1. Copyright 2006 John Wiley & Sons, Inc. Beni Asllani University of Tennessee at Chattanooga Waiting Line Analysis for Service Improvement Operations Management - 6 th Edition Chapter 5 Roberta Russell & Bernard W. Taylor, III

2. Copyright 2006 John Wiley & Sons, Inc.17-2 Waiting Line Analysis  Operating characteristics average values for characteristics that describe the performance of a waiting line system average values for characteristics that describe the performance of a waiting line system  Queue A single waiting line A single waiting line  Waiting line system consists of Arrivals Arrivals Servers Servers Waiting line structures Waiting line structures

3. Copyright 2006 John Wiley & Sons, Inc.17-3 Elements of a Waiting Line  Calling population Source of customers Source of customers Infinite - large enough that one more customer can always arrive to be served Infinite - large enough that one more customer can always arrive to be served Finite - countable number of potential customers Finite - countable number of potential customers  Arrival rate (λ) Frequency of customer arrivals at waiting line system Frequency of customer arrivals at waiting line system Typically follows Poisson distribution Typically follows Poisson distribution

4. Copyright 2006 John Wiley & Sons, Inc.17-4 Elements of a Waiting Line (cont.)  Service time Often follows negative exponential distribution Often follows negative exponential distribution Average service rate = μ Average service rate = μ  Arrival rate (λ) must be less than service rate (μ) or system never clears out

5. Copyright 2006 John Wiley & Sons, Inc.17-5 Elements of a Waiting Line (cont.)  Queue discipline Order in which customers are served Order in which customers are served First come, first served is most common First come, first served is most common  Length can be infinite or finite Infinite is most common Infinite is most common Finite is limited by some physical Finite is limited by some physical

6. Copyright 2006 John Wiley & Sons, Inc.17-6 Basic Waiting Line Structures  Channels are the number of parallel servers Single channel Single channel Multiple channels Multiple channels  Phases denote number of sequential servers the customer must go through Single phase Single phase Multiple phases Multiple phases  Steady state A constant, average value for performance characteristics that system will reach after a long time A constant, average value for performance characteristics that system will reach after a long time

7. Copyright 2006 John Wiley & Sons, Inc.17-7 Single-Channel Structures Single-channel, single-phase Waiting lineServer Single-channel, multiple phases Servers Waiting line

8. Copyright 2006 John Wiley & Sons, Inc.17-8 Multi-Channel Structures Servers Multiple-channel, single phase Waiting line Servers Multiple-channel, multiple-phase

9. Copyright 2006 John Wiley & Sons, Inc.17-9 Operating Characteristics NOTATIONOPERATING CHARACTERISTIC LAverage number of customers in the system (waiting and being served) L q Average number of customers in the waiting line WAverage time a customer spends in the system (waiting and being served) W q Average time a customer spends waiting in line P 0 Probability of no (zero) customers in the system P n Probability of n customers in the system ρUtilization rate; the proportion of time the system is in use Table 16.1

10. Copyright 2006 John Wiley & Sons, Inc.17-10 Cost Relationship in Waiting Line Analysis Expected costs Level of service Total cost Service cost Waiting Costs

11. Copyright 2006 John Wiley & Sons, Inc.17-11 Single-server Models  All assume Poisson arrival rate  Variations Exponential service times Exponential service times General (or unknown) distribution of service times General (or unknown) distribution of service times Constant service times Constant service times Exponential service times with finite queue length Exponential service times with finite queue length Exponential service times with finite calling population Exponential service times with finite calling population

12. Copyright 2006 John Wiley & Sons, Inc.17-12 Basic Single-Server Model: Assumptions  Poisson arrival rate  Exponential service times  First-come, first-served queue discipline  Infinite queue length  Infinite calling population  = mean arrival rate   = mean service rate

13. Copyright 2006 John Wiley & Sons, Inc.17-13 Formulas for Single- Server Model L =   -   - Average number of customers in the system Probability that no customers are in the system (either in the queue or being served) P 0 = 1 -  Probability of exactly n customers in the system P n = P 0 n = 1 -  n  Average number of customers in the waiting line L q =   (  - )

14. Copyright 2006 John Wiley & Sons, Inc.17-14 Formulas for Single- Server Model (cont.)  =  Probability that the server is busy and the customer has to wait Average time a customer spends in the queuing system W = = 1  -  L Probability that the server is idle and a customer can be served I = 1 -   = 1 - = P 0 Average time a customer spends waiting in line to be served W q =  (  - )

15. Copyright 2006 John Wiley & Sons, Inc.17-15 A Single-Server Model Given = 24 per hour,  = 30 customers per hour Probability of no customers in the system P 0 = 1 - = 1 - = 0.20  2430 L = = = 4 Average number of customers in the system  -  24 30 - 24 Average number of customers waiting in line L q = = = 3.2 (24) 2 30(30 - 24) 2  (  - )

16. Copyright 2006 John Wiley & Sons, Inc.17-16 A Single-Server Model Average time in the system per customer W = = = 0.167 hour 1  -  1 30 - 24 Average time waiting in line per customer W q = = = 0.133  (  -  ) 24 30(30 - 24) Probability that the server will be busy and the customer must wait  = = = 0.80  2430 Probability the server will be idle I = 1 -  = 1 - 0.80 = 0.20

