PROJECT ON QUEUING THEORY
- 1. STOCHASTIC PROJECT ON QUEING THEORY M/M/1 MODEL -TULIKA GARG (1933116) -AMAN SAINNI (1933128) -LORIKA KAPOOR(1933146) -RITIK ARYA (1933149)
- 2. WE THE STUDENTS OF DEPARTMENT OF STATISTICS, TAKE THIS MOMENT TO ACKNOWLEDGE EFFORTS OF EVERY ONE WHO HAS CONSTANTLY GUIDED US IN THE SUCCESSFUL COMPLETION OF THIS PROJECT. WE WOULD LIKE TO THANK OUR PRINCIPAL MRS. VIBHA SINGH CHAUHAN AND PROFF. ALKA SABHARWAL. FOR PROVIDING US WITH THE OPPORTUNITY TO EXPERIMENT AND CONDUCT THE CONCEPT OF QUEUING THEORY. THIS PROJECT PROVIDED US WITH THE OPPORTUNITY TO CLOSELY UNDERSTAND THE CONCEPT AS WELL AS ITS APPLICATION . ALSO WE REALIZED THE IMPORTANCE OF TEAMWORK, COOPERATION AND TIME MANAGEMENT IN OUR LIVES. IN THE PANDEMIC TIMES, IT WAS ALSO A CHALLENGE FOR US TO COORDINATE AMONGST US, WE WOULD LIKE TO THANK OUR PARENTS AND FRIENDS WHO TRULY MOTIVATED US GUIDED US TO DO THE BEST. AT LAST, WE WOULD LIKE TO THANK OUR PRESTIGIOUS INSTITUTION –KIRORI MAL COLLEGE FOR BELIEVING IN US. WE WOULD ALSO LIKE TO HAVE YOUR FEEDBACK ON OUR EFFORTS AS WE BELIEVE IN CONSTANTLY IMPROVING OUR SKILLSET. Acknowledgement This Photo by Unknown author is licensed under CC BY-NC-ND.
- 3. INDEX o Theory o Characteristics o Symbols and Formulas o Assumptions o Data Simulation o Observations From Simulation o Conclusion
- 4. THEORY
- 5. QUEUE A LINE OR A SEQUENCE OF PEOPLE, VEHICLES, ETC. AWAITING THEIR TURN TO BE ATTENDED TO OR TO PROCEED. This Photo by Unknown author is licensed under CC BY-NC.
- 6. QUEING THEORY It determines the measure of performance of waiting lines which can be used to design services. It is not an optimization technique since queuing cannot be eliminated completely without incurring inordinate expenses. The major goal is to reduce the adverse impact of a queue to tolerable levels.
- 7. Input/arrival distribution Output/departure (service) distribution Service Channels Service Discipline Maximum number of customers allowed in the system Calling source CHARACTERISTICS
- 8. SOURCE
- 10. SYMBOLS AND FORMULAS ( M/M/1)(FCFS,∞,∞) Ws Expected waiting time per customer in the system Wq Expected waiting time by customer in the queue Ls Expected no. Of customers in the system Lq Expected no. Of customers in the queue λ Mean arrival rate µ Mean service rate ρ=λ/μ Utilization factor n No. Of units in the system Pn Probability of exactly n customers in the system.
- 11. ASSUMPTIONS ➢The rate by which customers arrive is given by Poisson process ➢The rate by which services are provided is given by exponential distribution. ➢As a result of first 2 assumptions, the inter-arrival time also becomes exponential. ➢The service is provided on FCFS( first come first served) basis. ➢All the customers are patient, I.e. no one leave the queue in middle.
- 12. DATA SIMULATION
- 13. INPUT Arrival rate (λ) 10 per hour Average service time 5 minutes Service rate(µ) 12 per hour The data is stimulated for given/observed arrival and service rate of a queue system. The simulated data in per-minutes and has been generated for a period of 2 weeks. Assume it to be the case of some emergency service/ call-center where the counter is open 24*7 with no space limit. Also, Arrival rate is always less than service rate in an efficient system, if not the system will fail in ling run as their will be non- ending queues.
- 14. AIM The aim of our project is to firstly check is appropriateness/coincidence with theoretical values and then to check If there is any scope of improvement in the system I.e we want o idenfy if there is any possible value of arrival rate or service rate such that queues are small? the system effficency is at its peak?
