represents an M/M/1 queue with arrival rate λ and service rate μ.
represents an M/G/1 queue with arrival rate λ and service distribution sdist.
represents a G/M/1 queue with arrival distribution adist and service rate μ.
represents a G/G/1 queue with arrival distribution adist and service distribution sdist.
represents a queueing process with c service channels.
represents a queueing process with system capacity k.
represents a queueing process with initial state x0.
- QueueingProcess is a continuous-time and discrete-state process.
- QueueingProcess at time t is the number of customers in the system at time t.
- The times between arrivals are independent and follow the distribution adist, with λ indicating ExponentialDistribution[λ].
- The times taken to serve customers are independent and follow the distribution sdist, with μ indicating ExponentialDistribution[μ].
- QueueingProcess allows c and k to be any positive integers. x0 can be any non-negative integer and the distributions adist and sdist can be any continuous distributions with positive domain.
- QueueingProcess can be used with such functions as QueueProperties, StationaryDistribution, and RandomFunction.
Examplesopen allclose all
Basic Examples (2)
M/M Queues (14)
M/G Queues (6)
G/M Queues (3)
Ph/Ph Queues (3)
Customer Service Queues (6)
The arrival pattern of cars to the M. M. One oil change center follows a Poisson process at the rate of four per hour. If the time taken to perform an oil change is exponentially distributed and requires an average of 12 minutes to carry out, find the probability of finding more than three cars waiting for the single available mechanic to service their car. Find the steady-state distribution for the oil change center:
Patients arrive at an eye clinic according to a Poisson process with a mean of six per hour. There are three doctors on duty and the testing times for patients are distributed exponentially with a mean of 20 minutes. Find the average number of people waiting, the average amount of time spent by a patient at the clinic, and the percentage of time when at least one doctor is idle:
Arrivals at the checkout counter of a store are observed to follow a Poisson process with a rate of eight customers per hour. The service times for customers follow an exponential distribution with a mean of four minutes. Simulate the queue for 30 minutes. Also find the mean and the variance for the steady-state queue at the checkout:
Cars arrive at the drive-through window of a bank according to a Poisson process with a mean of 16 cars per hour. The service times follow an Erlang distribution with a mean of minutes and a standard deviation of minutes. Find the average waiting time until a customer reaches the window for service. The service distribution can be found using method of moments:
Arrivals to the takeout counter of a restaurant appear to follow a Poisson process with a mean of 10 per hour. Assuming that the service distribution is exponential, find the average rate at which a customer should be served so that the total time spent by a customer is less than 7.5 minutes:
An estimated 65,000 videos are uploaded every 12 hours on a popular online video channel. Each uploaded video is converted from the MPEG to SWF format and is then available for viewing on the channel. Find the minimum conversion rate so that there are no more than 5 videos on average undergoing the conversion in the system:
Communication System Queues (3)
A cable modem has a maximum transmission rate of 500,000 characters per second. Given that traffic arrives at the rate of 450,000 characters per second, compute the standard performance measures when the system is modeled as an M/M/1 queue:
A router receives packets from a group of users and transmits them over a single transmission line. Suppose that packets arrive according to a Poisson process at a rate of one packet every 4 milliseconds, and suppose that packet transmission times are exponentially distributed with mean 3 milliseconds. Find the mean number of packets in the system and the mean total delay in the system:
Subscribers connect to a university's online catalog at a rate of four subscribers per minute. Sessions have an average duration of 5 minutes. Find the probability that there are more than 25 users online:
Queueing Theory (5)
Compare the mean queue system size for the performance of two identical servers, each with its own separate queue, to the case when there is only a single queue in which to hold customers for both servers, assuming Poisson arrivals and exponential service times:
Derive the steady-state probabilities for an ample-server Markovian queueing process having arrival rate three and service rate five as the limit of the probabilities for the corresponding process with a finite number of servers:
PDF for the queue with finitely many servers:
PDF for the queue with infinitely many servers:
Phase Type Queues (4)
Trucks arrive at a storage facility according to an Erlang type-2 distribution with mean interarrival time of 30 minutes. The single attendant at the facility unloads them in a mean time of 25 minutes and the unloading times are exponentially distributed. Find the mean time for which trucks must wait in queue at the facility:
Messages arrive at a communication line according to a Poisson process, at an average rate of 1 message every 3 milliseconds. The transmission process can be represented by a two-phase hyperexponential distribution with phase probabilities 0.4 and 0.6. The average service times for the two phases are 4.8 milliseconds and 0.8 milliseconds, respectively. Find the mean number of messages and the mean time spent by a message in this communication system:
Cars arrive at the Super Car Wash according to a Poisson process with a mean interarrival time of 10 minutes. These cars are successively vacuumed, washed, and hand-dried, and the time to perform each of the three tasks is exponentially distributed with means of 1 minute, 3 minutes, and 1.5 minutes, respectively. Find the time that an arriving customer should expect to wait before vacuuming begins:
Messages arrive at a communication line according to a Poisson process, at an average rate of 1 message every 4 seconds. Five percent of the arriving messages require compression before they can be transmitted. The compression times follow an exponential distribution with a mean of 5 milliseconds while the message transmission times follow an exponential distribution with a mean of 3 milliseconds. Find the mean number of messages and the mean time spent by a message in this communication system:
Properties & Relations (10)
The stationary (system size) distribution for an M/M/1 queue follows GeometricDistribution:
The loss probability for an M/M/c/c queue is given by ErlangB:
The nonzero waiting probability for an M/M/c queue is given by ErlangC:
The mean queue length of an M/M/c queue is related to ErlangC:
Possible Issues (1)
Wolfram Research (2012), QueueingProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/QueueingProcess.html.
Wolfram Language. 2012. "QueueingProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/QueueingProcess.html.
Wolfram Language. (2012). QueueingProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/QueueingProcess.html