Copyright 2006 John Wiley & Sons, Inc. Beni Asllani University of Tennessee at Chattanooga Waiting Line Analysis for Service Improvement Operations Management - 5 th Edition Chapter 17 Roberta Russell & Bernard W. Taylor, III

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

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

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

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

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

Copyright 2006 John Wiley & Sons, Inc.17-7 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

Copyright 2006 John Wiley & Sons, Inc.17-8 Cost Relationship in Waiting Line Analysis Expected costs Level of service Total cost Service cost Waiting Costs

Copyright 2006 John Wiley & Sons, Inc.17-9 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

Copyright 2006 John Wiley & Sons, Inc 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

Copyright 2006 John Wiley & Sons, Inc 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 =   (  - )

Copyright 2006 John Wiley & Sons, Inc 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 =  (  - )

Copyright 2006 John Wiley & Sons, Inc 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  -  Average number of customers waiting in line L q = = = 3.2 (24) 2 30( ) 2  (  - )

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

Copyright 2006 John Wiley & Sons, Inc 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

Copyright 2006 John Wiley & Sons, Inc Constant Service Times  Constant service times occur with machinery and automated equipment   Constant service times are a special case of the single-server model with undefined service times

Copyright 2006 John Wiley & Sons, Inc Operating Characteristics for Constant Service Times  P 0 = 1 - Probability that no customers are in system Average number of customers in system L = L q +  Average number of customers in queue Lq =Lq =Lq =Lq = 2 2  (  - )

Copyright 2006 John Wiley & Sons, Inc Operating Characteristics for Constant Service Times (cont.)   = Probability that the server is busy Average time customer spends in the system W = W q + 1 Average time customer spends in queue Wq =Wq =Wq =Wq =  L q

Copyright 2006 John Wiley & Sons, Inc Constant Service Times: Example Automated car wash with service time = 4.5 min Cars arrive at rate = 10/hour (Poisson)  = 60/4.5 = 13.3/hour W q = = 1.14/10 =.114 hour or 6.84 minutes L q (10) 2 2(13.3)( ) L q = = = 1.14 cars waiting 2 2  (  - )

Copyright 2006 John Wiley & Sons, Inc Finite Queue Length   A physical limit exists on length of queue   M = maximum number in queue   Service rate does not have to exceed arrival rate (  ) to obtain steady-state conditions P 0 = Probability that no customers are in system 1 - /  1 - ( /  ) M + 1 Probability of exactly n customers in system  P n = (P 0 ) for n ≤ M n L = - Average number of customers in system /  /  1 - /  (M + 1)( /  ) M ( /  ) M + 1

Copyright 2006 John Wiley & Sons, Inc Finite Queue Length (cont.) Let P M = probability a customer will not join system Average time customer spends in system W =W =W =W = L  (1 - P M ) L q = L -  (1- P M )  Average number of customers in queue Average time customer spends in queue W q = W - 1

Copyright 2006 John Wiley & Sons, Inc Finite Queue: Example First National Bank has waiting space for only 3 drive in window cars. = 20,  = 30, M = 4 cars (1 in service + 3 waiting) Probability that no cars are in the system P 0 = = = / (20/30) /  1 - ( /  ) M + 1 P n = (P 0 ) = (0.38) = Probability of exactly 4 cars in the system  n=Mn=Mn=Mn=M L = - = 1.24 Average number of cars in the system /  /  1 - /  (M + 1)( /  ) M ( /  ) M + 1

Copyright 2006 John Wiley & Sons, Inc Finite Queue: Example (cont.) Average time a car spends in the system W = = hr L  (1 - P M ) L q = L - = 0.62  (1- P M )  Average number of cars in the queue Average time a car spends in the queue W q = W - = hr 1

Copyright 2006 John Wiley & Sons, Inc Finite Calling Population Arrivals originate from a finite (countable) population N = population size Probability of exactly n customers in system P n = P 0 where n = 1, 2,..., N n  N! (N - n)! Average number of customers in queue L q = N - (1- P 0 )  +  Probability that no customers are in system P 0 = n = 0 n = 0 N! (N - n)!  N n 1

Copyright 2006 John Wiley & Sons, Inc Finite Calling Population (cont.) Wq =Wq =Wq =Wq = L q L q (N - L) (N - L) Average time customer spends in queue L = L q + (1 - P 0 ) Average number of customers in system W = W q + Average time customer spends in system 1

Copyright 2006 John Wiley & Sons, Inc trucks which operate an average of 200 days before breaking down ( = 1/200 day = 0.005/day) Mean repair time = 3.6 days (  = 1/3.6 day = /day) Probability that no trucks are in the system P 0 = Average number of trucks in the queue L q = Average number of trucks in system L = ( ) =.520 Finite Calling Population: Example Average time truck spends in queue W q = 1.74 days Average time truck spends in system W = 5.33 days

Copyright 2006 John Wiley & Sons, Inc 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.

Copyright 2006 John Wiley & Sons, Inc 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 s!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.)

Copyright 2006 John Wiley & Sons, Inc 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.)

Copyright 2006 John Wiley & Sons, Inc 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 = Probability no students are in the system Number of students in the service area L = 6

Copyright 2006 John Wiley & Sons, Inc 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 = Waiting time in the service area W = L / = 0.60

Copyright 2006 John Wiley & Sons, Inc  Add a 4th server to improve service  Recompute operating characteristics  P 0 = 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.)