Development of Time-Dependent Queuing Models. in Non-Stationary Condition

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1 The 11th Asia Pacific Industrial Engineering and Management Systems Conference Development of Time-Dependent Queuing Models in Non-Stationary Condition Nur Aini Masruroh 1, Subagyo 2, Aulia Syarifah Hidayatullah 3, Katrin Rifanni Pamella 4 Mechanical and Industrial Engineering Department, Gadjah Mada University Jl. Grafika 2 Yogyakarta 55281, Indonesia aini@ugm.ac.id 1, subagyo@ugm.ac.id 2, aulia.syarifah.h@gmail.com 3, cutrin_901@yahoo.com 4 Abstract - The assumption of stationary increment used in conventional queuing theory has limitation in its applications. Some queuing systems such as university canteen, post office, and bank have different pattern at different time interval. In this case the stationary assumption does not apply. In the present work, models on non-stationary time-dependent queuing system are introduced. Two types of queuing environments are studied; single channel single phase and multiple channel single phase. The models are developed based on different types of data input; (i) arrival rate and service rate (ii) inter-arrival time and service rate (iii) combination of arrival rate, inter-arrival time, and service rate. The data input is modeled by using the best fitted distribution approach and the average value based on piecewise linear approximation. Performance evaluation of the models using several cases of queuing systems for both queuing environments reveals; (i) the error for non-stationary time dependent queuing models in flexible interval is smaller than time-independent queuing and non-stationary time dependent queuing in fixed interval, (ii) queuing model using combination of inter-arrival time and service rate as input provides smaller queuing parameter error, and (iii) input using average value is suitable for systems with identical inter-arrival time, while system with varied and significantly different in inter-arrival time is more suitable to use mean distribution as input. Keywords: single channel single phase, multi channel single phase, queuing theory, time-dependent queuing, non-stationary interval 1. INTRODUCTION Time dependent queuing model is commonly developed based on stationary condition, i.e. assuming to have identical independent distribution all the time. This assumption cannot be applied in some systems such as university canteen, post office, and bank. In these systems the arrival rate is a function of time and it has different pattern for different time interval. Some work on time-dependent queuing system has been done. Keller (1982) proposed time-dependent queuing algorithm for single server system. The arrival rate and service rate are assumed to follow exponential distribution with slowly changing time-dependent rates. The solution algorithm can be used to solve some queuing systems with small changing in arrival and service rate. Wulandari (2007) develop time-dependent queuing model for single : Corresponding Author. Tel Fax address: aini@ugm.ac.id channel-single phase system. The model is developed based on order 8 th Fourier series. Widiatmoko (2007) develop time-dependent queuing model for multiple channel-single phase system. The model is also developed based on order 8 th Fourier series. The drawbacks for both models are the possibility to get negative value for some queuing parameters that is impossible. Liani (2008) and Woko (2008) developed timedependent queuing models for multiple channel-single phase system and multiple channel-multiple phase system consequently. Both models are developed based on spreadsheet software. However, the previous models are developed under the stationary assumptions. As mention previously, in some cases this assumption is invalid. Hence in this work we proposed some time dependent queuing models for the system characterized by non-stationary time interval. The

2 Number of The 11th Asia Pacific Industrial Engineering and Management Systems Conference models allow each time interval to have different distribution. 2. METHODOLOGY This work consists of five main steps. The first step is determining the data pattern. Two way ANOVA test is conducted to see the effect of days and period within a day to the arrival rate. The second step is setting up the time interval. Two types of time interval are introduced; fixed interval and flexible interval. In the fixed interval, a day is divided into equal length of time, while in the flexible interval the length of interval is set based on the similarity of the condition of the respective interval. The model is developed for each time interval. The third step is model development. Two queuing environments are considered; single channel-single phase and multiple channel-single phase. The forth step is finding the best suited distribution of the arrival rate and service rate for each time interval. Three types of data input are used; (i) arrival rate and service rate (ii) inter-arrival time and service rate (iii) combination of arrival rate, inter-arrival time, and service rate. The queuing parameters are also determined using the developed models. The last step is model evaluation and implementation. In this work, four case studies (university canteen, post office, and two banks) are used to develop the model. 