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1Queuing ModelsLesson 10 LESSONs NINE and TEN QUEUINGMODELS.

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1 1Queuing ModelsLesson 10 LESSONs NINE and TEN QUEUINGMODELS

2 2Queuing ModelsLesson 10 Queuing Models Queuing is the study of waiting lines, or queues. Queuing is the study of waiting lines, or queues. The objective of queuing analysis is to design systems that enable organizations to perform optimally according to some criterion. The objective of queuing analysis is to design systems that enable organizations to perform optimally according to some criterion. Possible Criteria Possible Criteria –Maximum Profits. –Desired Service Level.

3 3Queuing ModelsLesson 10 Elements of the queuing process A queuing system consists of five basic components: Calling population Arrival process Queue configuration Queue discipline Service process

4 4Queuing ModelsLesson 10 CALLING POPULATION

5 5Queuing ModelsLesson 10 CALLING POPULATION CALLING POPULATION Homogeneous or heterogeneous Homogeneous or heterogeneous Can be finite by limiting calls or infinite if population is big Can be finite by limiting calls or infinite if population is big

6 6Queuing ModelsLesson 10 ARRIVAL PROCESS

7 7Queuing ModelsLesson 10 ARRIVAL PROCESS ARRIVAL PROCESS Static no control Static no control Dynamic control by machines or man Dynamic control by machines or man Balking not joining the queuing Balking not joining the queuing Renegingleave line before service Renegingleave line before service

8 8Queuing ModelsLesson 10 QUEUE CONFIGURATION

9 9Queuing ModelsLesson 10 QUEUE CONFIGURATION QUEUE CONFIGURATION Refers to number of queues, locations, spatial requirement and effect on customer behavior Refers to number of queues, locations, spatial requirement and effect on customer behavior Jockeying – line switching activity Jockeying – line switching activity

10 10Queuing ModelsLesson 10 QUEUE DISCIPLINE

11 11Queuing ModelsLesson 10 SERVICE PROCESS

12 12Queuing ModelsLesson 10 Possible Service Measurements Average time a customer spends in line. Average time a customer spends in line. Average length of the waiting line. Average length of the waiting line. The probability that an arriving customer must wait for service. The probability that an arriving customer must wait for service.

13 13Queuing ModelsLesson 10 The Arrival Process The random process is more common in businesses. The random process is more common in businesses. Under three conditions a Poisson Distribution can describe the random arrival process. Under three conditions a Poisson Distribution can describe the random arrival process.

14 14Queuing ModelsLesson 10 The three conditions required for the existence of the Poisson arrival process: The three conditions required for the existence of the Poisson arrival process: –Orderliness : one customer, at most, will arrive during any time interval. –Stationarity : for a given time frame, the probability of arrivals within a certain time interval is the same for all time intervals of equal length. –Independence : the arrival of one customer has no influence on the arrival of another. These conditions are unrestrictive and are approximately satisfied in many situations. These conditions are unrestrictive and are approximately satisfied in many situations.

15 15Queuing ModelsLesson 10 The Poisson Arrival Distribution PXk e k !  t) k t () (  Where = mean arrival rate per time unit. t = the length of the interval. e = (the base of the natural logarithm). k! = k (k -1) (k -2) (k -3) … (3) (2) (1).

16 16Queuing ModelsLesson 10 HANK’s HARDWARE (An illustration of the Poisson distribution) Customers arrive at Hank’s Hardware according to a Poisson distribution. Between 8:00 a.m. and 9:00 a.m., an average of 6 customers arrive at the store. What is the probability that k = 0, 1, 2, … customers will arrive between 8:00 and 8:30 in the morning.

17 17Queuing ModelsLesson 10 SOLUTION                            Input to the Poisson distribution Input to the Poisson distribution = 6 customers per hour. = 6 customers per hour. t = 0.5 hour. t = (6)(0.5) = 3. t = (6)(0.5) = 3. 

18 18Queuing ModelsLesson 10 Measures of Queuing System Performance P 0 = Probability that there are no customers in the system. P n = Probability that there are “n” customers in the system. L = Average number of customers in the system. L q = Average number of customers in the queue. W = Average time a customer spends in the system. W q = Average time a customer spends in the queue. P w = Probability that an arriving customer must wait for service. r = Utilization rate for each server (the percentage of time that each server is busy).

