SPREADSHEET SIMULATION: ICT ENABLED TEACHING OF OPERATIONS RESEARCH

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1 Bulletin of the Marathwada Mathematical Society Vol. 14, No. 1, June 2013, Pages SPREADSHEET SIMULATION: ICT ENABLED TEACHING OF OPERATIONS RESEARCH S. B. Kalyankar S.A.S. Mahavidylaya, Mukhed, Dist. Nanded India snjvkalvankar@yahoo.co.in and H. S. Acharya Allana Institute of Management Science, Azam Campus, Camp, Pune Abstract ICT enabled teaching has its own benefits. On shear strength of practice on computers, students can be made to visualize and get a proper first hand feel of many concepts. In this article we have tried to demonstrate this aspect of teaching OR to students of engineering management. We have chosen queuing as the concept to illustrate. It is true that these students accept the use of computers very easily. Many of these students carry laptops and are quite computer savvy. This attitude of students can be effectively exploited to expose them to a pedagogy with which they are naturally comfortable. This article uses SSQMT as the tool to use. However there can be other tools and the discussion on pedagogy will apply universally. 1 1 INTRODUCTION ICT enabled teaching has its own benefits. If a student happens to be computer savvy, many of his learning disabilities can be effectively masked and the process of learning can be made quite enjoyable. On shear strength of practice on computers students can be made to visualize and get a proper first hand feel of many concepts. Teaching OR to students of engineering, management has always been a challenge mainly because of their resistance to the traditional methods of class room teaching. At the same time it is true that these students accept the use of computers very easily. Many of these students carry laptops and are quite computer savvy. This attitude of students can be effectively exploited to expose them to a pedagogy with which they are naturally comfortable. Routing and queuing theories are quite complex. In this paper concepts of digital simulation of queues and models for its effective use in teaching of Queuing theory are discussed. 1 Key Words: Spreadsheet Simulation, Probability Distribution functions, Queueing Marathwada Mathematical Society, Aurangabad, India, ISSN

2 24 S. B. Kalyankar and H. S. Acharya 2 DIGITAL SIMULATION Simulation can be defined as the process of creating an abstract representation of a system in order to identify and understand those factors which control the system or to predict the future behavior of the system. Almost any system which can be quantitatively described by using equations or rules can be simulated. Simu1ation is an experimental problem solving technique. Simulations make possible the study of, and experimentation with, the interactions of a complex system or a subsystem thereof [10]. Simulations can be either event driven or time driven [3]. One can look at a Queuing system as driven by arrival of a customer which is a discrete event driving system. 2.1 Use of Speared sheet for simulation of queuing system. Queuing problems though are quite complex in nature can be easily simulated. To begin with, a student can simulate very simple queues using a Spreadsheet. In fact scenarios created on spread sheets and what if analysis are very popular with researchers in management and business analysis. Looking at the simplicity of the tool its effectiveness and adaptability are amazing. Example 2.1 Consider a queuing system where interarrival rate is, constant say 2 unit time per customer, and it takes exactly 1 unit of time to serve a customer. The problem can be trivially simulated on a spreadsheet(see Fig 1). What was given in Ex.2.1 is a deterministic queue with constant arrival rate and constant service rate. It is possible to simulate more complex models too. Operations Research an Introduction by Hamdy A. Taha [3] provides a CD which contains ready spread-sheet based simulation model for learners to practice with.

3 Spreadsheet Simulation: ICT Enabled Teaching Of THE MODEL REPRESENTATIONS USING KENDAL S NOTATION The illustrative example (Example 2.1) is a trivially simplistic model. Visualizing various queuing models ([1], [2], [7], [10], [8], [11]) and understanding how they are compactly represented using the Kendall s notation is the next step. A queuing system can be represented uniquely using a six character combination A/B/C/D/E/F. Kendall [6] has carefully planned the first three, where as fourth and fifth was introduced by A. M. Lee in 1966, and sixth notation was added by Taha in The attributes represented by these characters is given in Table 1. Table1: Symbols used to Represent Attributes of a Queuing Models Sr. No. Symbol Meaning Values / Options 1 A Represents the distribution of the inter-arrival time. Constant, Uniform Dist (D), Triangular Dist, 2 B Represents the distribution of the service time. Exponential Dist (M), General Distr. (G). 3 C Number of service channels 1, 2, 3,, M in 4 D Queue discipline FCFS, LIFO, Priority, Random etc. 5 E System capacity Infinite, or finite 6 F Size of calling source Infinite, finite Once the above notations are understood, one can straightway learn to use the simulation SSQMT (Speared sheet simulation of queuing model by Taha) as discussed in the next section. 4 SIMULATION OF SINGLE SERVER MODELS USING SSQMT In this section step by step procedures to simulate two of the selected models are explained. Students can follow the steps to simulate more models as given in Table 5, and gain confidence. SSQMT is a loadable excel work book, named SingleServer.xls, loadable from the CD containing TORA, which comes as an accompaniment to the text book [3]. Student has to simply open the SingleServer.xls file. Then follow instructions as given under. Example 4.1 Simulate a queuing model with deterministic inter-arrival time, exponential service time, single service channel and customers limited to 500. Inter-arrival time is 10 units per customer, Service time 11 unit; number of customers is limited to 200. Solution: Note that the model would be D/M/1/200 according to Kendall s notation.