3. MODEL DEVELOPMENT The models are developed based on spreadsheet software. Based on two-way ANOVA test using α=0.05, for all cases there is no significant difference between days but the difference between time period within a day is statistically significant. It confirms the non-stationary assumption to be applied. Hence, the models are developed for each time period within a day. Kolmogorov-Smirnov test and Anderson-Darling test are used to test the best suited distribution Flexible time interval The idea of flexible time interval is combining several fixed intervals into single interval. For this purpose, a piecewise linear approximation method is used. The idea is to combine the adjacent intervals that have similar trend. The advantage of this method is; (i) it can reduce the number of interval, hence the number of model developed is reduced (ii) as linear approximation is used, it is possible to use average value instead of mean distribution as model s input. 3.2 Single channel-single phase model Two models are developed based in fixed interval and six models are based on flexible interval. The university canteen and post office are used as the case studies. Based on the statistical test for the data, the best length of time interval for the fixed time interval for the university canteen is 30 minutes and 45 minutes for the post office. The flexible time interval is determined by using the relationship between period and the number of arrived during the respective time period. Figure 1 shows an example of piecewise linear approximation to determine the length of interval for the university canteen queuing system. It can be seen that the first interval can be merged with the second interval hence the length of interval becomes 1 hour (instead of 30 minutes in fixed interval case). The same condition also applies to period 4 to 6 and 9 to Setting time interval Two types of time interval are considered; fixed interval and flexible interval Fixed time interval Fixed time interval is set by dividing a day into 2 to 12 intervals. For example, if a day is divided into 3 intervals it means the length of interval is 2 hours if it is assumed that the system operates 6 hours a day. For each interval the distribution of the arrival rate, inter-arrival time and service time are assessed. Interval which has the best suited distribution is selected, i.e. the spread of the data within this interval can be described well using certain distribution. Figure 1. Piecewise linear approximation for determining the length of interval (University Canteen case) The models are developed for each time interval for both fixed interval and flexible interval. Table 1 shows the description for the queuing models.

3 The 11th Asia Pacific Industrial Engineering and Management Systems Conference Table 1. Model description Model Interval Data input Arrival X Fixed Average arrival rate, Y Fixed arrival rate, A Flexible Average arrival rate, B Flexible arrival rate, C Flexible Average inter arrival time, second D Flexible inter arrival time, second E Flexible - Average arrival rate, - Average inter arrival time, second F Flexible - arrival rate, - inter arrival time, second Service rate, rate, rate, rate, The model is designed to be executed in spreadsheet (spreadsheet model) so users can use it easily. The queuing parameters considered are expected number of in the system (L), expected number of in the queue (L q ), expected waiting time per in the system (W s ), and expected waiting time per in the queue (W q ). Arrival rate for each input type can be defined as follows. a. Model with arrival rate and service rate as input (1) b. Model with inter-arrival time and service rate as input (2) c. Model with combination of arrival rate, interarrival time, and service rate input (3) The queuing parameter can be determined as follows. a. The number of in the server at the end of (4) b. The number of in the queue at the end of (5) c. The number of in the system at the end d. Expected waiting time per in the queue (6) e. Expected waiting time per in the system (7) (8) The inputs used in the spreadsheet model are: 1., duration of the observation, minutes 2., average (or mean distribution) of arrival rate, 3., average (or mean distribution) of 4., average (or mean distribution) of interarrival time, minutes 3.3 Multiple channel-single phase model As the case for single channel-single phase model, the models are also developed based on fixed interval and flexible