# QueueingNetworkProcess

QueueingNetworkProcess[γ,r,μ,c]

represents an open (Jackson) queueing network process with arrival vector γ, routing probability matrix r, service vector μ, and service channel vector c.

QueueingNetworkProcess[γ,r,μ,c,k]

represents a closed (Gordon–Newell) queueing network process with k jobs in the system.

# Details

- QueueingNetworkProcess is a continuous-time and discrete-state process.
- QueueingNetworkProcess at time t is the number of customers in the network at time t.
- The arrivals at node i in the network follow PoissonProcess[γ
_{i}]. - The service times at node i in the network follow ExponentialDistribution[μ
_{i}]. - QueueingNetworkProcess allows c to be any vector of positive integers, k any positive integer, and the entries of the routing probability matrix r must lie between 0 and 1.
- QueueingNetworkProcess can be used with such functions as QueueProperties, StationaryDistribution, and RandomFunction.

# Examples

open allclose all## Basic Examples (2)

## Scope (10)

### Simple Feedforward Networks (2)

Define a network with three queues in series:

Routing probability matrix for the network:

Compute a probability for the steady state of the network:

Verify the result using Burke's theorem:

Performance measures for the first node in the network:

Define a feedforward queueing network:

Routing probability matrix for the network contains no back routes:

Compute a probability for the steady state of the network:

Verify the result using Poisson aggregation and decomposition:

Performance measures at the fourth and fifth nodes in the network:

### Simple Feedback Networks (3)

Define a network process corresponding to a queue with feedback probability p:

PDF for the steady state of the network:

Network with two nodes that feed back into each other:

Routing probability matrix for the network:

Mean time spent in the system:

Define a cyclic queue with three nodes:

### Jackson Networks (2)

### Gordon–Newell Networks (3)

Define a closed queueing network:

There are no arrivals from outside the network:

Properties of the second node in the network:

Plot the simulated values at the first two nodes in the network:

A closed network with seven nodes:

Steady-state performance measures at each node in the network:

## Applications (12)

### Machine Repair (3)

An electronics manufacturer uses five robots when manufacturing its circuit boards. The breakdown times for the robots are exponentially distributed with a mean of 30 hours. The company has two repairmen who can repair the robots, and the repair times are exponentially distributed with a mean of three hours. Find the average number of robots that are operational at any given time by using a queueing network with two states, "working" and "broken":

The average number of operational robots:

Two machines in a factory are desired to be operational at all times. The machines break down according to an exponential distribution with mean failure rate . Upon breakdown, a machine has a probability that it can be repaired locally by a single repair person who works according to an exponential distribution with parameter . With probability , the machine must be repaired by a specialist who works according to an exponential distribution with parameter . There is a probability that a machine will also require the special service after completing the local service. Find the percentage of time that both machines are operational by using a queueing network with states "working", "locallyRepairable", and "specialistRepairable":

Both machines are operational for around 23% of the time:

A repair facility shared by a large number of machines has two sequential stations with service rates one per hour and two per hour, respectively. The cumulative failure rate of the machines is 0.5 per hour. Find the probability that both service stations in the repair facility are idle:

Probability that both service stations are idle:

Average repair time for a machine going through both repair stages:

### Computer Systems (4)

In a series network of three routers, the packets arrive at the rate of 100 packets per second. The service rates of the three routers are 250 packets per second, 150 packets per second, and 200 packets per second, respectively. Find the probability of having two packets at each of the three routers using a series network:

Probability of having two packets at each of the three routers:

Obtain the same result using Probability:

Find the probability that there are three packets in the network:

Ten requests circulate in a three-node central server system. The central server (node 1) sends requests with probabilities 0.3 and 0.7 to the remaining two nodes. The exponential service times at the three nodes are 1, 2, and 0.8, respectively. Find the bottleneck device in this closed network of servers:

The first node is the bottleneck device:

Ten requests circulate in a three-node central server system. The central server (node 1) sends requests with probabilities 0.3 and 0.5 to the remaining two nodes. External requests arrive only at the central server and follow a Poisson process with a rate of 0.15, while the exponential service times at the three nodes are 1, 2, and 0.8, respectively. Find the bottleneck device in this open network of servers:

The first node (central server) is the bottleneck device:

New programs arrive at a CPU according to a Poisson process with rate γ. A program spends an exponentially distributed execution time of mean 1/μ 1 in the CPU. At this stage, the program execution is either complete with probability p or it requires additional information from secondary storage with probability 1-p. The retrieval of information from secondary storage requires an exponentially distributed amount of time with mean 1/μ 2. Find the mean time that each program spends in the system:

Using Little's formula, this is the mean time spent by a program:

### Customer Service (1)

Customers arrive at a supermarket according to a Poisson process with a mean rate of 40 per hour. They take an average of hour to fill their shopping carts before proceeding to the four checkout counters in the store. The checkout times are exponentially distributed with a mean of four minutes. Find the average number of customers in the store at any given time:

### Communication Networks (2)

A transmitter has two permits for message transmission. As long as the transmitter has a permit, it generates messages at an exponential rate of λ. The messages enter the transmission system and are sent at an exponential rate of μ. As soon as a message arrives on the other side of the transmission system, the corresponding permit is sent back to the transmitter at an exponential rate of μ. Find the steady-state probability mass function (pmf) for the network:

Steady-state distribution for the network:

Steady-state pmf for the network:

Probability that there are no messages at the transmitter:

The following shows the telephone system at an insurance company. Calls come in to the 800 number (black) and are then directed to claims (red) or policy service (blue):

The arrival rates from the outside to the three nodes:

The service rates at the three nodes:

The routing probability matrix for the network:

The diagram indicates that the service channel vector is as follows:

Hence, this open network is described as follows:

Compare the mean queue lengths at the three nodes:

The total average time spent by a customer in the system is around 17 minutes:

### Airport Terminal (1)

Passengers arrive at an airport and proceed to one of the four check-in counters at the terminal. Next, they proceed to the security check, and finally they board their flights from one of the three gates in the terminal:

The terminal network is described by:

Study the performance at the security check and at the third gate:

## Properties & Relations (3)

A queueing network process with a single node is equivalent to a queueing process:

Find the arrival rate for the equivalent queueing process:

These have the same stationary distribution:

Define a network with two queues in series:

PDF for the steady state of the network:

Obtain the same result using a product of the individual queueing processes:

A zero external arrival vector γ corresponds to a closed queueing network:

#### Text

Wolfram Research (2012), QueueingNetworkProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/QueueingNetworkProcess.html.

#### CMS

Wolfram Language. 2012. "QueueingNetworkProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/QueueingNetworkProcess.html.

#### APA

Wolfram Language. (2012). QueueingNetworkProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/QueueingNetworkProcess.html