- 15. Simulated Data (1st 10 stimulations ) Cell Ref. B C D E F G H 8 Time random arrival Number in queue Busy Time left Exit 9 00:00 0.614413 0 0 0 0 0 10 00:01 0.647613 0 0 0 0 0 11 00:02 0.026604 1 0 1 2 0 12 00:03 0.79067 0 0 1 1 1 13 00:04 0.881719 0 0 0 0 0 14 00:05 0.793045 0 0 0 0 0 15 00:06 0.024242 1 0 1 3 0 16 00:07 0.776826 0 0 1 2 0 17 00:08 0.131736 1 1 1 1 1 =IF(C9<$B$2/60,1,0) =IF(G9>1,E9+D10,IF(H9=1,MAX(0,E9 +D10-H9),E9)) =IF(D9=1,1,0) =IF(OR(G9>1,E9>0,D10=1), 1,0) =IF(G9>1,G9- 1,IF(F10=1,MAX(1,ROUND(- $B$3*(LN(1-RAND())),0)),0)) =IF(G9=1,1,0) =IF(F9=1,MAX(1,ROUND(- $B$3*LN(1-RAND()),0)),0) ▪ Cell B2 consists the arrival rate ▪ Cell B3 consists the service time.
- 16. Adding another column of "Total no. of customers in the system " by adding people in queue + people getting served. X frequency probability Expected 0 3244 0.166667 3360.167 1 3177 0.138889 2800.139 2 2515 0.115741 2333.449 3 2186 0.096451 1944.541 4 1852 0.080376 1620.451 5 1486 0.06698 1350.376 6 1197 0.055816 1125.313 7 819 0.046514 937.7608 8 608 0.038761 781.4674 9 651 0.032301 651.2228 10 569 0.026918 542.6857 11 585 0.022431 452.2381 12 445 0.018693 376.8651 13 293 0.015577 314.0542 14 122 0.012981 261.7118 15 174 0.010818 218.0932 16 121 0.009015 181.7443 17 70 0.007512 151.4536 18 23 0.00626 126.2113 19 9 0.005217 105.1761 20 15 0.004347 87.64677 TOTAL 20161 0.978263 19722.77 Our observed and expected frequencies are approximately distributed as negative exponential distribution thus matching the theoretical results.
- 17. OUTPUT/OBSERVATION FROM STIMULATION OUTPUT observed theoretical arrivals (λ) 9.764880952 10 server utilization(λ/µ) 83.88888889 83.33333 average no. of customers in queue(Lq) 5.622222222 4.166667 Average service time 4.589752492 5 average length of non-empty queue(Ln) 4.797482171 6 variance of no. of customers in the system 15.1973579 30 average no .of customers in the system (Ls) 4.109121571 5 o Arrival rate and service time indicate that aur generations coincide with the model assumed. o The rest assure that are result match with the theoretical results making the model more reliable and comparable. o The server utilizations here show that the staff/server is busy almost 83% of the time.
- 18. SCOPE FOR IMPROVEMENT? Arrival rate (λ) 12 per hour average service time 5 minutes service rate(µ) 12 per hour OUTPUT observed theoretical arrivals (λ) 11.81845 12 server utilization(λ/µ) 98.84921 100 (Lq) 34.87059 Not defined (Ln) 51.15296 Not defined Customer fluctuations 35.85957 Not defined (Ls) 0.012549 Not defined OUTPUT observed theoretical arrivals (λ) 10.97619 11 server utilization(λ/µ) 91.94444 91.66666667 (Lq) 10.62904 10.08333333 (Ln) 15.59184 12 customer fluctuations 140.6133 132 (Ls) 11.54913 11 Arrival rate (λ) 11 per hour average service time 5 minutes service rate(µ) 12 per hour If the company tries to increase customers by marketing aur any other campaigns such that : If the company tries to increase customers by marketing aur any other campaigns such that : Ln new > Ln old Lq new > Lq old Ln new > Ln old Lq new > Lq old SU new > SU old
- 19. Arrival rate (λ) 10 per hour average service time 4 minutes service rate(µ) 15 per hour If the company find a new technology/workforce such that : OUTPUT observed theoretical arrivals (λ) 9.919643 10 server utilization(λ/µ) 68.61607 66.66666667 (Lq) 1.053767 1.333333333 (Ln) 1.545554 3 customer fluctuations 3.656231 6 (Ls) 1.740588 2 Lq new < Lq old Ln new < Ln old SU new < SU old
- 20. FINAL COMMENTS AND CONCLUSIONS The model seems to be working appropriately and according to (M /M/1). The improvement in the system can occur if either arrival rate increases but does not become equal to service rate, or if service rate falls. But if the later happens it results the server utilization to go severely down which may lead to problems of its own. So, the most ideal way might be to increase customers and reduce the service rate. However, doing any of these in short periods is not possible thus the best way to overcome inefficiency of server will be to increase customer arrival rate by 1 as shown in case 2.