interval. For this queuing environment, the cases of two national banks in Indonesia are used; Bank Mandiri (Bank A) and Bank BPD DIY (Bank B). The statistical test shows that for the fixed interval, the optimal length of time interval for both banks is 1 hour. The length of interval for the flexible time interval is determined by using piecewise linear approximation approach. The queuing parameters considered in multiple channel is the same as the parameters used in single channel. The only difference is in this case s number of server is used. However, because multiple server has s number of server, two approaches in determining the service rate ( are introduced. The first approach is to use the average and the second approach is to use the individual service rate. The queuing parameters for the first approach (average service rate) can be defined as follows. a. The number of in the server at the end of

4 The 11th Asia Pacific Industrial Engineering and Management Systems Conference (9) b. The number of in the queue at the end of With (18) is the actual number of in queue at time t is the predicted value of in queue n is the number of experiments (10) c. The number of in the system at the end (11) d. Expected waiting time per in the queue (12) e. Expected waiting time per in the system (13) The queuing parameters for the second approach (individual service rate) are as follows. a. The number of in the server at the end (14) b. The number of in the queue at the end (15) c. Expected waiting time per in the system = IF ( = 0, 0, ) (16) d. Traffic intensity at n (t) = IF ( = 0, 0, ) (17) The model is developed for each time interval for both fixed interval and flexible interval with the description is shown in Table MODEL EVALUATION The models accuracy is determined through the parameter s values. These values are compared to the actual values. The parameter error used is Mean Absolute Deviation (MAD) that is defined as Table 2 shows the MAD for single channel single phase model. It can be seen that models using arrival rate (number of per minute) as input enable to provide better accuracy than those using inter-arrival time as input. However, the models those use combination of arrival rate and inter-arrival time as input is more accurate than other models. It can be explained as follows. The error for model C and model D in early period is very high. This is because in the early he inter-arrival of the is very diverse and only a small number of s arrive. It can be seen that the model uses distribution (model D) provide less error than the model uses average value (C). Nevertheless the result also shows that the use of interarrival time when its value is too diverse is not recommended. For that reason model E and F are developed. The inter-arrival input is used in the period when the arrival of is very diverse otherwise arrival rate is used. Table 2. MAD for single channel-single phase model Model A B C D E F X Y Post office Univ. canteen As model E is recommended to model the post office and model F is recommended for university canteen, paired-t-test will be conducted to test the model validity. Using α=0.05, it is not sufficient evidence to say that both model are statistically different from the real value. Time independent queuing model is also used as benchmark model. The formula used is as follows. L (19) 2 L (20) q ( ) L 1 W (21) L W q (22) q Unfortunately, if equation (19) (22) are used to solve

5 ρ(t), traffic intensity Expected number of in queue The 11th Asia Pacific Industrial Engineering and Management Systems Conference these cases, it will result negative value for some parameter that is impossible. The time independent also assumed that the arrival rate follows Poisson distribution and the service rate follows exponential distribution all the time. However, in this study this assumption does not hold. In the case of multiple channel-single phase models, the models are developed based on equal service rate assumption and different service rate (individual service rate). The equal service rate assumptions said that all servers have the same service time. Table 3 shows the parameter error for the models. Table 4. MAD for time independent model Case MAD Post Office University canteen Bank A Bank B Table 3. MAD for multiple channel-single phase models Model A B C D E F X Y Equal service rate Bank A Bank B Individual service rate Bank A Bank B DETERMINING NUMBER OF SERVERS As the models have been validated, it will be used to evaluate the number of servers required. Suppose let us use the case of Bank A. In this case, model E is used to estimate the queuing parameters. Figure 2 shows the number of s in queue for Table 3 shows that the use of equal service rate assumptions looks better than the use of individual service rate. However, statistical test using α = 0.05 concludes that both approaches are not significantly different with