Waiting Line Models. 4EK601 Operations Research. Jan Fábry, Veronika Skočdopolová

Transcription

1 Waiting Line Models 4EK601 Operations Research Jan Fábry, Veronika Skočdopolová

2 Waiting Line Models Examples of Waiting Line Systems Service System Customer Server Doctor s consultancy room Patient Doctor Bank Client Clerk Crossing Car Traffic lights Airport Airplane Runway Fire station Fire Emergency unit Telephone exchange Call Switchboard Service station Car Petrol pump

3 Waiting Line Models Waiting Line System Source Arrival Process Waiting Area Service Exit {Potential Customers} {Customers} {Queue} {Server}

4 Arrival Process Waiting Line Models Source {Potential Customers} Infinite tourists Finite machines in factory

5 Arrival Process Waiting Line Models Arrivals Arrivals In batches BUS of tourists Individually patients Scheduled trams, trains Unscheduled patients

6 Waiting Line Models Arrival Process Arrival Arrival Arrival Arrival rate number of arrivals per time unit (POISSON distribution) Time Average arrival rate = average number of arrivals per time unit (mean of POISSON distribution)

7 Waiting Line Models Arrival Process Arrival Arrival Arrival Time Interarrival time time period between two arrivals (EXPONENTIAL distribution) Average interarrival time = 1/ average time period between arrivals (mean of EXPONENTIAL distribution)

8 Service Process Waiting Line Models Service {Server} Service rate number of customers served per time unit (POISSON distribution) Average service rate = average number of customers served per time unit (mean of POISSON distribution)

9 Service Process Waiting Line Models Service {Server} Service time time customer spends at service facility (EXPONENTIAL distribution) Average service time = 1/ average time customers spend at service facility (mean of EXPONENTIAL distribution)

10 Service Process Waiting Line Models Service configurations (type, number and arrangement of service facilities) 1. Single facility Arrival Exit Queue Server

11 Service Process Waiting Line Models 2. Multiple, parallel, identical facilities (SINGLE queue) Arrival Exit Queue Servers

12 Service Process Waiting Line Models 2. Multiple, parallel, identical facilities (MULTIPLE queue) Arrival Exit Queues Servers

13 Service Process Waiting Line Models 3. Multiple, parallel, but not identical facilities Arrival Exit Queues Servers

14 Service Process Waiting Line Models 4. Serial facilities Arrival Exit Queue Server Queue Server Queue Server 5. Combination of facilities

15 Waiting Line Waiting Line Models Discipline of the queue FCFS (First-Come, First-Served) Service {Server}

16 Waiting Line Waiting Line Models Discipline of the queue LCFS (Last-Come, First-Served) Service {Server}

17 Waiting Line Waiting Line Models Discipline of the queue PRI (Priority system) Service {Server}

18 Waiting Line Waiting Line Models Discipline of the queue SIRO (Selection In Random Order) Service {Server}

19 Waiting Line Models Analysis of Waiting Line Models Cost Waiting cost Service cost (facility cost) cost of construction cost of operation cost of maintenance and repair other costs (insurance, rental)

20 Waiting Line Models Analysis of Waiting Line Models Time characteristics Average waiting time in the queue Average waiting time in the system Number of customers Average number of customers in the queue Average number of customers in the system

21 Waiting Line Models Analysis of Waiting Line Models Probability characteristics Probability of empty service facility Probability of the service facility being busy Probability of finding N customers in the system Probability that N > n Probability of being in the system longer than time t

22 Waiting Line Models Classification of Waiting Line Models Kendall s notation A / B / C / D / E / F Size of customer s source Maximum length of queue Queue discipline Number of parallel servers Probability distribution of service time Probability distribution of interarrival time

23 Standard Single-Server Exponential Model Waiting Line Models

24 Waiting Line Models Standard Single-Server Exponential Model (M/M/1/FCFS/ / ) Arrival Exit Queue Server

25 Assumptions Single server Standard Single-Server Exponential Model Interarrival times exponential probability distribution with the mean = 1/λ Service times exponential probability distribution with the mean = 1/μ Infinite source Unlimited length of queue Queue discipline is FCFS

26 Standard Single-Server Exponential Model Example Grocery One shop assistant serves 25 customers per hour (on the average) From 8 a.m. to 6 p.m. 18 customers per hour arrive (on the average)

27 Standard Single-Server Exponential Model Example Grocery Average arrival rate Average service rate λ = 18 customers per hour μ = 25 customers per hour

28 Standard Single-Server Exponential Model Example Grocery Utilization of the system probability that the server is busy probability that there is at least 1 customer in the system Probability of an empty facility (server is idle) P(0) 1 P( 0) 0. 28

29 Standard Single-Server Exponential Model Example Grocery Average waiting time in the system W 1 W 0.143hours 8.6 minutes Average waiting time in the queue W q W 1 ( ) W 0.103hours 6.2 minutes

30 Standard Single-Server Exponential Model Example Grocery Average number of customers in the system L W L 2.57 customers Average number of customers in the queue L q W q 2 ( ) L q 1.85customers

31 Probabilistic Model with Continuous Demand Example Grocery Probability of finding exactly N customers in the system P( N) P(0) N (1 ) N P(0) P(1) P(2) P(3) 0.105

32 Probabilistic Model with Continuous Demand Example Grocery Probability that N > n P n 1 N n P{N > 0} P{N > 1} P{N > 2} P{N > 3} 0.269

33 Probabilistic Model with Continuous Demand Example Grocery Probability of being in the system longer than time t ( t T t e P ) P{T > 1 min} P{T > 2 min} P{T > 3 min} P{T > 4 min} 0.627