Simulation of  Queuing problems using  Random numbers -- Renuka Narang
Simulation <ul><li>Simulation  is imitation of some real thing, or a process.  </li></ul><ul><li>The act of simulating som...
Simulation Model <ul><li>A  simulation model  is a mathematical model that calculates the impact of  uncertain  inputs and...
Simulation techniques <ul><li>Simulation techniques can be used to assist management decision-making, where analytical met...
Simulation and Queuing problems. <ul><li>A major application of simulation has been in the analysis of waiting line, or qu...
Queuing problems. <ul><li>For queuing systems, it is usually not possible to develop analytical formulas, and simulation i...
Components of queuing systems <ul><li>A queue system can be divided into four components </li></ul><ul><ul><li>Arrivals:  ...
Structures of queuing system <ul><li>There are a number of structures of queuing systems in practice.  </li></ul><ul><li>W...
Random Numbers <ul><li>What is the purpose of random numbers? </li></ul><ul><ul><li>There is randomness in the way custome...
Random Numbers <ul><li>Such numbers can be computer generated, and are often listed in published statistical tables.  </li...
Example Problem <ul><li>The arrival time of a customer at a retail sales depot is according to the following distribution ...
Solution Calculation of Basis of random allocation Inter arrival time Probability Cumulative Probability Basis of random a...
Solution Customer Random Number Inter arrival time Random number Service time Time of arrival Service starts at Service en...
Solution <ul><li>Average waiting time per customer is  </li></ul><ul><ul><li>4/10 = .4 minutes </li></ul></ul><ul><li>Perc...
<ul><li>Waiting time should be as less as possible!! </li></ul><ul><li>  Thank you!   </li></ul>
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Simulation Techniques

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Simulation For Queuing Problems Using Random Numbers