4 26 S. B. Kalyankar and H. S. Acharya Step 1: Open Single Server.xls Sheet 1. This is configured to work as simulator with all spread sheets formulae (Fig. 2 )See Table 5 for locations/cell references to give input values. Step 2: To specify the number of arrivals type the value of N = 200 in C2. Step 3: Type X in A4, this selects Constant as the lnterarrival process Step 4: Type 10 in C4 which is the value of interarrival time. Step 5: Type X in A10 to select Exponential service. Step 6: Type 11 in D10 which is the value of service time. Step 7: Press F 9 to start simulation. Output: You can instantly see the output generated on the Spread Sheet. (See Fig 2 for a Snap Shot) 5 A BRIEF INSIGHT INTO THE PROCESS OF SIMULATION The process of simulation and computation of model statistics is explained in brief here ([3, pp ]. 1. First arrival occures at time T = 0, and service facility is idle at the start. 2. Arrival time of customer will be obtained as follows. Departure time of customer 1 = (arrival time of customer 1 + service time of customer 1)

5 Spreadsheet Simulation: ICT Enabled Teaching Of Arrival time of customer 2 = (arrival time of customer 1 + interarrival time of Customer 1 ) 3. Departure time of customer will be obtained as fallows Departure time of customer i = max{(arrival time i), (departure time i 1)}+ {Service time of i}. 6 MODEL STATISTICS The model statistics like waiting time in queue, waiting time in system, average waiting time in queue, average waiting time in system, and utilization factor will be considered as follows. Waiting time in queue of a customer i is W q (i) and in system W s (i) can be computed as W q (I) = (Departure time of Customer i) (arrival time of customer i) (service time of i) W s (i) = (Departure time of Customer i) (arrival time of customer i) Average waiting time in queue, W q is a sum of W q (i) divided by number of arrivals. n i=1 W q = w q (i) n. Average waiting time in system, w s, be a sum of W s (i) divided by number of arrivals, W s = n i=1 w s (i). n The average facility utilization will be a sum of service times divided by time of departure time of last arriving customer. Sum of Service Times Average Facility Utilization(U) = Departure Time of Last Arriving Customer Percent Idleness Facility Utilization(U i ) = (1 U) 100%. At every time the mentioned model statistics has different nature for different probability distribution selected. These model statistics will also be called as Performance measure of the simulated queueing model. The next section will be used for understanding these performance measures. 7 UNDERSTANDING THE PERFORMANCE MEASURES A Steady state queue is one, where no more changes occur with advance time. Any steady state queue is properly understood, if one can compute six performance measures for the queue ( Table 2). The last two columns in Table 2 are specific to models with Poisson arrival and Exponential service.

6 28 S. B. Kalyankar and H. S. Acharya Table 2: Performance Measures of a Generalized Poisson Queueing Model Sr.No Symbol used Performance Measure Formula Using Little s Formula 1 L s Expected number of L s = np n L s = λ eff W s customers in system n=0 2 L q Expected number of L q = (n c)p n L q = λ eff W q customers in queue n=0 3 W s Expected waiting time in system 4 W q Expected waiting time in queue W s = W q + 1 µ W q = W s + 1 µ W s = Ls λ eff W q = L q λ eff 5 c Expected number of c = L s L q = λ eff µ busy servers 6 U Facility Utilization U = c c In general λ is the arrival rate and µ the service rate per server. Here symbol c is the number of servers in queueing model, λ eff be effective arrival rate [3, p 559]. The performance measures for generalized Poisson queueing model can also be obtained by using simulation method. Different distributions can also be used for simulation of queueing models using SSQMT. 8 THE DISTRIBUTIONS USED IN SIMULATION The probability distributions mentioned in Section 7 are used while simulating queues using SSQMT for arrival as well as service. These distributions can be found in standard text books on probability theory. Table 3 shows these probability distributions with their Mean, Variance and Standard Deviation. Table 3: Probability Distributions with Mean, Variance and Standard Deviation Name of Distribution Probability Density Function Mean Variance Standard Diviation { (PDF) λe λx, x > 0 Exponential Distribution f(x) = 1 1 0, otherwise λ λ 2 1 λ Uniform Distribution f(x) = b a 1, a x b a+b = 0, otherwise 2 2(x a) (b a)(c a) Triangular Distribution P (x) =, a x c a+b+c 2(b x) (b a)(b c), c < x b 3 (b a) 2 12 a 2 +b 2 +c 2 ab ac bc 18 (b a) 2 12 a 2 +b 2 +c 2 ab ac bc 18 The probability distribution stated above is used in SSQMT simulator. The combinations of sixteen different queueing models for inter-arrival and service time distributions will be possible. 9 SIMULATIONS USING SSQMT How to simulate a queueing model using SSQMT is explained in Section 4. In this section the details of cell references for choice of different probability distributions is given. Table 4 shows the Cell Reference and choice of distributions for inter-arrival

