Queueing Models: Data Collection and Hand Simulation

1. Queueing Models:Data Collection and Hand Simulation from Prof. Goldsman’s lecture notes

2. Outline • Queueing Models • Data Collection • Hand Simulation

3. Queueing Models • Probabilistic and stochastic models • Important aspects: • Interarrival time • Service time • Waiting time • Mathematics not reality! • Several assumptions: • Arrival process, service process • Queue size and discipline • Time horizon • Calling population

4. Queueing Models (Cont’d) • M/M/1, M/M/n, M/G/1, etc • Use theoretical model to estimate the behavior of a real system (this is only an approximation!) • Queues are a model archetype • Queue system entities: • Flow unit • Queue • Server • Where each entity has its own state space and transitions

5. Hand Simulation

6. Time-Flow Mechanism • Time-step incrementation • Stepping through time in equal (e.g. 1 hour) increment • Event-step incrementation • Calls for a simulation to proceed from one event to the next • Incremental time steps are uneven • Simulation begins at time zero • Occurrence times of the events resulting from the simulated performance of all system components are determined • Master clock is updated to the time of the earliest event occurrence • Commonly used

7. Concepts • (Pseudo-) Random number generator • Most generator uses U[0, 1] • Steady state • Variability of runs

8. How to simulate? • State variables: describe state of system, give a “system snapshot” • Event: anything that can change the state of the system • FEL (Future Event List) • Clock • Keep all these and any desired cumulative statistics in a simulation table

9. Clock and Socks Example • I have n pairs of socks in my dryer. I remove socks one at a time and place them on top of the dryer. When I get a matching pair, I fold them and place them in my laundry basket • How much room do I need on top of my dryer?

10. Example (Cont’d) • Assume n pairs distinct socks • Notation: Li, Ri, i = 1, …, n: the two socks • Assume: • Move a sock from dryer: 1 sec • Check for match and fold: 2 sec

11. Example (Cont’d) State variables: • For each i = 1, …, n, Li and Ri • D: if in dryer • T: if on top of the dryer • DT: if being moved to dryer top • B: if in basket • The values of the variables Li and Ri (together with the clock time t) provide a complete system snapshot

12. Example (Cont’d) Initial Snapshot • Li = Ri = D for all i = 1, …, n • t = 0 • Extra state variable: • #D = #socks in dryer #D = 2n initially • #T = # socks on dryer top #T = 0 initially

13. Example (Cont’d) Event types • Grab: • A sock leaves the dryer • State changes from D to DT • #D decreases • Arrive top: • A sock arrives at the dryer top • State changes from DT to T • Fold: • A pair of socks is matched, folded, and put in basket • Both socks: state changes from T to B

14. Example (Cont’d) Activities • A time interval which triggers an event • Moving sock from dryer to top • Checking for a match

15. Simulation • Assume we have 2 pairs • t = 0- (start up) • State: • L1 = D • R1 = D • L2 = D • R2 = D • #D = 4 • #T = 0 t = 0, grab a sock from dryer Random integer for socks

16. Verification and Validation • Verify: • Building the model right • Valid: • Building the right model

17. Game Time! • Hand Simulation: NISP ATM Simulation Assumption: 1 ATM Machine • Flip two coins (Arrival and Service Times): • 1st Coin: • Head  0 customer enters the NISP ATM • Tail  1 customer enters the NISP ATM • 2nd Coin: • Head  service time 10 minutes • Tail  service time 5 minutes