Economic Optimization Model

1. Waiting Line Models For Service Improvementby:Shannon DuffyKatie McPartlinJason JacklowAmanda HoltzB.J. Ko

2. Economic Optimization Model • EOM • Which has developed using queuing analysis

3. EOM • Used at L.L. Bean for telemarketing operations • To determine the optimal number of telephone trunks for incoming calls • The number of agents scheduled • The queue capacity • The maximum number of customers who are put on hold to wait for an agent

4. Queuing models are used • To determine the economic impact of busy signals • Customer waiting time • Lost orders

5. Decision about waiting lines • Are based on averages for customer arrivals and service times • They are used to computer operation characteristics • Which are average of values for characteristics that describe the performance of a waiting line system

6. Elements of a waiting line • Basic element is queue • Which is a single waiting line • Which consists of arrivals, servers, and waiting line structure • Single-channel queuing system

7. The Calling Population • Is the source of the customers to the queuing system, and it can be either infinite or finite • Infinite • Calling population assumes such a large number of potential customers that is always possible for one more customer to arrive to be served • Ex. – grocery store, bank

8. Finite calling population • Has a specific, countable number of potential customers • Ex. – repair facility in a shop

9. Arrival Rate • Is the rate at which customers arrive at the service facility during a specified period of time • This rate can be estimated from empirical data derived from studying the system or a similar system, or it can be average of these empirical data

10. Service times • The time required to serve a customer, is more frequently described by the negative exponential distribution • Service must be expressed as a rate to be compatible with the arrival rate • Customers must be served faster than they arrive or an infinitely large queue will build up

11. Queue Discipline and Length • Queue discipline is the order in which waiting customers are served • Most common type is first come, first served

12. Infinite Queue • Can be of any size with no upper limit and is the most common queue structure • Ex. Movie theater line

13. Finite Queue • Is limited to size • Ex. Driveway at bank

14. Basic Waiting Line Structures • There are four basic structures according to the nature of the service facilities • Single-channel, single-phase • Single-channel, multiple-phase • Multiple-channel, single-phase • Multiple-channel, multiple-phase

15. Channels and Phases • Channel • Is the number of parallel servers for servicing arriving customers • Phases • Denotes the number of sequential servers each customer must go through to complete service

16. Poisson Distribution • The Poisson Distribution is a discrete distribution which takes on the values X=0,1,2,3… • It is often used as a model for the number of events (such as the number of telephone calls at a business or the number of accidents at an intersection) in a specific time period.

17. Single-Channel, Single-Phase Models • Most basic of the waiting line structures • Frequently used variation • Poisson arrival rate, exponential service times • Poisson arrival rate, general distribution of service times • Poisson arrival rate, constant service items • Poisson arrival rate, exponential service times with a finite queue and a finite calling population

18. Basic Single-Server Model • Assume the following • Poisson arrival rate • Exponential service times • First-come, first-served queue discipline • Infinite queue length • Infinite calling population

19. Constant Service times • The single-server model with Poisson arrivals and constant service times is a queuing variation that is of particular interest in operations management, since the most frequent occurrence of constant service times is with automated equipment and machinery. • This model has direct applications for many manufacturing operations

20. Finite Queue Length • Since some waiting lines systems the length of the queue may be limited by the physical area in which the queue forms; • Space may permit only a limited number of customers to enter the queue • Variation of the single-phase, single-channel queuing model

21. Finite Calling Populations • The population of customers from which arrivals originate is limited, such as the number of police cars at a station to answer calls

22. Multi-Server Models • Two or more independent servers in parallel serve a single waiting line • The number of servers must be able to serve customers faster than they arrive

23. Definition of Variables • Pn = Probability of n Units in System • = Mean Number of Arrivals per Time Period •  = Mean Number of People or Items Served per Time Period • Ls = Average Number of Units in the System • Ws = Average Time a Unit Spends in the System (Wait Time + Service Time)

24. Definition of VariablesContinued • Lq = Average Number of Units in the Waiting Line • Wq = Average Time a Unit Spends Waiting in the Line •  = Utilization Factor for the System (Percent of Time the Servers are Busy) • P0 = Probability of 0 Units in the System

25. Single Channel, Single Phase • Ls = /( • Ws = 1/( • Lq = • Wq = •  • P0 = • Pn =n • In all Cases, 

26. Single Channel, Single Phase Example • For cars arriving/hour • cars serviced/hour • Ls = /(= 2/(3-2) = 2 cars in System • Ws = 1/(= 1/(3-2) = 1 hour average time spent in system • Lq == 22/3(3-2) = 1.33 cars waiting

27. Single Channel, Single Phase Example Cont. • Wq = = 2/(3(3-2)) = 2/3 hour or 40 minute average time waiting •  = 2/3 = 66.7% Utilization of Mechanic • P0 = = 1-(2/3) = 1/3 = 0.33 probability there are no cars in system = 0.33 • P1 =1 = (2/3)12/3)) = 1/9 probability there is 1 car in system = 0.11

28. The End Any Questions?