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Simulation Techniques

  1. 1. Simulation of Queuing problems using Random numbers -- Renuka Narang
  2. 2. Simulation <ul><li>Simulation is imitation of some real thing, or a process. </li></ul><ul><li>The act of simulating something generally involves representation of certain </li></ul><ul><ul><li>key characteristics or </li></ul></ul><ul><ul><li>behaviours </li></ul></ul><ul><li>of a selected physical or abstract system. </li></ul><ul><li>Simulation involves the use of models to represent real life situation. </li></ul>
  3. 3. Simulation Model <ul><li>A simulation model is a mathematical model that calculates the impact of uncertain inputs and decisions we make on outcomes that we care about, such as profit and loss, investment returns, etc. </li></ul><ul><li>A simulation model will include: </li></ul><ul><ul><li>Model inputs that are uncertain numbers/ uncertain variables </li></ul></ul><ul><ul><li>Intermediate calculations as required </li></ul></ul><ul><ul><li>Model outputs that depend on the inputs -- These are uncertain functions </li></ul></ul>
  4. 4. Simulation techniques <ul><li>Simulation techniques can be used to assist management decision-making, where analytical methods are either not available or inappropriate. </li></ul><ul><li>Typical business problems where simulation could be used to aid management decision-making are </li></ul><ul><ul><li>Inventory control. </li></ul></ul><ul><ul><li>Queuing problems. </li></ul></ul><ul><ul><li>Production planning. </li></ul></ul>
  5. 5. Simulation and Queuing problems. <ul><li>A major application of simulation has been in the analysis of waiting line, or queuing systems. </li></ul><ul><li>Since the time spent by people and things waiting in line is a valuable resource, the reduction of waiting time is an important aspect of operations management. </li></ul><ul><li>Waiting time has also become more important because of the increased emphasis on quality. Customers equate quality service with quick service and providing quick service has become an important aspect of quality service </li></ul>
  6. 6. Queuing problems. <ul><li>For queuing systems, it is usually not possible to develop analytical formulas, and simulation is often the only means of analysis. </li></ul><ul><li>Simulation can hence be used to investigate problems that are common in any situation involving customers, items or orders arriving at a given point, and being processed in a specified order. </li></ul><ul><li>For ex: </li></ul><ul><ul><li>Customers arrive in a bank and form a single queue, which feeds a number of service desks. The arrival rate of the customers will determine the number of service desks to have open at any specific point in time </li></ul></ul>
  7. 7. Components of queuing systems <ul><li>A queue system can be divided into four components </li></ul><ul><ul><li>Arrivals: Concerned with how items (people, cars etc) arrive in the system. </li></ul></ul><ul><ul><li>Queue or waiting line: Concerned with what happens between the arrival of an item requiring service and the time when service is carried out. </li></ul></ul><ul><ul><li>Service: Concerned with the time taken to serve a customer. </li></ul></ul><ul><ul><li>Outlet or departure: The exit from the system. </li></ul></ul><ul><li>A queuing problem involves striking a balance between the cost of making reductions in service time and the benefits gained from such a reduction </li></ul>
  8. 8. Structures of queuing system <ul><li>There are a number of structures of queuing systems in practice. </li></ul><ul><li>We will study only one i.e. single queue – single service point. </li></ul><ul><ul><li>Single queue – single service point </li></ul></ul><ul><ul><li>Queue discipline is first come – first served. </li></ul></ul><ul><ul><li>Arrivals* are random and for simulation this randomness must be taken into account. </li></ul></ul><ul><ul><li>Service times ** are random and for simulation this randomness must be taken into account </li></ul></ul><ul><li>*Inter-arrival time : Is the time between the arrival of successive customers in a queuing situation. </li></ul><ul><li>**Service time: Is the length of time taken to serve customers </li></ul>
  9. 9. Random Numbers <ul><li>What is the purpose of random numbers? </li></ul><ul><ul><li>There is randomness in the way customers are likely to arrive. </li></ul></ul><ul><ul><li>The service time in most of the cases is also variable. </li></ul></ul><ul><ul><li>The purpose of the random numbers is to allow you to randomly select an arrival or service time from the appropriate distribution. </li></ul></ul><ul><ul><li>To account for randomness, random numbers are used. </li></ul></ul>
  10. 10. Random Numbers <ul><li>Such numbers can be computer generated, and are often listed in published statistical tables. </li></ul><ul><li>Here we have a set of random numbers. </li></ul><ul><ul><ul><li>89 07 37 29 28 08 75 01 21 63 </li></ul></ul></ul><ul><ul><ul><li>34 65 11 80 34 14 92 48 83 91 </li></ul></ul></ul><ul><ul><ul><li>52 49 98 44 80 04 42 37 87 96 </li></ul></ul></ul><ul><li>The random numbers are displaced as two-digit numbers in the range between 00 and 99. </li></ul><ul><li>Every number is equally likely to occur and there is no pattern, and thus no way of predicting what number will be next in the sequence. </li></ul>
  11. 11. Example Problem <ul><li>The arrival time of a customer at a retail sales depot is according to the following distribution </li></ul><ul><li>Simulate the process for 10 arrivals and estimate the average waiting time for the customer and percentage idle time for the server. </li></ul><ul><li>Use the following random numbers: </li></ul><ul><li>For IAT: 25, 19, 64, 82, 62, 74, 29, 92, 24, 23, 68, 96. </li></ul><ul><li>For ST: 92, 41,66,07,44,29,52,43,87,55,47,83 </li></ul><ul><li>Assume that the shop opens at 9:00 am in the morning. </li></ul>Inter-arrival time Probability   Service time Probability (in minutes)   (in minutes) 3 0.1   3 0.3 4 0.2   4 0.5 5 0.5   5 0.1 6 0.1   6 0.1 7 0.1      
  12. 12. Solution Calculation of Basis of random allocation Inter arrival time Probability Cumulative Probability Basis of random allocation 3 0.1 0.1 0.0 -- 0.09 4 0.2 0.3 0.1 -- 0.29 5 0.5 0.8 0.3 -- 0.79 6 0.1 0.9 0.9 -- 0.89 7 0.1 1 0.9 -- 0.99 Service time Probability Cumulative Probability Basis of random allocation 3 0.3 0.3 0.0 -- 0.29 4 0.5 0.8 0.3 -- 0.79 5 0.1 0.9 0.8 -- 0.89 6 0.1 1 0.9 -- 0.99
  13. 13. Solution Customer Random Number Inter arrival time Random number Service time Time of arrival Service starts at Service ends at Waiting customer Idle time Inter arrival time 3 4 5 6 7 Service time 3 4 5 6 Basis of random allocation 0.0 -- 0.09 0.1 -- 0.29 0.3 -- 0.79 0.9 -- 0.89 0.9 -- 0.99 Basis of random allocation 0.0 -- 0.29 0.3 -- 0.79 0.8 -- 0.89 0.9 -- 0.99 1 25 4 92 6 9.04 9.04 9.10 -- 4 2 19 4 41 4 9.08 9.10 9.14 2 -- 3 64 5 66 4 9.13 9.14 9.18 1 -- 4 82 6 07 3 9.19 9.19 9.22 -- 1 5 62 5 44 4 9.24 9.24 9.28 -- 2 9.29 9.29 9.32 -- 1 9.33 9.33 9.37 -- 1 9.40 9.40 9.44 -- 3 9.44 9.44 9.49 -- -- 9.48 9.49 9.53 1 -- Total time 53 mins Total waiting time 4 mins Total idle time 12 mins 6 74 5 29 3 7 29 4 52 4 8 92 7 43 4 9 24 4 87 5 10 23 4 55 4
  14. 14. Solution <ul><li>Average waiting time per customer is </li></ul><ul><ul><li>4/10 = .4 minutes </li></ul></ul><ul><li>Percentage average for the server is </li></ul><ul><ul><li>(12/53)*100 = 22.64% </li></ul></ul>
  15. 15. <ul><li>Waiting time should be as less as possible!! </li></ul><ul><li> Thank you!  </li></ul>
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