the twelve periods of the working day, the heuristic algorithm obtained S12 = {4,3,5,4,4,5,5,3,6,4,4,4} . The server

Table 5 Sorted service-time distributions and server availabilities throughout the working day (numerical example). Teller

Service station

Time

Work (minutes)

Average (seconds)

Std dev (seconds)

Distribution (seconds)

2

3 4 4 2 3 1 5 2 2 1

8:37–15:28 18:01–21:13 8:50–18:00 8:37–12:59 15:20–21:13 8:51–21:11 15:56–20:49 15:55–21:04 13:03–15:44 16:00–18:51

411 192 550 262 293 740 293 309 161 171 3382 56.37 h

106.3

59.06293254

4 + ERLA(34.1, 3)

107.84 119.8

61.6892211 61.66100875

26 + GAMM(46.5, 1.76) 13 + ERLA(35.6, 3)

126.648 153.9 172.6 174

82.18417366 9.994998749 110.7329219 11.91637529

24 54 16 32

3 7 4 5 6 1

66

+ + + +

GAMM(65.8, 1.56) EXPO(99.9) ERLA(78.3, 2) EXPO(142)

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Fig. 4. Layout of the simulation model. Table 6: Results of validation and sensitivity analysis. Case

Expected queue waiting time

Table 8 Capacity solutions of Wq and Lq with sorted server allocations but preserving the original server availability.

Expected queue length

(minutes) Actual system (allocation rule 1) Simulation model (allocation rule 1) a) All ST = 4 + ERLA(34.1, 3) b) No multi-servicing c) No reneging d) Allocation rule 2 e) No human interaction f) 15 s of human interaction

11.83 11.50 ± 0.853

22.09 20.59 ± 1.59

3.64 min ± 0.24 5.36 ± 0.35 14.31 ± 3.16 10.0 ± 0.73 2.23 ± 0.13 11.01 ± 0.76

6.50 ± 0.44 9.06 ± 0.62 25.57 ± 5.83 17.88 ± 1.36 3.99 ± 0.24 19.70 ± 1.41

working hours did not sum to 51 (as might be supposed), because not all servers were available during all intervals. For example, only servers 2,3,7 and 4 were available during the 11:00 to 12:00 period, meaning that the fifth service station opened by the model was not in service. In addition, the server allocation was difficult to identify because the server availability followed no clear pattern. The plan capacities, along with the accumulated Wq and Lq values in each interval from 9:00, are shown in Table 8. The second analysis assumed that all servers were available when needed. Table 9 shows the BSBH results in this instance. The heuristic

Starting time interval

Plan capacity regardless server availability (servers)

Wq (seconds)

Lq (customers)

9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00

4 3 5 4 4 5 5 3 6 4 4 4

104.28 263.93 299.26 280 ± 295.96 312.91 303.24 318.06 311.98 297.85 300.09 298.05

1.8458 ± 0.193 6.2086 ± 0.431 7.5253 ± 0.614 7.6332 ± 0.613 8.5770 ± 0.629 9.5862 ± 0.678 9.4264 ± 0.693 10.1017 ± 0.665 9.8209 ± 0.633 9.4748 ± 0.583 9.6507 ± 0.552 9.5421 ± 0.539

± 10.291 ± 17.025 ± 23.498 21.789 ± 20.909 ± 21.333 ± 21.441 ± 20.468 ± 19.207 ± 17.565 ± 16.428 ± 16.029

can now open up to six service stations with guaranteed occupancy (as 7 servers are available). Note that this table is identical to Table 7, but in this simulation all open service stations are functional and the proper roster can be obtained. The first, second and third columns of Table 10 list the start times of the intervals, the numbers of working servers and the rosters (server identifications associated to the service stations), respectively. For example, from 9:00 to 10:00, server 2 was allocated to

Table 7 Expected service time predicted by BSBH with sorted server allocations but preserving the original server availability (heuristic solution in bold font). Time 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00

2

3

4

5

6

393.44 ± 14.688

322.36 ± 17.081 263.93 ± 17.025

318.06 ± 20.468

104.28 ± 10.291 127.19 ± 12.435 384.41 ± 26.573 280 ± 21.789 295.96 ± 20.909 349.02 ± 21.416 330.81 ± 22.646 285.89 ± 20.053

321.17 ± 17.931 322.87 ± 16.406 318.25 ± 16.193

297.85 ± 17.565 300.09 ± 16.428 298.05 ± 16.029

32.6649 ± 3.351 83.0649 ± 8.864 299.26 ± 23.498 245.85 ± 20.217 258.53 ± 19.495 312.91 ± 21.333 303.24 ± 21.441 274.58 ± 19.608 327.03 ± 19.907 286.51 ± 17.514 284.81 ± 16.119 287.99 ± 15.532

