Open Queuing Networks
The Queuing add-in computes values for
the steady-state measures for a network of queues. The individual
stations of the network may be either Poisson or non-Markovian.
In the latter case the results are approximate. On this
page we discuss open queuing networks where flow arrivals
enter from outside the network and ultimately leave the network.
add a model for a network of queues, place the cursor at the
worksheet cell where the model is to be located and select Open
Network from the menu. The dialog box below allows entry
of the parameters of the model. Two arrangements are possible.
The Serial arrangement assumes that stations are traversed
by customers in sequence. Customers enter the system at the
first station. After processing, they move the second station,
and so on until the customer leaves the system after passing
through the last station. The General option allows
the flows to pass between stations in a very general way. We
illustrate the Serial option first and later discuss
the General option. Two options are also available for
the distributions of interarrival times and service times.
Clicking the Poisson button gives a model in which all
times are exponentially distributed. The results provided by
the model are accurate in this case. The General option
allows non-Markovian stations in the network. When some stations
are non-Markovian the results are approximate.
illustrate the Poisson case first. When
all arrival and service processes are Poisson, the resultant
network is called a Jackson network. It has been shown that
this kind of network can be analyzed by considering each station
independently using the formulas for single Poisson queues.
System steady-state results are accurately computed by adding
the results for the stations.
check boxes on the dialog determine optional presentation of
the steady state results. The theory for the analysis of queuing
networks does not allow finite queues or finite populations,
so results related to these options are not available.
The Serial Case
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 3 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.
The add-in constructs the arrays shown below to model the
network with three stations. Data has already been filled in
for the example problem. Note that the upper-left corner of
the information is at cell B2, the location specified by the
dialog. In columns C through E in row 2, we have replaced the
computer generated names by 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.
This cell is outlined in maroon to indicate that it may be
changed by the user. 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
efficiency greater than 1 and an unstable system.
remainder of the rows of the display show the results that
are computed with the queuing functions provided by the add-in.
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 process 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 in process 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
illustrate the general network, we modify the problem slightly
by changing the proportions of the jobs passing through each
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.
Since the different numbers of jobs pass through each device,
we must use the General network option. The dialog below
The display for this option is shown below with the data entered
for the computer problem. The display has been placed with
its upper-left corner at cell B16.
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
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 values for the mean number in the system, queue and service
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 values associated
with the mean time in the system, queue and service are determined
by applying Little's Law using the Total Arrival rate in
column G. 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.