MNG221 - Management Science –

1. MNG221 - Management Science – Queuing Analysis

2. Queuing Analysis Introduction Queuing analysisis the probabilistic analysis of waiting lines. • Because time is a valuable resource, the reduction of waiting time is an important topic. • Providing quick service is an important aspect of quality customer service.

3. Queuing Analysis Introduction • Aware of this, more and more companies are focusing on reducing waiting time as an important component of quality improvement. • Increased service capacity will reduce waiting time and provide faster service, which usually means adding more servers.

4. Queuing Analysis Introduction • However, increasing service capacity has a monetary cost, and therein lies the basis of waiting line analysis: the trade-off between the cost of improved service and the cost of making customers wait.

5. Queuing Analysis Elements of Waiting Line Analysis

6. Elements of Waiting Line Analysis • Waiting lines form because people or things arrive at the servicing function, or server, faster than they can be served. • Waiting lines result because customers do not arrive at a constant, evenly paced rate, nor are they all served in an equal amount of time.

7. Elements of Waiting Line Analysis • Thus, a waiting line is continually increasing and decreasing in length (and is sometimes empty), and it approaches an average rate of customer arrivals and an average time to serve the customer in the long run.

8. Elements of Waiting Line Analysis • Decisions about waiting lines and the management of waiting lines are based on the average customer arrival and service time. • Operating Characteristics are average values for characteristics that describe the performance of a waiting line system:

9. Elements of Waiting Line Analysis • Example: The average time a customer must wait in line, the average number of customer waiting in line etc. NOTE: The operating characteristics of a queuing system are steady states.

10. Queuing Analysis The Single-Server Waiting Line System

11. The Single-Server Waiting Line System The Fast Shop Market waiting line system Components of a waiting line system include arrivals, servers, and the waiting line structure.

12. The Single-Server Waiting Line System The most important factors to consider in analyzing a queuing system: • The Queue Discipline - is the order in which waiting customers are served. (In what order customers are served?)

13. The Single-Server Waiting Line System • The Queue Discipline Example: Customers - can be processed on a First Come, First Serve, A Predetermine Appointment, or Alphabetically according to their last name. A machine queue discipline, however can have “last-in, first-out” or “random selection”.

14. The Single-Server Waiting Line System The most important factors to consider in analyzing a queuing system: • The nature of the calling population (where customers come from). The calling population is the source of the customers, which may be finite or infinite. • Queuing systems that have an assumed infinite calling population are more common.

15. The Single-Server Waiting Line System The most important factors to consider in analyzing a queuing system: • The arrival rate (how often customers arrive at the queue). The arrival rate (λ) is the frequency at which customers arrive at a waiting line during a specified period of time according, to a probability distribution and is most frequently described by a Poisson distribution.

16. The Single-Server Waiting Line System The most important factors to consider in analyzing a queuing system: • This rate can be estimated from empirical data derived from studying the system or a similar system, or it can be an average of these empirical data.

17. The Single-Server Waiting Line System The most important factors to consider in analyzing a queuing system: • The service rate (how fast customers are served). The service rate (μ) is the average number of customers who can be served during a specified time period and is often described by the negative exponential distribution.

18. The Single-Server Waiting Line System The most important factors to consider in analyzing a queuing system: • A service rate is similar to an arrival rate in that it is a random variable. • However, to analyze a queuing system, both arrivals and service must be in compatible units of measure. Thus, service time must be expressed as a service rate to correspond with an arrival rate.

19. The Single-Server Waiting Line System • Poisson Distribution a probability distribution that describes the occurrence of a relatively rare event in a fixed period of time; often used to define arrivals at a service facility in a queuing system • Exponential Distribution a probability distribution often used to define the service times in a queuing system

20. Single-Server (SS) Model Assumptions of the basic SS Model • An infinite calling population • A first-come, first-served queue discipline • Poisson arrival rate • Exponential service times NOTE: customers must be served faster than they arrive (λ>μ) , or an infinitely large queue will build up.

21. Single-Server (SS) Model Assumptions of the basic SS Model Given that • λ= the arrival rate (average number of arrivals per time period) • µ = the service rate (average number served per time period) • λ= mean arrival rate; • µ = mean service rate; We can state the following formulas for the operating characteristics of a single-server model.

22. Single Server Queuing Formulas The probability that no customer is in the queuing system (either in the queue or being served) The probability that ncustomer are in the queuing system

23. Single Server Queuing Formulas The average number of customers in the queuing system (i.e., customers being served and in the waiting line) The average number of customers in the waiting line is

24. Single Server Queuing Formulas The average time a customer spends in the total queuing system (i.e., waiting and being served) is: The average time a customer spends waiting in the queue to be served is:

25. Single Server Queuing Formulas The probability that the server is busy (i.e., the probability that a customer has to wait), known as the utilization factor, is: The probability that the server is idle (i.e., the probability that a customer can be served) is:

26. Single Server Queuing Formulas • This last term, 1 - (λ /µ), is also equal to P0. That is, the probability of no customers in the queuing system is the same as the probability that the server is idle. • We can compute these various operating characteristics for Fast Shop Market by simply substituting the average arrival and service rates into the foregoing formulas.

27. Queuing Analysis The Single-Server Waiting Line SystemA Worked Example

28. Single Server Queuing AWorked Example If λ = 24 customers per hour arrive at checkout counter µ = 30 customers per hour can be checked out

29. Single Server Queuing AWorked Example If λ = 24 customers per hour arrive at checkout counter µ = 30 customers per hour can be checked out

30. Single Server Queuing AWorked Example If λ = 24 customers per hour arrive at checkout counter µ = 30 customers per hour can be checked out

31. Single Server Queuing AWorked Example If λ = 24 customers per hour arrive at checkout counter µ = 30 customers per hour can be checked out

32. Single Server Queuing AWorked Example Important Aspects of General Model and Example • The operating characteristics are averages. • They are assumed to be steady-stateaverages. • Steady state is a constant average level that a system realizes after a period of time. • For a queuing system, the steady state is represented by the average operating statistics, also determined over a period of time.

33. Queuing Analysis The Effect of Operating Characteristics on Managerial Decisions

34. Effects of Operating Characteristics on Managerial Decisions • Alternative 1: The addition of an employee • Alternative 2: Addition of a new check out counter

35. Effects of Operating Characteristics on Managerial Decisions Alternative 1: The addition of an employee • The addition of an extra employee will cost the store manager $150 per week. • For each minute that average customer waiting time is reduced, the store avoids a loss in sales of $75 per week.

36. Effects of Operating Characteristics on Managerial Decisions Alternative 1: The addition of an employee • If a new employee is hired, customers can be served in less time, that is the service rate will increase to • µ = 40 customers served per hour • The arrival rate will remain the same (λ = 24 per hour) • λ = 24 customers per hour arrive at checkout counter

37. Effects of Operating Characteristics on Managerial Decisions

38. Effects of Operating Characteristics on Managerial Decisions Alternative 1: The addition of an employee • The average waiting time per customer has been reduced from 8 minutes to 2.25 minutes. • The savings (that is, the decrease in lost sales) is computed as follows: 8.00 min. - 2.25 min. = 5.75 min. 5.75 min. x $75/min. = $431.25

39. Effects of Operating Characteristics on Managerial Decisions Alternative 1: The addition of an employee • Because the extra employee costs management $150 per week, the total savings will be: $431.25 - $150 = $281.25 per week

40. Effects of Operating Characteristics on Managerial Decisions Alternative 2: The addition of a new checkout counter • Constructing a new checkout counter will cost $6,000, plus an extra $200 per week for an additional cashier. • The new checkout counter would be opposite the present counter.

41. Effects of Operating Characteristics on Managerial Decisions Alternative 2: The addition of a new checkout counter • There would be several display cases and racks between the two lines so that customers waiting in line would not move back and forth between the lines. • Such movement, called jockeying, would invalidate the queuing formulas we already developed.

42. Effects of Operating Characteristics on Managerial Decisions Alternative 2: The addition of a new checkout counter • We will assume that the customers would divide themselves equally between the two lines, so the arrival rate for each line would be half of the prior arrival rate for a single checkout counter.

43. Effects of Operating Characteristics on Managerial Decisions Alternative 2: The addition of a new checkout counter • Thus, the new arrival rate for each checkout counter is: λ = 12 customers per hour arrive at checkout counter And the service rate remains the same for each of the counters: µ =30 customers per hour can be checked out

44. Effects of Operating Characteristics on Managerial Decisions

45. Effects of Operating Characteristics on Managerial Decisions • Using the same sales savings of $75 per week for each minute's reduction in waiting time, we find that the store would save: 8.00 min. - 1.33 min. = 6.67 min. 6.67 min. x $75/min. = $500.00 per week • Next we subtract the $200 per week cost for the new cashier from this amount saved: $500 - 200 = $300

46. Effects of Operating Characteristics on Managerial Decisions • Because the capital outlay of this project is $6,000, it would take 20 weeks ($6,000/$300 = 20 weeks) to recoup the initial cost (ignoring the possibility of interest on the $6,000). • Once the cost has been recovered, the store would save $18.75 ($300.00 - 281.25) more per week by adding a new checkout counter rather than simply hiring an extra employee.

47. Effects of Operating Characteristics on Managerial Decisions • However, we must not disregard the fact that during the 20-week cost recovery period, the $281.25 savings incurred by simply hiring a new employee would be lost.

48. Operating Characteristics for each Alternative System

49. Queuing Analysis Undefined and Constant Service Times

50. Single-Server (SS) Model Assumptions of the basic SS Model • An infinite calling population • A first-come, first-served queue discipline • Poisson arrival rate • Exponential service times NOTE: customers must be served faster than they arrive (λ>μ) , or an infinitely large queue will build up.