p- value=0.112 for Bank A and p-value=0.728 for Bank B. Hence, the use of equal service rate will simplify the model. Table 3 also shows that for Bank A, model E together with model A provides smaller MAD. In this case the length of interval for fixed and flexible interval is the same, therefore model X turns out to model A. But it is not the case for Bank B. Based on the MAD value, model Y is recommended; fixed interval with mean inter-arrival distribution is used as input. It seems that join interval does not work as the model requires more detail (i.e. smaller interval required). In addition, the use of mean distribution is needed instead of average value. Further, the paired-t-test is conducted with α=0.05 and it concludes that model X and model Y are not statistically different from the real systems they described. As the case for single channel-single phase, time independent queuing model for multiple servers is also used as benchmark model. Similar to single channel-single phase environment, some parameter values have negative values that is impossible. Table 4 shows the MAD for the time independent model for both single channel and multiple channel models. Figure 2 Expected number of s in queue It can be seen from Figure 2 that the expected number of s in queue is varied. In this case the number of server required for each period is determined based on the expected number of. If time independent model is used to determine the number of server, 3 servers are recommended. However, the proposed time dependent model recommends using 3 servers for period 1-2 (8 am to 10 am) and period 6 (1 pm to 2 pm), and using 2 servers for other periods. Figure 3 and Figure 4 show the traffic Perbandingan Server Utilization Factor Model intensity and the expected number of in the queue Time Dependent dan Time Independent respectively after revising the number of server. 1,05 1 0,95 0,9 0,85 0,8 0,75 0,7 0,65 0, time independent time dependent mandiri

6 Expected number of s in queue The 11th Asia Pacific Industrial Engineering and Management Systems Conference Perbandingan Lq untuk Model Time Dependent dan Figure 3. Traffic intensity for the new scenario Time Independent 3 to use mean distribution as input. 2 1 time dependent time independent REFERENCES Figure 4. Expected number of for the new scenario Figure 3 shows that the traffic intensity for the time dependent model is higher than the time dependent but it is still no more than 1. It means that the service rate is higher than arrival rate, hence all s can be served at the respective period. Figure 4 shows an interesting point. Although time independent recommends using 3 servers all the time, but the expected number in queue for this model is higher than the time independent model. It is because in time independent, it is assumed that the arrival rate and the service rate are identical all the time. 5. CONCLUSION This work focuses on developing mathematical model for non-stationary time dependent queuing. Two queuing environments are considered; single channel-single phase and multiple channel-single phase. The model is designed to be applied in spreadsheet software. The numerical example shows that the time dependent queuing models outperform the time independent queuing model. Furthermore model with combination of arrival rate and inter arrival time as input provides better accuracy. The use of average value as input is suitable for systems with identical inter-arrival time, while system with varied and significantly different in inter-arrival time is more suitable Eick, S.G., Massey, W.A., Whitt, W., (1993), Mt/G/ With Sinusoidal Arrival Rate, AT &T Bell Laboratories, New Jersey. Hidayatullah, A.S., (2010), Development of Time Dependent Queuing Model for Single Channel Single Phase for Non-Stationary Condition, Thesis, Industrial Engineering, Gadjah Mada University, Indonesia. Jensen, P.A. dan Bard, J.F., (2003), Operations Research: Models and Method, John Willey and Sons, New York. Keller J.B., (1982), Time-Dependent Queues, SIAM Review, Vol. 24, No. 4, pp Lapin, L.L., Whisler, W.D., (2002), Quantitative Decision Making With Spreadsheet Application, Thompson Learning, Stamford. Liani, S., (2008), Development of Time Dependent Queuing Model for Single Channel Multi Phase, Thesis, Industrial Engineering, Gadjah Mada University, Indonesia. Taha, H.A., (2003), Operation Research An Introduction, 7 th edition, Prentice-Hall, Inc, New Jersey. Widiatmoko, L.D., (2007), Development of Time Dependent Queuing Model for Multi Channel Multi Phase, Thesis, Industrial Engineering, Gadjah Mada University, Indonesia. Woko, B.S., (2008), Spreadsheet Based Queuing Model for Multi Channel Multi Phase, Thesis, Industrial Engineering, Gadjah Mada University, Indonesia. Wulandari, N. E., (2007), Mathematical Model Development for Time Dependent Queuing Model, Thesis, Industrial Engineering, Gadjah Mada University, Indonesia.