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 QueueServer 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: L i, R i, 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, L i and R i 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 L i and R i (together with the clock time t) provide a complete system snapshot

12. Example (Cont ’ d) Initial Snapshot L i = R i = 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 eventA time interval which triggers an event Moving sock from dryer to topMoving sock from dryer to top Checking for a matchChecking 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): 1 st Coin: Head  0 customer enters the NISP ATM Tail  1 customer enters the NISP ATM 2 nd Coin: Head  service time 10 minutes Tail  service time 5 minutes