1. 1 Overview of Queueing Systems Michalis Faloutsos Archana Yordanos The web

2. 2 Overview of queueing concepts Exponential Distribution Memoryless: Poisson Process. with mean arrival rate : The probability of having an arrival(departure) at time x. The probability of having k events. Interarrival time in a Poisson process is exponential:  n is the interarrival t n+1 -t n

3. 3 Little’s Theorem N T : customer arrival rate N: average number of customers in system T: average delay per customer in system Little’s Theorem: System in steady-state  

4. 4 Queueing Systems n : number of customers in the system (including queue + server) p n : steady state probability of finding n customers in the system  /  : Traffic rate (traffic intensity) M stands for ``Markovian'',

5. 5 Modeling a queueing system What goes left, must come right: # of transitions  = # of transitions  Pi are probabilities:

6. 6 Basic formulas Expected number of customers in the system: Expected time a customer spends in the system: Expected time a customer spends in the queue: Expected number of customers in the system: