M/G/1 queue

In queueing theory, a discipline within the mathematical theory of probability, an M/G/1 queue is a queue model where arrivals are Markovian (modulated by a Poisson process), service times have a General distribution and there is a single server. The model name is written in Kendall's notation, and is an extension of the M/M/1 queue, where service times must be exponentially distributed. The classic application of the M/G/1 queue is to model performance of a fixed head hard disk. A queue represented by a M/G/1 queue is a stochastic process whose state space is the set {0,1,2,3...}, where the value corresponds to the number of customers in the queue, including any being served. Transitions from state i to i + 1 represent the arrival of a new customer: the times between such arrivals have an exponential distribution with parameter λ. Transitions from state i to i − 1 represent a customer who has been served, finishing being served and departing: the length of time required for serving an individual customer has a general distribution function. The lengths of times between arrivals and of service periods are random variables which are assumed to be statistically independent. Customers are typically served on a first-come, first-served basis, other popular scheduling policies include processor sharing where all jobs in the queue share the service capacity between them equally last-come, first served without preemption where a job in service cannot be interrupted last-come, first served with preemption where a job in service is interrupted by later arrivals, but work is conserved generalized foreground-background (FB) scheduling also known as least-attained-service where the jobs which have received least processing time so far are served first and jobs which have received equal service time share service capacity using processor sharing shortest job first without preemption (SJF) where the job with the smallest size receives service and cannot be interrupted until service completes preemptive shortest job first where at any moment in time the job with the smallest original size is served shortest remaining processing time (SRPT) where the next job to serve is that with the smallest remaining processing requirement Service policies are often evaluated by comparing the mean sojourn time in the queue.