

Model
Classes


Queuing Networks


In many applications, rather than
passing through a single queue, a calling unit will pass though
a series of queues. An example is the registration procedure
at a college where students visit a number of stations as required
for the approval of their academic program. Each station involves
one or more servers and a queue of waiting students. Another
example is a job shop where machines are grouped by function.
A collection of machines of the same type can be modeled as
a queuing station. A job entering the shop is processed at
one or more stations before it leaves. The analysis of these
types of networks can be quite difficult unless they have some
special structure.
Analytical results are available when all external arrivals
to the network are Poisson, each queue within the network has
unlimited capacity and exponential service times, and transfers
from one queue to another are made randomly according to given
probabilities. This type of system is called a Jackson network,
and some of its steadystate behavior can be described using
the same formulas developed for single station Markovian queues.
As we have seen, these formulas are very easy to apply and
yield numerical results that are critical to the design of
a system.
When the individual queuing stations are not Markovian the
results are approximate. 

Example: Computer Center
Batch jobs submitted to a computer center pass through
three steps: an input processor, the central processor,
and an output printer. Jobs arrive to the computer center
at random with an average rate of 10 per minute. To handle
the load of jobs submitted, the computer center may have
several of the three types of processors operating in
parallel. The times for the three steps have exponential
distributions with mean values as follows: input processor
10 seconds, central processor 5 seconds, and printer
70 seconds. When processors are not immediately available
for a job, the job must wait in a queue. There is an
unlimited queue for each processor. Our task is to find
the minimum number of each type of processor and compute
the average time required for a job to pass through the
system. 
This example illustrates a serial queuing network where the
external input enters the first station and the remaining stations
process the entire amount. The Queuing addin constructs the
arrays shown below to model the network with three stations.
Data is for the example problem. Column B holds titles for
the parameters and results of the analysis. In columns C through
E in row 2, we have placed the names for the three network
devices. Row 3 shows the arrival rates. The yellow color in
columns D through E indicates that these cells hold formulas.
In fact, they all refer to the entry in cell G2, where the
arrival rate to the first station of the network is specified.
Row 4 holds the service rates for the devices, which are computed
from the mean processing times given in the problem statement.
Row 5 holds the number of servers of each device. We have entered
the minimum number of servers that can handle the load. Any
smaller number results in an unstable system. 



The remainder of the rows of the
display show the results that are computed with the queuing
functions provided by the addin. The rows in columns C, D,
and E describe the results for the individual stations. The
station results are added together to obtain the system results
in column G. For a network with arrivals and services governed
by exponential distributions, each queuing station may be analyzed
independently with the formulas derived for Poisson processes.
Reviewing the efficiencies for the three stations reveals
that the central processor is the least busy, while the printers
are busy 97% of the time. Of the almost 50 jobs in the system,
on the average, 43 of them are waiting or inprocess at the
printers. From the system results we see that on the average
a job spends about 5 minutes in the system.
Any number not colored yellow may be changed by the user to
obtain new results. For example, one might wonder if increasing
the number of printers would result in significantly lower
throughput times for jobs. That can be immediately determined
by increasing the number in cell E5. 

The General Network
To illustrate the general network, we modify the problem slightly
by changing the proportions of the jobs passing through each
device.
All jobs must go through the input processor.
Because of errors, only 80% of the jobs go through the
central processor. Only 40% of the jobs passing through
the central processor go to the output printer. 
The Queuing addin provides a General network option
for entering the more complex data for this system. The display
for this option is shown below with the data entered for the
example. The display has been placed with its upperleft corner
at cell B16.
For this option, a new data item appears in row 17, the Independent
Arrival Rate. Each station may have arrivals from outside the
system as indicated by the entries in this row. For the current
example, all arrivals occur to the Input processor, so the other
entries in this row are 0. The arrival rate in row 18 is a computed
quantity and the contents of the row should not be changed. As
before, the service rates and number of servers are controlled
by the user. The numbers of servers shown here are the smallest
numbers that will satisfy the demand at the three stations.
Data concerning the proportions of jobs passing from one station
to another is entered in the table in the range C31:E33. We call
this the transfer matrix because it shows the proportions of
the flow transferred from one station to another. In row 31 we
see that 80% of the flow transfers from the Input processor to
the central processor. The remaining 20% leaves the network.
From row 32 we see that 40% of the flow passing through the central
processor goes to the printer.
The arrival rate shown in row 18 is computed with an equation
involving the Inverse of the Augmented matrix. The Augmented matrix
is the transfer matrix subtracted from the identity matrix The
entries in the Transfer matrix are entirely general as long as
the Inverse exists. An entry on the diagonal represents a recycling
through the station. In many cases the numbers in the matrix
represent proportions of flow transferred. A requirement of the
Jackson network is that the routing be probabilistic. That is,
a particular entity leaving a station will transfer the next
with the given probability distribution. 



The numbers in column G are computed
by summing the measures in columns C through E. The Mean Number entries
represent the average number of jobs in the service, queue and
system. The Mean Time entries often are not meaningful
for individual jobs because the jobs follow different routes,
but they do represent the average time considering all jobs. 


