Mitchell Jareo MAT4340 – Operations Research Dr. Bauldry Queueing Theory Mitchell Jareo MAT4340 – Operations Research Dr. Bauldry

Queue Characteristics 5 Characteristic Components Arrival Pattern Service Pattern Number of Servers Service Facility Capacity Service Order

Arrival Patterns Interarrival Time Balking – queue is too long Time between customer arrivals to the facility Deterministic Random by a known probability distribution May be state dependent or not Single arrivals or in batches Balking – queue is too long Reneging – service wait is too long

Service Patterns Service Time Time for one server to serve one customer Deterministic Random by a known probability distribution May be state dependent or not One server or sequence of servers

System Capacity Maximum number of customers service and in queues which the facility can hold Customers cannot enter full facilities Cannot wait outside; must leave Capacity can be finite or infinite

Queue Disciplines Order in which customers are served FIFO (first-in, first-out) order of arrival LIFO (last-in, last-out) last one first Random basis Priority basis Whatever other basis you can cook up

Kendall’s Notation Specifies queue characteristics v/w/x/y/z where v = interarrival pattern (D/M/Ek/G) w = service pattern (D/M/Ek/G) x = number of available servers y = system capacity z = queue discipline (FIFO,LIFO,SIRO,PRI,GD) y and z default to infinite and FIFO

Examples Grocery Store with 6 checkout lanes M/M/6/24/FIFO Car Wash with 4 car waiting lane M/M/1/4/FIFO Work for a 12 typist typing pool M/M/12/500/LIFO Emergency Room w/2 docs and 20 seats M/M/2/20/PRI

A Deterministic Example A bus cleaning facility. D/D/1 Buses arrives one the hour five at a time Cleaning takes 11 minutes Simulated queue for one hour Average buses in facility Average queue length Average time in facility

M/M/1 Systems Exponentially distributed interarrival times with parameter, λ. Exponentially distributed service times with parameter, μ. 1 server. λ is the average customer arrival time μ is the average service rate. Both are customers per unit time.

M/M/1 Systems Given the exponentially distributed interarrival and service times. Expected interarrival time = 1/ λ Expected serice time = 1/ μ Utilization factor, or traffic intensity Expected number of arrivals per mean service time is denoted as ρ = λ/ μ If ρ < 1, then steady state probabilities exist and are given by ρ n = ρ n (1 – ρ)

M/M/1 Measures L – average number of customers Lq – average length of queue W – average time customer is in system Wq – average customer queue wait time W(t) – probability customer spends more than t time units in system Wq (t) – probability customer spends more than t time units in queue