represents an open (Jackson) queueing network process with arrival vector γ, routing probability matrix r, service vector μ, and service channel vector c.
represents a closed (Gordon–Newell) queueing network process with k jobs in the system.
- 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[γi].
- The service times at node i in the network follow ExponentialDistribution[μi].
- 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.
Examplesopen allclose all
Basic Examples (2)
Simple Feedforward Networks (2)
Simple Feedback Networks (3)
PDF for the steady state of the network:
Jackson Networks (2)
Gordon–Newell Networks (3)
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":
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":
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:
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:
Obtain the same result using Probability:
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:
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:
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:
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:
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:
Properties & Relations (3)
PDF for the steady state of the network:
Wolfram Research (2012), QueueingNetworkProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/QueueingNetworkProcess.html.
Wolfram Language. 2012. "QueueingNetworkProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/QueueingNetworkProcess.html.
Wolfram Language. (2012). QueueingNetworkProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/QueueingNetworkProcess.html