QueueTraffic and queuing theory

Transcription

1 QueueTraffic and queuing theory +

2 Queues in everyday life You have certainly been in a queue somewhere. Where? How were they different? At ticket vending machines, cash desks, at the doctors, at printers, in a call center, We encounter queues all the time! 2

3 Examples: supermarket, doctor Supermarket: many cash desks, many queues They even might have an express queue line! Doctor: one doctors (~cash desk), one queue Similar: queues in a self service restaurant, queues in front of a cable car 3

4 What is it about today? Learn a mathematical model How are queues analyzed? What are important concepts and terms for this? Learning goals with the use of QueueTraffic Being able to explain the concepts arrival rate, throughput, utilization and calculate those for given situations with the help of QueueTraffic Being able to interpret different values for utilization: Do we get a queue or not? Being able to explain the difference between Poisson and uniform distribution in a illustrative way 4

5 The best system? What is better, what worse? Many cash desks? Fast lines? Number tickets?...? Cannot be decided like this! What can be decided: How well works a system in a certain situation. 5

6 Waiting queues and me: 2 points of view 1. I am responsible for the setting of queues Manager of a store Doctor in a medical practice Operator of a cable car... I can control the system 2. I am waiting in a queue Client Patient Hiker... I am depending on the system 6

7 Problems of a store manger When and why do we get queues? Too many people, too few desks What can we do about it? More cash desks What if there are too many desks open? costs, boredom, Let on more people in First serve the people who need little service time e.g. printers that sorts job by their size Limit the time during which someone is being served e.g. limited treatment time per client at the doctor 7

8 Problems of a customer Why does it always feel to us like all other queues move faster? Which queue should I take? Where shall I append? How long do I have to wait? Where there are the fewest people in queue? Where the people have the least products to dispatch? Where the fastest cashier is? Where one can only pay cash? Where someone helps me bagging?...? 8

9 We observe From the left he queue gets extended From the right the queue gets shortened 9

10 Arrival and service-process The act of new clients permanently putting theirselves in queue is called arrival process. The number of clients adding theirselves to the queue within a certain period of time is called the arrival rate λ. The act of clients being cleared from queue from the right is called service process. The time one service process takes for one client is called the service time b. 10

11 Arrival rate λ (lambda) und service time b Arrival rate λ: How often do new clients arrive? In a supermarket 5 clients come through the door per hour, on average. Arrival rate: λ = 5/60 clients/minute Service time b: how long takes a service station for one client? It take the man at the cashier on average 15 minutes to serve a client. service time: b = 15 minutes/client 11

12 throughput μ (Mu) Throughput μ: How many clients are served per time unit? The throughput is the reciprocal of the service time: μ = 1/b Example: service time b = 15 minute/client Throughput: μ = 1 / 15 clients/minute = 1 client / 15 minutes = 4 clients/hour 12

13 Utilization ρ (Rho) Utilization ρ = λ/μ: How much of the capacity of the system is used? Important parameter for the analysis of queuing systems! 3 examples: - λ 1 = 10 clients/hour, μ 1 = 30 c./h - λ 2 = 30 c./h, μ 2 = 30 c./h - λ 3 = 60 c./h, μ 3 = 30 c./h Utilization: ρ 1 = λ 1 /μ 1 = 10 / 30 = 0.33 ρ 2 = λ 2 /μ 2 = 30 / 30 = 1 ρ 3 = λ 3 /μ 3 = 60 / 30 = 2 13

14 Distributions at the arrival Arrivals in QueueTraffic are uniform or Poisson distributed. or 14

15 QueueTraffic: Demo Source: 15

16 QueueTraffic Situation and help Simulation area Traffic control Volume of traffic Simulation control Data and charts 16

17 You solve some first exercises Solve exercises A until 3. 17

18 Theoretical and effective throughput in in QueueTraffic So far, we only considered the theoretical throughput μ t, since we took at look at how many clients (cars) can be served (drive through) under optimal conditions. QueueTraffic is a simulation and to calculate the throughput it counts the number of cars driving over the crossroad per round. This is called the effective throughput μ e. 18

19 Why do we need the effective throughput? Fact: How many car could theoretically get through cannot easily be measured or counted. The number of served cars can easily be counted! This gives us the effective throughput μ e μ e = μ t is reached, if λ > μ t, i.e. if enough cars arrive. 19

20 Sample - calculation Arrival rate: λ = 10 car/60s Effective throughput: μ e = 10 cars/60s Theoretical throughput: μ t = (28 s/60s) * (1 car/1s) = 28 cars/60s Utilization:!!! ρ = λ / μ t = (10 cars/60 s) / (28 cars/60 s) = 0.36 calculated! simulated! (counted) 20

21 Remarks to the calculations in QueueTraffic Traffic volume is per 60 s Formula for theoretical throughput μ t = proportion green time * capacity per lane e.g.: μ t = 28s/60s * 1 car/s given! Use the theoretical throughput μ t to calculate the utilization: ρ = λ / μ t 21

22 now: your turn! 22

23 The most important terms term example Arrival rate λ Service time b throughput μ t = 1/b 12 cars/minute 0.1 minute/car 10 cars/minute utilization ρ = λ / μ t

24 Connection between utilization ρ = λ / μ t and traffic jam. In general holds: ρ < 1 no (or little) jam: system under-worked ρ 1 a little jam: system loaded ρ > 1 growing jam: system overloaded Practical Interpretation: Jam if λ > μ t, i.e. if the arrival rate is bigger than the (theoretical) throughput. Thus: If more cars arrive than what can be served, we get jam! 24

25 What was it about today? Get to know a mathematical model How are waiting queues analyzed? What are important terms and concepts? Concrete goals: Being able to explain the concepts arrival rate, throughput, utilization and calculate those for given situations with the help of QueueTraffic Being able to interpret different values for utilization: Do we get a queue or not? Being able to explain the difference between Poisson and uniform distribution in a illustrative way 25

26 THE END QueueTraffic und queuing theory Remarks and feedback please to: 26