13.9157 ± 1768 68.1321 ± 7.191 254.78 ± 19.627 230.99 ± 18.516 237.93 ± 18.404 280.73 ± 20.506 290.56 ± 20.765 270.82 ± 19.314 311.98 ± 19.207 282.4 ± 17.283 277.88 ± 15.99 284.56 ± 15.455

349.97 ± 22.863 350.46 ± 21.147 341.07 ± 20.889

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Table 9 Expected service times obtained by BSBH, with sorted server allocations and unlimited server availability (heuristic solution in bold font). Time 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00

2

3

4

393.44 ± 14.688

322.36 ± 17.081 263.93 ± 17.025

104.28 ± 10.291 384.41 ± 26.573 280 ± 21.789 295.96 ± 20.909 349.02 ± 21.416 330.81 ± 22.646

349.97 ± 22.863 350.46 ± 21.147 341.07 ± 20.889

5

318.06 ± 20.468 321.17 ± 17.931 322.87 ± 16.406 318.25 ± 16.193

6

299.26 ± 23.498 312.91 ± 21.333 303.24 ± 21.441 327.03 ± 19.907

297.85 ± 17.565 300.09 ± 16.428 298.05 ± 16.029

311.98 ± 19.207

Table 10 Roster solutions of Wq and Lq with sorted server allocations and unlimited server availability. Starting time

Plan capacity

Roster

Wq

Lq

interval

(servers)

by server

(seconds)

(customers)

9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 Total server hrs.

4 3 5 4 4 5 5 3 6 4 4 4 51

{2,3,7,4} {2,3,7} {2,3,7,4,5} {2,3,7,4} {2,3,7,4} {2,3,7,4,5} {2,3,7,4,5} {2,3,7} {2,3,7,4,5,6} {2,3,7,4} {2,3,7,4} {2,3,7,4}

104.28 263.93 299.26 280 ± 295.96 312.91 303.24 318.06 311.98 297.85 300.09 298.05

1.8458 ± 0.193 6.2086 ± 0.431 7.5253 ± 0.614 7.6332 ± 0.613 8.5770 ± 0.629 9.5862 ± 0.678 9.4264 ± 0.693 10.1017 ± 0.665 9.8209 ± 0.633 9.4748 ± 0.593 9.6507 ± 0.552 9.5421 ± 0.539

server station 1, server 3 to station 2, server 7 to station 3 and server 4 to station 4. The fourth and fifth columns of Table 10 give the confidence intervals of Wq and Lq (accumulative from 9:00), respectively. Fifty-one real teller working hours were spent in this scenario, saving 9.52% (1-51/56.37) of the payroll. Meanwhile one can not reject the hypothesis of Wq = 300 seconds during each hour of the working day. In real banking operations, teller employees are absent from their stations during periods of the working day. Some multi-skilled employees may work as tellers during some hours and as executives with other responsibilities during the remainder of the workday. This compensates for uniquely busy periods such as 17:00 to 18:00 h, which required 6 tellers. Instead of hiring one employee for only one hour a day, workers with other responsibilities can be recruited as tellers during this period. The study findings are summarised below.

± 10.291 ± 17.025 ± 23.498 21.789 ± 20.909 ± 21.333 ± 21.441 ± 20.468 ± 19.207 ± 17.565 ± 16.428 ± 16.029

4. After optimising the analysed situation, the average queueing time was reduced by 57.73% (1-5/11.83). This fact improves the level of service quality for customers. 5. The operation was significantly enhanced by controlling the average and distribution of the service times per server. This preliminary analysis suggested the following definitions: (a) Server consistency: the s th server can be regarded as consistent if the random variable related to his/her total service time can be fitted to an exponential distribution with mean 1/ µs . If all servers realise exponential distributions, the queueing system gets closer to satisfying the assumptions of Burke's theorem. (b) Server speed: The server speed is related to the mean total service time (1/ µs ) of each server. Some companies reward fast workers with monetary compensation. Thus, if all servers process their customers at the same average speed, and are encouraged to work as fast as possible, the queueing system gets closer to satisfying the assumptions of Burke's Theorem. Therefore, a specific standard could be developed for service speed. To set this company standard, we recommend considering the smallest, median and average total service times among all servers in the company (or among all servers in the branch of interest). For server consistency and monitoring of the mean service time, a company may need to revise its servers training program to include best practices on human customer-server interactions. (c) If Burke's theorem holds, the distributions of time between customer arrivals can be estimated from the distributions of time between customer exits. An automated sampling plan is then possible.