19 19Queuing ModelsLesson 10 Classification of Queues Queuing system can be classified by: Queuing system can be classified by: –Arrival process. –Service process. –Number of servers. –System size (infinite/finite waiting line). –Population size. Notation Notation –M (Markovian) = Poisson arrivals or exponential service time. –D (Deterministic) = Constant arrival rate or service time. –G (General) = General probability for arrivals or service time. Example: M / M / 6 / 10 / 20 Example: M / M / 6 / 10 / 20

20 20Queuing ModelsLesson 10 M / M / 1 Queuing System Characteristics Poisson arrival process. Poisson arrival process. Exponential service time distribution. Exponential service time distribution. A single server. A single server. Potentially infinite queue. Potentially infinite queue. An infinite population. An infinite population.

21 21Queuing ModelsLesson 10 Performance Measures for the M / M /1 Queue P 0 = 1- ( /  ) P 0 = 1- ( /  ) P n = [1 - ( /  )] ( /  ) n P n = [1 - ( /  )] ( /  ) n L = / (  - ) L = / (  - ) L q = 2 / [  (  - )] L q = 2 / [  (  - )] W = 1 / (  - ) W = 1 / (  - ) W q = / [  (  - )] W q = / [  (  - )] P w = /  P w = /   = / 

22 22Queuing ModelsLesson 10 MARY’s SHOES Customers arrive at Mary’s Shoes every 12 minutes on the average, according to a Poisson process. Customers arrive at Mary’s Shoes every 12 minutes on the average, according to a Poisson process. Service time is exponentially distributed with an average of 8 minutes per customer. Service time is exponentially distributed with an average of 8 minutes per customer. Management is interested in determining the performance measures for this service system. Management is interested in determining the performance measures for this service system.

23 23Queuing ModelsLesson 10 SOLUTION Input Input = 1 / 12 customers per minute = 1 / 12 customers per minute = 60 / 12 = 5 per hour.  = 1 / 8 customers per minute  = 1 / 8 customers per minute = 60 / 8 = 7.5 per hour. Performance Calculations Performance Calculations P 0 = 1- ( /  ) = 1 - (5 / 7.5) = P n = [1 - ( /  )] ( /  ) = (0.3333)(0.6667) n L = / (  - ) = 2 L q = 2 / [  (  - )] = W = 1 / (  - ) = 0.4 hours = 24 minutes W q = / [  (  - )] = hours = 16 minutes P 0 = 1- ( /  ) = 1 - (5 / 7.5) = P n = [1 - ( /  )] ( /  ) = (0.3333)(0.6667) n L = / (  - ) = 2 L q = 2 / [  (  - )] = W = 1 / (  - ) = 0.4 hours = 24 minutes W q = / [  (  - )] = hours = 16 minutes P w = /   = / 

24 24Queuing ModelsLesson 10 WINQSB Input Screen  

25 25Queuing ModelsLesson 10 Performance Measurements

26 26Queuing ModelsLesson 10 M / M / k Queuing Systems Characteristics Customers arrive according to a Poisson process at a mean rate  Customers arrive according to a Poisson process at a mean rate  Service time follow an exponential distribution. Service time follow an exponential distribution. There are k servers, each of which works at a rate of  customers. There are k servers, each of which works at a rate of  customers. Infinite population, and possibly infinite line. Infinite population, and possibly infinite line.

27 27Queuing ModelsLesson 10 Performance Measure

28 28Queuing ModelsLesson 10 The performance measurements L, L q, W q,, can be obtained from Little’s formulas. from Little’s formulas.

29 29Queuing ModelsLesson 10 Revision Question At a Food Lion Provision Shop, customers spend an average of 25 mins selecting their groceries and checking out by entering a single line queue served by two cashiers. The service times required for the cashiers to check out customers follow an exponential distribution and average four minutes. Customers arrive at the cashier counter according to a Poisson distribution at the average rate of eight customers per hour. The table below shows part of the computer calculation: Determine the followings : –average time a customer spends in the store (in minutes). –average number of customers waiting in line prior to being checked out. –proportion of customers who will have to wait in line. What assumption(s) did you make in part (a)? Comment whether the assumption(s) is/are realistic.

