Queuing Theory. Using the Math. Management Science

Transcription

1 Queuing Theory Using the Math 1

2 Markov Processes (Chains) A process consisting of a countable sequence of stages, that can be judged at each stage to fall into future states independent of how the process arrived at the previous state. Memoryless Many Queuing Models are Markov Chains 2

3 Queuing Models A Queuing Model Comprises Markovian Arrivals A Line or a Queue A Number of Servers And A Service Method 3

4 Arrivals Jobs arrive for service. Parameter is Mean Arrival Rate - λ (lamda) (so many jobs arriving per period) Key Statistic is the Interarrival Time, the average time between the arrivals of a job. Average time between arrival of jobs = mean interarrival time = 1/λ 4

5 Arrivals This statistic can be: Deterministic (e.g. 1 every time without fail) (Use Simulation Methods) period, According to a Probability Distribution: General (Use Simulation Methods) Markovian use Queuing Models 5

6 Service Jobs are serviced. Parameter is Mean Service Rate - µ (mu) (So many jobs serviceable per period) Key Statistic is the Mean or Average Potential Service Time, the average time it takes to service a job. Average Potential Service Time = 1/µ 6

7 Service This statistic can be: Deterministic (1 every time period, without fail) According to a Probability Distribution: Markovian General (we only use µ and σ) Number of servers can be one or many. 7

8 Queues Queue Length may be limitless (infinite) or finite. Job (customer) Population - may be infinite of finite. Queue Discipline jobs are served on FIFO. Time Horizon system foes on an on over and infinite or finite time frame. Warning if λ µnone of this works 8

9 Performance Measurement P 0 - Probability that there are no jobs in the system. P n - Probability that there are n jobs in the system. L - Average number of jobs in the system. L q - Average number of jobs in the queue. 9

10 Performance Measurement W - Average time a job spends in the system. W q - Average time a job spends in the queue. P w - Probability that all servers are busy. ρ - The Utilization rate for each server (percentage of time each server is busy). 10

11 Queue Classification Arrival / Service Process Process / Number of Servers / Finiteness of Queue Length / Finiteness of Job Population. M = exponential and reverse exponential (Poisson) distribution D = deterministic G = any general distribution of service times 1 = 1 server k = more than one server F = finite length of queue m = job population 11

12 Which gives us a number of choices to look at: M / M / 1 / M / M / k / M / G / 1 M / D / 1 M / M / k / F M / M / 1 / F λ µ M / M / 1 / F λ =µ M / M / 1 / / m 12

13 M / M / 1 Arrivals Markovian use Poisson distribution. Service Markovian use Exponential distribution. 1 Server Infinite Queue length Infinite Customer population 13

14 Truck Loading and Unloading from Quantitive Methods for Business Anderson, Sweeney and Williams M / M / 1 Trucks using a single-channel loading dock arrive according to a Poisson distribution. The time required to load/unload follows an exponential probability distribution. The mean arrival rate is 12 trucks per day, and the mean service rate is 18 trucks per day. 14

15 M / M / 1 Truck Loading and Unloading Parameters are Mean Arrival Rate - λ (12 trucks per day) and Mean Service Rate - µ (mu) (18 trucks per day) Queue type is M/M/1 a. What is the probability that no trucks are in the system? One truck? a. is asking for calculation of P 0 and for P n where n=1 = ( ) = ( ) P 1 λ / µ 0 ( 1 / )( / ) P λ µ λ µ n n 15

16 M / M / 1 Truck Loading and Unloading b. What is the average number of trucks waiting for service? In the queue? b. is asking for the calculation of L and Lq c. What is the average time a truck waits for the loading/unloading service to begin? Ave time in total at loading dock? c. is asking for the calculation of Wq and W = λ 1 L = λ µ λ L q µ ( µ λ) ( ) W q µ µ λ ( ) ( ) = λ W 2 = ( ) µ λ ( ) 16

17 M / M / 1 Truck Loading and Unloading d. What is the probability that an arriving truck will have to wait for service? d. is looking for the calculation of P w the probability that all the servers are busy. P = λ w µ Finally what is the utilization rate ρ for the server? ρ = λ µ 17

18 M / M / k Truck Loading and Unloading Due to the incredible business acumen of the company s Management Scientists the number of loading docks has increased to 3; the mean arrival rate is 40 trucks per day, and the mean service rate on each line is 18 trucks per day. Answer the questions a. through d. and calculate ρ. 18

19 M / M / k Arrivals Markovian use Poisson distribution. Service Markovian use Exponential distribution. k Servers Infinite Queue length Infinite Customer population 19

20 M / M / k Truck Loading and Unloading What is λ? What is µ? What is k? P P 0 n 1 = n k n= k 1 1 λ 1 λ kµ n= 1 n! + k! kµ λ µ µ ( ) n λ µ for n k 0 n! P = ( ) n λ µ for n k ( n k) P > 0 k! k 20

21 M / M / k Truck Loading and Unloading L λµ ( ) ( )( ) 2 0 k 1! k λµ λ = P + µ kµ λ L q = λµ ( ) ( )( ) 2 0 k 1! k λµ kµ λ P W λµ ( ) µ = + ( )( ) 2 0 k 1! k kµ λ P 1 µ W q = λµ ( ) ( )( ) 2 0 k 1! k µ kµ λ P P w λ µ 1 kµ = k! kµ λ k P 0 ρ = λ kµ 21

22 Reading and Homework. Homework 11, due in class on Thursday 9 th of May. Redo M/M/1 with Lamda = 20 and Mu = 30 Redo M/M/k with k = 2 22