Queueing Theory: Recap Starting point: M/M/1 Poisson arrivals Exponential service times Markov Chain analysis Memoryless property Elegant closed-form results Key insights into system performance

Variations on the M/M/1 Queue M/G/1 - generalized service time M/D/1 - deterministic service time M/M/1/K - finite buffer system M/M/c - up to c servers concurrently M/M/c/c - Erlang loss model M/M/∞ - infinite server system G/G/1 - generalized arrivals + service

Even More Variations (1 of 2) Balking (discouraged arrivals) As the queue becomes longer, new arrivals are less likely to join it (e.g., restaurant) Aborted jobs (e.g., call center tech support) If waiting too long, customers might leave queue Variable rate servers Service rate changes with time, either randomly, or based on load or queue length (e.g., Safeway) Vacationing servers Server disappears for a while, so that no one receives service (e.g., Post Office)

Even More Variations (2 of 2) Server failures (e.g., power outage) Independent failures or catastrophes reduce rate Multiple queues vs single shared queue Multiple servers, with either separate or shared central queue (e.g., bank) Jockeying Customers can change to a different queue at any time (e.g., customs, lane-changing) Multi-class priority queues Different service classes (e.g., airplane)

Queueing Network Models So far we have been talking about a queue in isolation In a queueing network model, there can be multiple queues, connected in series or in parallel (e.g., CPU, disk, teller) Two versions: Open queueing network models Closed queueing network models

Open Queueing Network Models Assumes that arrivals occur externally from outside the system Infinite population, with a fixed arrival rate, regardless of how many in system Unbounded number of customers are permitted within the system Departures leave the system (forever)

Open Queueing Network Example CPU Disk A Disk B Jobs In Jobs Out

Closed Queueing Network Models Assumes that there is a finite number of customers, in a self-contained world Finite population; arrival rate varies depending on how many and where Fixed number of customers (N) that recirculate in the system (forever) Can be analyzed using Mean Value Analysis (MVA) and balance equations

Closed Queueing Network Example CPU Disk A Disk B

Open Queueing Network Analysis Analysis makes use of response time relationship, Little’s Law, visit ratios, Jackson’s Theorem, M/M/1 results, etc. For a fixed-capacity service center i in an open queueing network, the response time Ri is given by Ri = Si (1 + Qi) where Ri is the mean response time, Si is the mean service time, and Qi is the mean number of customers in the queue

Closed Queueing Network Analysis Self-contained system: finite customer population, no external inputs/outputs Finite population implies that arrival rates at different queues depend on the distribution of customers in the network Analysis makes use of iterative and/or recursive solution to compute mean values of performance measures (MVA)

Mean Value Analysis (MVA) A clever analysis technique for closed queueing network models (only) Provides information about the mean values of performance measures (e.g., queue size, response time), but not their variance, etc The crux of the analysis is given by Ri (N) = Si (1 + Qi (N-1) ) where Ri is the mean response time for N customers, Si is the mean service time, and Qi is the mean number of customers in the queue when there are N-1 in the system