30 30Queuing ModelsLesson 10 Multiple queue advantages : Multiple queue advantages : Service can be differentiated Division of labour possible Selection option for customer Deterred balking Queuing Process

31 31Queuing ModelsLesson 10 Multiple queue disadvantages : Multiple queue disadvantages :Anxiety Lack of fairness Lack of privacy Types of services clearly stated Queuing Process

32 32Queuing ModelsLesson 10 Single queue advantages : Single queue advantages : First come first serve (fairness) No anxiety to select fastest line Reneging difficult Queue cutting resolved Privacy enhanced Reducing average waiting time Queuing Process

33 33Queuing ModelsLesson 10 Single queue disadvantages : Single queue disadvantages : No specialisation Unnecessary held up in waiting line Possible balking Queuing Process

34 34Queuing ModelsLesson 10 Take a number advantages : Take a number advantages : No need for formal line Free to wander about and browse items More relaxing Queuing Process

35 35Queuing ModelsLesson 10 Take a number disadvantages : Take a number disadvantages :Reneging Require large waiting area Queuing Process

36 36Queuing ModelsLesson 10 QUEUE DISCIPLINE QUEUE DISCIPLINE Policy of selecting next customer from the queue for service Policy of selecting next customer from the queue for service First come first serve most popular static method (selection depends on position in queue only) First come first serve most popular static method (selection depends on position in queue only) Queuing Process

37 37Queuing ModelsLesson 10 Dynamic disciplines selection involve some attribute or status in selection Dynamic disciplines selection involve some attribute or status in selection Shortest processing time minimise average time of customers Shortest processing time minimise average time of customers Queuing Process

38 38Queuing ModelsLesson 10 Preemptive priority interrupt current person service for newly arrived customer with higher priority e.g. fire or ambulance services Preemptive priority interrupt current person service for newly arrived customer with higher priority e.g. fire or ambulance services Round robin service give customer partial service and then movers to next waiting customer – alternating between waiting and being served Round robin service give customer partial service and then movers to next waiting customer – alternating between waiting and being served Queuing Process

39 39Queuing ModelsLesson 10 Service person begins to take orders while customers are still waiting in line is a direct approach to avoid reneging Service person begins to take orders while customers are still waiting in line is a direct approach to avoid reneging Queuing Process

40 40Queuing ModelsLesson 10 SERVICE PROCESS SERVICE PROCESS Distribution of service times, arrangement of servers, management policies and server behaviour contribute to performance Distribution of service times, arrangement of servers, management policies and server behaviour contribute to performance Queuing Process

41 41Queuing ModelsLesson 10 WAITING PERCEPTION Unoccupied time feels longer than occupied time Unoccupied time feels longer than occupied time Preprocess waits feel longer than in process waits Preprocess waits feel longer than in process waits Anxiety makes waits seem longer Anxiety makes waits seem longer

42 42Queuing ModelsLesson 10 Uncertain waits are longer than known finite waits Uncertain waits are longer than known finite waits Unexplained waits are longer than explained wait Unexplained waits are longer than explained wait Unfair waits are longer than equitable waits Unfair waits are longer than equitable waits WAITING PERCEPTION

43 43Queuing ModelsLesson 10 People more willing to wait for valuable service People more willing to wait for valuable service Solo waiting feels longer than group waiting Solo waiting feels longer than group waiting Customer attitudes Customer attitudes WAITING PERCEPTION

44 44Queuing ModelsLesson 10 Environment Environment Unused facilities and idle staff increase annoyance Unused facilities and idle staff increase annoyance Unfamiliar music makes perceived time seem longer than familiar background music Unfamiliar music makes perceived time seem longer than familiar background music WAITING PERCEPTION

45 45Queuing ModelsLesson 10 REVIEW QUESTIONS 1. What does M/D/1 refers to? 2. What are the characteristics of M / G / 1? 3. How can you measure the performance of a queue?

46 46Queuing ModelsLesson 10 REVIEW QUESTIONS 4. List and discuss the factors affecting queue perception. 5. What are the five features of a queue? 6. How can you configure queue?


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