22 Bank Teller Problem

22.1 Introduction

This is a classical problem in queuing theory: should there be multiple queues,one for each teller in a bank, or should we have one single queue for all thetellers? We have adopted this version of the problem from an example in [2].The two situations are shown in Figs. 22.1 and 22.2.

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Fig. 22.2. Bank Tellers: Single Queue

First look at Fig. 22.1. Customers arrive according to the exponentialdistribution (which produces inter-arrival times) and they immediately go toany teller that is free (not busy). If all tellers are busy, then the customerchooses the shortest queue to wait for service. Once in a queue the customerwaits there, and does not jump to other queues, in a first-come first-servedbasis, until he/she has completed their business. Customers require differenttypes of transactions numbered #1 through #5 in Table 22.1. This table givesthe probabilities and approximate mean service times for each of these typesof transactions. For example, type #2 occurs with probability 0.3 and has

James J. Buckley: Simulating Fuzzy Systems, StudFuzz 171, 161–164 (2005)www.springerlink.com c© Springer-Verlag Berlin Heidelberg 2005

162 22 Bank Teller Problem

approximate mean service time 2 minutes. Service times are governed by theexponential distribution. So, for transaction type #2 the service time will becomputed using the exponential distribution having mean approximately 2minutes. Customers only have one transaction, from Table 22.1, to completeand after service they leave the bank.

In Fig. 22.2 customers arrive according to the exponential distributionbut they have one queue for all the tellers. The queue discipline is also first-come, first-served. The customers still have the same types of transactions,probabilities and approximate mean service times as shown in Table 22.1.Service times are computed as in the multiple queue case.

Table 22.1. Business Transactions at the Bank

Type Probability Mean Service Time (minutes)

1 0.1 ≈12 0.3 ≈23 0.3 ≈34 0.2 ≈45 0.1 ≈5

The bank manager is interested in only one statistic, R = response time,mean time in the system, for these two models. Time units will be in minutes.Our study period is a four hour period on Monday through Friday. We willsimulate for one half a year, or 31,200 minutes, in order to be sure to get intosteady-state since all simulations begin with the bank empty (the swampingmethod in Chap. 7). All run times were less than one second.

Using historical data we estimate arrival rates and the mean service timesfor the different bank transactions in Table 22.1. We assume that the proba-bilities for the various transactions in Table 22.1 are known and crisp. Theycould be estimated from data and then we would model them as a discretefuzzy probability distribution; but then this would be another study which weshall not do in this book. As in Chap. 3 we obtain fuzzy estimators given inTable 22.2. Notice that in this table λ for arrivals is given as a rate (numberof arrivals per minute), so 1/λ is the mean time between arrivals. But theµi are the mean service times in minutes, for the i-th business transaction,1 ≤ i ≤ 5. Fuzzy estimators give fuzzy distributions and fuzzy systems andthen response time will be a fuzzy number R. We wish to compare R for thetwo systems: multiple queues vs single queue.

We assume there is a continuous function F so that R = F (all parameters)and then by the extension principle R = F (fuzzy parameters). However, wedo not know this function, nor can we derive it or find it in the literature. Sowe use crisp simulation to estimate the alpha-cuts of R (Chap. 7). Actually,except for the complications of choosing the smallest queue and the different

22.4 Summary 163

Table 22.2. Fuzzy/Crisp Probability Distributions for the Bank

Item Distribution Details

Arrivals Exponential λa = (0.5/1/1.5) rateService #1 Exponential µ1 = (0.5/1/1.5) timeService #2 Exponential µ2 = (1/2/3) timeService #3 Exponential µ3 = (2/3/4) timeService #4 Exponential µ4 = (3/4/5) timeService #5 Exponential µ5 = (4/5/6) timeTransactions Discrete 0.1/1, 0.3/2, 0.3/3, 0.2/4, 0.1/5

transactions in Table 22.1, an F could be found in most operations researchtext books [1].

The last thing we need to do before simulation is to decide on how tochoose the parameters in their alpha-cuts to estimate the end points of theintervals R[α]. In this problem this is easily solved: (1) for the left end pointof R[α] use λ the left end point of λ[α] and use µi the left end point of µi[α],1 ≤ i ≤ 5; and (2) for the right end point of R[α] use λ the right end pointof λ[α] and use µi the right end point of µi[α], 1 ≤ i ≤ 5.

22.2 First Simulation: Multiple Queues

The simulation results are in Table 22.3 and graphs of R are in Fig. 22.3.For the rest of this chapter we assume that there are six tellers. The bankmanager does not want to hire more tellers.

Table 22.3. Multiple Queues Simulation Alpha-Cuts of R

Item α = 0 Cut α = 0.5 Cut α = 1 Cut

R [1.976, 16.504] [2.475, 4.569] 3.180

22.3 Second Simulation: Single Queue

The simulation results are in Table 22.4 and the graph of R is in Fig. 22.3.

22.4 Summary

We see from Fig. 22.3 that there is not much difference in the time spent inthe system for the two models. The base of these fuzzy numbers is like a 99%

164 22 Bank Teller Problem

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Fig. 22.3. Graphs of R (Minutes)

Table 22.4. Single Queue Simulation Alpha-Cuts of R

Item α = 0 Cut α = 0.5 Cut α = 1 Cut

R [1.976, 21.690] [2.456, 4.105] 3.036

confidence interval. However, multiple queue R has less uncertainty than thesingle queue R. It looks like we almost have multiple queue R a fuzzy subsetof the single queue R (Sect. 2.2.3 of Chap. 2). We would recommend thesingle queue method (see below).

There are situations where one model may do better (less response time)that the other model. In fact, for the multiple queue model, if we let customerschange queues to the faster moving line, then its response time should de-crease and the multiple queue model may be doing a little better than thesingle queue model.

Many banks in the US have changed to the single queue model. Whenasked why the change the answer was usually that it gives the customermore privacy with the teller. This bank manager sees that both models giveapproximately the same response time, the multiple queue method has lessuncertainty but the single queue allows more privacy, so the bank managerchanges to the single queue. A GPSS program for the single queue model isin Chap. 28.

References

1. H.A. Taha: Operations Research, Fifth Edition, Macmillan, New York, 1992. 1632. Thomas J. Schriber: Simulation Using GPSS, John Wiley and Sons, New York,

1974. 161