ISM 270. Service Engineering and Management Lecture 7: Queuing Systems, Capacity Management

Transcription

1 ISM 270 Service Engineering and Management Lecture 7: Queuing Systems, Capacity Management 1

2 Queuing Systems CHARACTERISTICS OF A WAITING LINE SYSTEM Arrival Characteris=cs Wai=ng Line Characteris=cs Service Facility Characteris=cs Measuring the Queue s Performance Queuing Costs 2

3 You ve Been There Before! The other line always moves faster. Thank you for holding. Hello...are you there? If you change lines, the one you lei will start to move faster than the one you re in Corel Corp.

4 WaiTng Line Examples SituaTon Arrivals Servers Service Process Bank Customers Teller Deposit etc. Doctor s PaTent Doctor Treatment office Traffic Cars Light Controlled intersecton passage Assembly line Parts Workers Assembly Tool crib Workers Clerks Check out/in tools 4

5 WaiTng Lines First studied by A. K. Erlang in 1913 Analyzed telephone facilites Body of knowledge called queuing theory Queue is another name for waitng line Decision problem Balance cost of providing good service with cost of customers waitng 5

6 Cost Deciding on the OpTmum Level of Service Minimu m total cost Total expected cost Cost of providing service Low level of service OpTmal service level NegaTve Cost of waitng Tme to company High level of service 6

7 WaiTng Line Terminology Queue: WaiTng line Arrival: 1 person, machine, part, etc. that arrives and demands service Queue discipline: Rules for determining the order that arrivals receive service Channel: Number of waitng lines Phase: Number of steps in service 7

8 Three Parts of a Queuing System at Dave s Car Wash 8

9 Input CharacterisTcs Input Source (PopulaTon) Size Arrival Pahern Infinite Finite Random Non Random Poisson Other 9

10 Poisson DistribuTon Number of events that occur in an interval of Tme Example: Number of customers that arrive in 15 min. Mean = λ (e.g., 5/hr.) Probability: λ = 0.5 λ =

11 Input CharacterisTcs Input Source (PopulaTon) Size Arrival Pahern Behavior Infinite Finite Random Non Random PaTent ImpaTent Poisson Other Balk Renege 11

12 WaiTng Line CharacterisTcs WaiTng Line Length Queue Discipline Unlimited Limited FIFO (FCFS) Random Priority 12

13 Service CharacterisTcs Service Facility ConfiguraTon Single Channel MulT Channel Single Phase 13

14 Single Channel, Single Phase System Arrivals Queue Service system Service facility Served units Ships at sea Ship unloading system WaiTng ship line Empty ships Dock 14

15 Single Channel, MulT Phase System Arrivals Queue Service system Service facility Service facility Served units Cars in area McDonald s drive through WaiTng cars Pay Pick up Cars & food Anil Sahai ISM 270 D 15 Spring 2009

16 MulT Channel, Single Phase System Arrivals Queue Service system Service facility Served units Service facility Example: Bank customers wait in single line for one of several tellers. 16

17 MulT Channel, MulT Phase System Arrivals Queue Service system Service facility Service facility Served units Service facility Service facility Example: At a laundromat, customers use one of several washers, then one of several dryers. 17

18 WaiTng Line Performance Measures Average queue Tme, W q Average queue length, L q Average Tme in system, W s Average number in system, L s Probability of idle service facility, P 0 System utlizaton, ρ Probability of k units in system, P n > k 18

19 AssumpTons of the Basic Simple Queuing Model Arrivals are served on a first come, first served basis Arrivals are independent of preceding arrivals Arrival rates are described by the Poisson probability distributon, and customers come from a very large populaton Service Tmes vary from one customer to another, and are independent of one and other; the average service Tme is known Service Tmes are described by the negatve exponental probability distributon The service rate is greater than the arrival 19 rate

20 Types of Queuing Models Simple (M/M/1) Example: InformaTon booth at mall MulT channel (M/M/S) Example: Airline Tcket counter Constant Service (M/D/1) Example: Automated car wash Limited PopulaTon Example: Department with only 7 drills 20

21 Simple (M/M/1) Model CharacterisTcs Type: Single channel, single phase system Input source: Infinite; no balks, no reneging Arrival distributon: Poisson Queue: Unlimited; single line Queue discipline: FIFO (FCFS) Service distributon: NegaTve exponental RelaTonship: Independent service & arrival Service rate > arrival rate 21

22 Simple (M/M/1) Model EquaTons Average number of units in queue Average Tme in system Average number of units in queue Average Tme in queue System utlizaton W W q s L L q = s = µ λ 1 = µ λ λ 2 λ = µ (µ λ ) λ µ (µ λ ) ρ = λ µ 22

23 Simple (M/M/1) Probability EquaTons Probability of 0 units in system, i.e., system idle: P 0 = 1 ρ = 1 λ µ Probability of more than k units in system: P n>k = ( λ ) µ k+1 Where n is the number of units in the system 23

24 MulTchannel (M/M/S) Model CharacterisTcs Type: MulTchannel system Input source: Infinite; no balks, no reneging Arrival distributon: Poisson Queue: Unlimited; multple lines Queue discipline: FIFO (FCFS) Service distributon: NegaTve exponental RelaTonship: Independent service & arrival Σ Service rates > arrival rate 24

25 Model B (M/M/S) EquaTons Probability of zero people or units in the system: Average number of people or units in the system: Average Tme a unit spends in the system: 25

26 Model B (M/M/S) EquaTons Average number of people or units waitng for service: Average Tme a person or unit spends in the queue Anil Sahai ISM 270 D 26 Spring 2009

27 Constant Service Rate (M/D/1) Model CharacterisTcs Type: Single channel, single phase system Input source: Infinite; no balks, no reneging Arrival distributon: Poisson Queue: Unlimited; single line Queue discipline: FIFO (FCFS) Service distributon: NegaTve exponental RelaTonship: Independent service & arrival Σ Service rates > arrival rate 27

28 Model C (M/D/1) EquaTons Average number of people or units waitng for service: Average Tme a person or unit spends in the queue Average number of people or units in the system: Average Tme a unit spends in the system: 28

29 Remember: λ & µ Are Rates λ = Mean number of arrivals per Tme period e.g., 3 units/hour µ = Mean number of people or items served per Tme period e.g., 4 units/hour 1/µ = 15 minutes/unit If average service Tme is 15 minutes, then μ is 4 customers/hour T/Maker Co.