7 Spreadsheet Simulation: ICT Enabled Teaching Of Table 4: Cell Reference in SSQMT [3] Sr. No. Inter-arrival Time ( ) ( 1 λ Service Time 1 µ) Distribution Numeric Value Distribution Numeric Value Cell address Cell address Cell address Cell address 1 A4: Constant C4 A9:Constant C9 2 A5: Exponential D5 A10:Exponential D10 3 A6: Uniform D6 and F6 A11:Uniform D11 and F11 4 A7: Triangular D7, F7 and H7 A12:Triangular D12, F12 and H12 time and service time. simulated. Using these different choices the queueing models will be 10 RESULTS OF SIMULATION QUEUEING MODELS USING SSQMT While simulating, the values of inter-arrival time has been varied in ascending order keeping the value of service time unchanged. The number of customers will also be kept constant. The results for three situations will be observed and recorded. 1. Inter-arrival time is less than service time. 2. Inter-arrival time is equal to service time. 3. Inter-arrival time is greater than service time. The simulation of sixteen different queueing models will be carried out and for example two Simulation Models i.e. Example 3 and, Example 2 are given in this paper accompanied with the results of simulation. Example 10.1 If the inter-arrival time (λ) and service time (µ) are constant and the value ( of λ is increased in steps keeping the value of µ unchanged then Case I: 1 λ µ) < 1 i.e. constant inter-arrival time is less than constant service time. ( Case II: 1 λ µ) = 1 i.e. constant inter-arrival time is equal to constant service time. ( ) Case III: 1 λ > 1 µ i.e.constant interarrival time is greater than constant service time. Table 5: Outcome for simulated model 1 Input 1. Constant Service time Sr. (C 9) = 10 Out Put Summary No 2. Nbr (C 2)= 50 Constant Interarrival Time (C4) U U i L q L s W q W s

8 30 S. B. Kalyankar and H. S. Acharya Following deductions can be made from the Table 5. Case I II III Deductions (a) Facility is busy and no idleness exists. (b) Queue length and system length are reasonably high. (c) Waiting time in queue and system will be very high. (a) Facility is busy and no idleness exists. (b) Queue length and system length are zero. (c) Waiting time in queue is zero and in system it is equal to µ. (a) Server will have to wait for customers and facility becomes towards idleness. (b) Queue length is zero and system length is negligible. (c) Waiting time in queue is zero and in system it is equal to µ. 11 CONCLUSIONS We have discussed the use of spread sheet simulation to enhance the students appreciation regarding queues and simulation. Such approach of teaching allows not only better understanding of queueing theory but also better introduction to speared sheet simulation. References [1] Frederick S. Hillier and Gerald J. Lieberman, Operational Research, Second Edition CBS Publishers and Distributors, 485, Jain Bhawn Nath Nagar, Shahdra,

9 Spreadsheet Simulation: ICT Enabled Teaching Of New Delhi , (India). [2] Gross, G. and Harris C. Fundamentals of Queueing Theory, New York: A Wiley-Interscience Publication, John Wiley & Sons, Inc.(1998). [3] Hamdy A. Taha, Operations Research: An Introduction, Pearson Prentice Hall, New Delhi, India (2009) ( books.google.com/books/about/ Operations research.html?id.) [4] Harry Perros, Computer Simulation Techniques. The Definitive Introduction, (2009)This book is available for free download from web site: [5] Kainti Swarup, P. K. Gupta and Man Mohan, Operations Research Sultan Chand & Sons, Educational Publishers, New Delhi,(2005). [6] Kendell D. G., Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of Imbedded Chain, Ann. Math. Stat., 24, (1953). [7] Kleinrock, L., Queueing Systems - Volume 1: Theory, John Wiley & Sons, New York, (1975). [8] Li Wen LIU, Koichi and Masashi, A Queueing Model of a Production System with two Machienes, one Operator and Priorities, Queueing System 4, (1989), [9] Maggu P. L., Operational Research and Linear Programming, Shalini Prakashan, 192, Khari Kuan, Meerut (U.P.), (1972). [10] Malwina J. et.al., On the Maximum Queue Length in the Supermarket Model, The Annals of Probability, Vo1.34, No.2, (2006), [11] Moez Draief, The Single Server Queue and the Storage Model: Large Deviation and Fixed Points, Journal of Applied Mathematics and Stochastic Analysis, Volume 2006, 1-22, (2005). [12] Narsingh Deo, System Simulation with Digital Computer, Prentice-Hall of India, Pvt. Ltd. New Delhi , (2006). [13] U Narayan Bhat, An Introduction to Queueing Theory, Modeling and Analysis in Applications, Brikhauser, Boston, (2008).