1. Server allocation rule. If more than one server is available, customers tend to walk to the nearest service station. Thus, service stations closer to the end of the queue tend to be more utilised than farther stations. For this reason, it is recommended that the fastest server (minimum mean service time) be assigned to the available service station nearest the end of the queue, and that the other servers be allocated following the FSCSS rule. 2. To minimise the number of multi-services, customers must understand the paperwork and other requirements of the service in advance. In practice, managers could appoint a queue manager who queries customers just entering the queue on their intended operations. Each customer can then be advanced to the queue (if all requirements are completed) or diverted (if some element is missing in their preparation). 3. The number of real server working hours varied over the analysed day. After optimising the number of server working hours in the model, the payroll was significantly reduced by 9.52l%.

Table 11 compares the average times between the customer arrivals in each time interval of the real system and the confidence intervals of the expected times between customers exists in the same real system 68

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applicable to other systems such as call centres, super market cashiers, retail stores, toll booth operations, and manufacturing stations. For complex systems such as real queueing operations, a comprehensive closed mathematical solution is not available. Such systems can be solved by simulation modelling and analysis, but the required information may be expensive to obtain. To reduce the analysis cost in large-scale problems, one requires an automated generalised sample plan that obtains the time between customer arrivals (perhaps as an extension of Burke's theorem) in a general queueing system (per company branch). Because the time between arrivals can change on daily, weekly, and seasonal timescales, the distribution of the times between arrivals (per time interval during the working day) must be accurately forecasted based on the history of the system. These problems are future research directions of the present work.

Table 11 Comparison of times between customer arrivals and time between customer exits. Time interval

8:37 to 9:00 9:00 to 10:00 10:00 to 11:00 11:00 to 12:00 12:00 to 13:00 13:00 to 14:00 14:00 to 15:00 15:00 to 16:00 16:00 to 17:00 17:00 to 18:00 18:00 to 19:00 19:00 to 20:00 20:00 to 21:00

Average TBA

Confidence interval TBE

(seconds)

(seconds)

87.40 31.00 32.80 31.18 29.80 28.70 25.70 31.10 31.80 33.00 29.40 29.40 35.30

125.13 ± 104.47 40.79 ± 7.73 39.02 ± 7.48 42.92 ± 8.02 31.03 ± 4.7 33.49 ± 5.78 31.41 ± 4.04 37.14 ± 5.99 33.56 ± 5.27 58.64 ± 13.38 41.57 ± 7.48 36.48 ± 7.42 28.64 ± 4.16

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(the confidence intervals were approximated, as multi-services were not explicitly reported in the file). When the average time between arrivals is beyond the confidence interval of the expected time between exits (bold-font intervals in Table 11), the hypothesis that the two results differ cannot be rejected. Therefore, in some time intervals of the present study, the distribution of times between customer exits did not well approximate the distribution of times between customer arrivals. 5. Conclusions, recommendations and further research This paper introduced a simulation-based heuristic that minimises the number of system server hours during a working day while constraining the mean customer queueing time. This heuristic finds the appropriate number of active servers among a set of individuals, and allocates them to service stations in each appropriately divided time interval during a working day. The special characteristics of complex queue systems (customer reneging, multi-servicing, the total-service time distributions of individual servers, and the dynamic times between customer arrivals) defy an analytical solution, and require a simulation model. For the benefit of practitioners, we also provide a practical sampling procedure that identifies the distributions of the random variables in the model. Under certain assumptions, the random distribution data input to the model can be obtained without manual sampling. In such cases, the application can be deployed to all branches of the company. The heuristic was demonstrated in a numerical example. The allocation of servers to service stations requires an efficient allocation rule. In our case, allocating the available servers based on their mean service time (fastest to slowest) significantly altered the solution from that of unordered server allocation. The average queueing time and number of serving hours were reduced by 57.73% and 9.52%, respectively, significantly reducing payroll and enhancing the business profit. The company involved in the study identified a need to review its servertraining program. Especially, it intends to introduce best practices on server-customer interactions, which will improve the servers consistency and reduce their mean total service times. In addition, when the queue system is interrupted by a significant proportion of multi-service operations, a queue executive (human, mechanical or logical) should be located at the beginning of the queue. This functionary will check customers' service needs and related elements (e.g., paperwork) before directing or diverting them to or from the queue. Companies that can continuously reduce their customer queue times gain a distinct advantage in the competitive service industry. The advantage is especially wide for companies serving customers at the base of the pyramid. Although this paper focussed on banking operations, its findings are 69

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