1. Queueing Theory What is a queue? Examples of queues: Grocery store checkout Fast food (McDonalds – vs- Wendy’s) Hospital Emergency rooms Machines waiting for repair Communications network

2. Queueing Theory Basic components of a queue: customers Input source Queue Service (calling population) mechanism

3. Queueing Theory Queueing System Characteristics Input source: population size (assumed infinite) customer generation pattern (assumed Poisson w rate or equivalently, exponential with an inter-arrival time ) arrival behavior (balking, blocking) Queue: queue size (finite or infinite) queue discipline (assumed FIFO, other include random, LIFO, priority, etc..)

4. Queueing Theory Service mechanism: number of servers service time and distribution (exponential is most common)

5. Queueing Theory Naming convention: a / b / c / d / e / f a – distribution of inter-arrival times b – distribution of service time c – number of servers Where M – exponential distribution (Markovian) D – degenerate distribution (constant times) E k – Erlang distribution (with shape k) G = general distribuiton Ex. M/M/1 or M/G/1

6. Queueing Theory Naming convention cont.: a / b / c / d / e / f d – capacity of the system (default – infinite) e – size of the calling population (default – infinite) f – queue discipline Where FIFO – first in, first out LIFO – last in, first out Random – random selection Ex. M/D/1/inf./20/LIFO

7. Queueing Theory Terminology and Notation: State of system – number of customers in the queueing system, N(t) Queue length – number of customer waiting in the queue - state of system minus number being served P n (t) – probability exactly n customers in system at time t. [p i ] s – number of servers [c] n – mean arrival rate (expected arrivals per unit time) when n customers already in system  n – mean service rate for overall system (expected number of customers completing service per unit time) when n customers are in the system.  – utilization factor (=  s , in general)

8. Queueing Theory Terminology and Notation: L = expected number of customers in queueing system = L q = expected queue length = W = expected waiting time in system (includes service time) W q = expected waiting time in queue

9. Queueing Theory Little’s Law: W = L / and W q = L q / Note: if n are not equal, then = W = W q + 1/  when  is constant.

10. Role of Exponential Distribution Property 1: f T (t) is a strictly decreasing function of t (t > 0). P[0 P[t < T < t + ] P[0 < T < 1/2  ] =.393 P[1/2  < T < 3/2  ] =.383 Property 2: lack of memory. P[T > t + | T > ] = P[T > t] f T (t) =  e -  t for t > = 0 F T = 1 - e -  t E(T) = 1/  V (T) = 1/  2  fT(t)fT(t) t Let T be a random variable representing the inter-arrival time between events.

11. Role of Exponential Distribution Property 3: the minimum of several independent exponential random variables has an exponential distribution. Let T 1, T 2,…. T n be distributed exponential with parameters  1,  2,…,  n and let U = min{T 1, T 2,…. T n } then: U is exponential with rate parameter  = Property 4: Relationship to Poisson distribution. Let the time between events be distributed exponentially with Parameter . Then the number of times, X(t), this event occurs over some time t has a Poisson distribution with rate  t: P[X(t) = n] = (  t) n e -  t /n!

12. Role of Exponential Distribution Property 5: For all positive values of t, P[T t] for small.  can be thought of as the mean rate of occurrence. Property 6: Unaffected by aggregation or dis-aggregation. Suppose a system has multiple input streams (arrivals of customers) with rate 1, 2,… n, then the system as a whole has An input stream with a rate =

13. Break for Exercise