17. Copyright 2006 John Wiley & Sons, Inc.17-17 Service Improvement Analysis  Possible Alternatives Another employee to pack up purchases Another employee to pack up purchases service rate will increase from 30 customers to 40 customers per hour service rate will increase from 30 customers to 40 customers per hour waiting time will reduce to only 2.25 minutes waiting time will reduce to only 2.25 minutes Another checkout counter Another checkout counter arrival rate at each register will decrease from 24 to 12 per hour arrival rate at each register will decrease from 24 to 12 per hour customer waiting time will be 1.33 minutes customer waiting time will be 1.33 minutes  Determining whether these improvements are worth the cost to achieve them is the crux of waiting line analysis

18. Copyright 2006 John Wiley & Sons, Inc.17-18 Waiting Line Cost Analysis Add extra employee to increase service rate from 30 to 40 customers per hour Add extra employee to increase service rate from 30 to 40 customers per hour Extra employee costs $150/week Extra employee costs $150/week Each one-minute reduction in customer waiting time avoids $75 in lost sales Each one-minute reduction in customer waiting time avoids $75 in lost sales Waiting time with one employee = 8 minutes Waiting time with one employee = 8 minutes W q = 0.038 hours = 2.25 minutes 8.00 - 2.25 = 5.75 minutes reduction 5.75 x $75/minute/week = $431.25 per week New employee saves $431.25 - $150.00 = $281.25/wk

19. Copyright 2006 John Wiley & Sons, Inc.17-19 Waiting Line Cost Analysis New counter costs $6000 plus $200 per week for checker New counter costs $6000 plus $200 per week for checker Customers divide themselves between two checkout lines Customers divide themselves between two checkout lines Arrival rate is reduced from  = 24 to  = 12 Arrival rate is reduced from  = 24 to  = 12 Service rate for each checker is  = 30 Service rate for each checker is  = 30 Example 16.2 W q = 0.022 hours = 1.33 minutes 8.00 - 1.33 = 6.67 minutes 6.67 x $75/minute/week = $500.00/wk - $200 = $300/wk Counter is paid off in 6000/300 = 20 weeks

20. Copyright 2006 John Wiley & Sons, Inc.17-20 Waiting Line Cost Analysis Adding an employee results in savings and improved customer service Adding an employee results in savings and improved customer service Adding a new counter results in slightly greater savings and improved customer service, but only after the initial investment has been recovered Adding a new counter results in slightly greater savings and improved customer service, but only after the initial investment has been recovered A new counter results in more idle time for employees A new counter results in more idle time for employees A new counter would take up potentially valuable floor space A new counter would take up potentially valuable floor space Example 16.2

21. Copyright 2006 John Wiley & Sons, Inc.17-21 Two or more independent servers serve a single waiting line Poisson arrivals, exponential service, infinite calling population s  >  P 0 = 11s!  s s  s  - s  -  n=s-1 n=0 1 11n!n!11n!n!  n + Basic Multiple-server Model Computing P 0 can be time-consuming. Tables can used to find P 0 for selected values of  and s.

22. Copyright 2006 John Wiley & Sons, Inc.17-22 Probability of exactly n customers in the system P n = P 0, for n > s 1 s! s n-s n P 0, for n > s 1 11n!n!11n!n!  n Probability an arriving customer must wait P w = P 0 1 11s!s!11s!s! s  s  - s  - s Average number of customers in system L = P 0 +  ( /  ) s (s - 1)!(s  - ) 2  Basic Multiple-server Model (cont.)

23. Copyright 2006 John Wiley & Sons, Inc.17-23 W = L Average time customer spends in system   =  /s  Utilization factor Average time customer spends in queue W q = W - = 1 L q L q = L -  Average number of customers in queue Basic Multiple-server Model (cont.)

24. Copyright 2006 John Wiley & Sons, Inc.17-24 Multiple-Server System: Example Student Health Service Waiting Room = 10 students per hour  = 4 students per hour per service representative s = 3 representatives s  = (3)(4) = 12 P 0 = 0.045 Probability no students are in the system Number of students in the service area L = 6

25. Copyright 2006 John Wiley & Sons, Inc.17-25 Multiple-Server System: Example (cont.) L q = L - /  = 3.5 Number of students waiting to be served Average time students will wait in line W q = L q / = 0.35 hours Probability that a student must wait P w = 0.703 Waiting time in the service area W = L / = 0.60

26. Copyright 2006 John Wiley & Sons, Inc.17-26  Add a 4th server to improve service  Recompute operating characteristics  P 0 = 0.073 prob of no students  L = 3.0 students  W = 0.30 hour, 18 min in service  L q = 0.5 students waiting  W q = 0.05 hours, 3 min waiting, versus 21 earlier  P w = 0.31 prob that a student must wait Multiple-Server System: Example (cont.)

27. Copyright 2006 John Wiley & Sons, Inc.17-27 Cost Evaluation  Service Cost = (Number of servers) x (wages per time unit) = s C s  Waiting Cost = (Number of customers waiting in the system) x (cost of waiting per time unit) = L C w  Total cost = service cost + waiting cost