QueueProperties

QueueProperties[qproc]

gives a summary of properties for the queueing process qproc.

QueueProperties[{qproc,i}]

gives a summary of properties for the i^(th) 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 steady-state) 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 steady-state properties refer to the long-term 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 all

Basic Examples  (3)

Properties of an M/M/1 queue:

Obtain the value for a specific property:

Description of a property:

Scope  (10)

Single Queues  (4)

Properties of an M/M/2/6/5 queue:

M/G/1 queue with Erlangian service:

Exact values of properties for a G/M/1 queue with Erlangian arrivals:

Approximate values for properties of a Ph/Ph/1 queue:

Queueing Networks  (2)

Define an open queueing network:

Properties of the first node in the network:

Define a closed queueing network:

Properties of the second node in the network:

Queue Simulation Data  (2)

Properties based on simulation data for a single queue:

Use the corresponding random path for the data to obtain the same results:

Properties based on simulation data for a queueing network:

Property Values  (2)

Property values for a single queue:

Property values for a node in a closed queueing network:

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:

Steady-state 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:

The mean total delay:

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 the method of moments:

The resulting drive-through 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 mean system time:

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:

Compare mean system sizes:

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:

Find the average time if there are three postal clerks:

Properties & Relations  (4)

Steady-state 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:

Neat Examples  (1)

Mean system time for m servers, fixed service rate, and utilization factor ρ:

Mean system time for m servers, fixed arrival rate, and utilization factor ρ:

Mean system time for one server and utilization factor ρ:

Introduced in 2012
 (9.0)