QueueProperties
QueueProperties[qproc]
gives a summary of properties for the queueing process qproc.
QueueProperties[{qproc,i}]
gives a summary of properties for the i node in the queueing network process qproc.
QueueProperties[data]
gives a summary of properties for queueing simulation data.
QueueProperties[…,"property"]
gives the specified "property".
Details
 The process qproc can represent a single queue, using QueueingProcess, or a network of queues, using QueueingNetworkProcess.
 In QueueProperties[data], data can be of the form generated by RandomFunction.
 QueueProperties[qspec,"property","Description"] gives a description of the property as a string.
 Basic properties for include:

"ArrivalRate" arrival rate for the queueing system "ArrivalDistribution" arrival distribution for the queueing system "DataSource" whether data is from the queue or queueing network "InitialState" initial state of the queueing system "NetworkType" type of queueing network "NodeCount" number of nodes in the queueing network "QueueDiagram" diagram of the queueing system "QueueNotation" Kendall notation for the queue "SelectedNode" selected node in the queueing network "ServiceChannels" number of service channels "ServiceRate" service rate for each server "ServiceDistribution" service distribution for each server "SummaryTable" summary of properties "SystemCapacity" maximum capacity of the queueing system "Throughput" departure rate for the queueing system "UtilizationFactor" fraction of time the servers are busy  Stationary (or steadystate) performance measures include:

"MeanSystemSize" mean number of jobs in the system "MeanSystemTime" mean time spent in the system "MeanQueueSize" mean number of jobs in the queue "MeanQueueTime" mean time spent in the queue "StationarySystem" whether a steady state will be reached  Stationary or steadystate properties refer to the longterm behavior of a queueing system. They are only available if the queueing system becomes stationary. Typically, a queueing system becomes stationary if the utilization factor is less than 1.
 If the queueing system is specified with exact or symbolic parameters, then the result will also be exact or symbolic. Some performance properties may only be computable when given approximate numeric parameters.
 If a property is not available, this is indicated by Missing["reason"] in the corresponding entry of the table.
Examples
open allclose allBasic Examples (3)
Scope (10)
Single Queues (4)
Queueing Networks (2)
Queue Simulation Data (2)
Applications (7)
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:
The average number of people waiting:
The average time in minutes spent by a patient at the clinic:
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:
Steadystate performance measures:
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:
The mean number of packets in the router:
Cars arrive at the drivethrough 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 the method of moments:
The resulting drivethrough process:
The average waiting time until the customer reaches the service window:
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:
Find the service rate μ that ensures the average time spent is less than 7.5 minutes:
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:
The ratio of system sizes indicates that a single queue will have a smaller system size:
Customers arrive at a rate of 11 per hour to a post office. Find the average time spent by a customer at the post office, if there are two clerks that each can handle 9 customers per hour:
Average time in minutes spent by a customer at the post office:
Properties & Relations (4)
Steadystate performance measures obey Little's laws:
Relation between mean system size and mean system time :
Relation between mean queue size and mean queue time :
Mean system size is the mean of the stationary distribution for a queue:
The stationary distribution for an M/M/c queue exists if the utilization factor is less than 1:
The mean queue size of an M/M/c queue is related to ErlangC:
Possible Issues (2)
Some property values may not exist or be available:
Since the process is not stationary, there is no mean system size:
The system size is not bounded:
The mean system size exists if the utilization factor () is less than 1:
The system size is now bounded:
Performance measures may be undefined for exact parameters:
Use inexact input to obtain performance measures based on the estimates from simulation:
Text
Wolfram Research (2012), QueueProperties, Wolfram Language function, https://reference.wolfram.com/language/ref/QueueProperties.html.
CMS
Wolfram Language. 2012. "QueueProperties." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/QueueProperties.html.
APA
Wolfram Language. (2012). QueueProperties. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/QueueProperties.html