OMP9334 Project, Term 1, 2023:
Priority queueing for multi-phase jobs
Due Date: 5:00pm Friday 21 April 2023
Version 1.00
Updates to the project, including any corrections and clarifications, will be posted on the
course website. Make sure that you check the course website regularly for updates.
Change log
Version 1.00. Issued on 20 March 2023.
1 Introduction and learning objectives
When you were learning about operational analysis earlier in the term, we talked about jobs that
require multiple visits to the CPU (or servers) to receive their service. In this project, you will
use simulation to study how priority queueing can be used to improve the performance of a multi-
server system that works on jobs that require multiple visits to the servers.
In this project, you will learn:
1. To use discrete event simulation to simulate a computer system
2. To use simulation to solve a design problem
3. To use statistically sound methods to analyse simulation outputs
2 Support provided and computing resources
If you have problems doing this project, you can post your question on the course forum. We
strongly encourage you to do this as asking questions and trying to answer them is a
great way to learn. Do not be afraid that your question may appear to be silly, the
other students may very well have the same question! Please note that if your forum post
shows part of your solution or code, you must mark that forum post private.
Another way to get help is to attend a consultation (see the Timetable section of the course
website for dates and times).
If you need computing resources to run your simulation program, you can do it on the VLAB
3 Multi-server system configuration and job characteristics
for this project
The configuration of the system that you will use in this project is shown in Figure 1. The sys-
tem consists of a dispatcher and n servers where n > 1. The dispatcher has two queues: a high
priority queue and a low priority queue. You can assume that both queues have infinite queueing
slots. You have not learnt about priority queues yet but the following description will explain how
priority queues are used.
We will use the word job to refer to a request that requires service from this system. A job
may require one or more visits to the servers in order to get all its work completed. These visits of
a job take place one after another with a possible time gap between two consecutive visits. Jobs
in this system do not use parallel processing so each job does not use more than one server at a time.
We will now explain how this system handles a new job. When a new job (i.e., an external
arrival) arrives at the system, the dispatcher will send the job to any one of the idle servers if
there is at least one idle server. If all the servers are busy, the dispatcher will place this job at the
end of the high priority queue.
After a job has completed a visit to the server, the job either requires or does not require further
visits to the servers. If the job does not require further visits to the servers, then the job will depart
from the system permanently. If the job requires further visits to the servers, then the job will be
sent back to the dispatcher. We will use the term re-circulated jobs to refer to those jobs that are
sent back to the dispatcher from the servers because these jobs require further visits to the servers.
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A job that arrives at the dispatcher can either be a new job or a re-circulated job, see Figure
1. We have already explained how the dispatcher handles new jobs. We will start to describe how
the dispatcher handles the re-circulated jobs. Since the dispatcher handles all re-circulated jobs
in the same way, the procedure therefore applies to a generic re-circulated job. We first need to
define some notation. First, when a re-circulated job arrives at the dispatcher, the job can have
completed 1, 2, 3 or more visits to the servers. We will use c to denote the number of completed
server visits when a re-circulated job arrives at the dispatcher. Second, the dispatcher uses a
threshold h, which is an integer bigger than or equal to 1, to decide on whether an arriving re-
circulated job should be considered a high or low priority job. Now we have defined the notation,
we can state the rule that the dispatcher uses: When a re-circulated job arrives at the dispatcher,
the dispatcher will classify this job as low priority if its value of c is greater than or equal to h;
otherwise the job is a high priority job. Let us consider an example.
Example 1 In this example, we assume the threshold h has a value of 2. Let us consider a job
which requires altogether 3 server visits before it will permanently depart from the system. So, this
job will re-circulate to the dispatcher two times: once with a value of c = 1 and the other with
c = 2.
When this job re-circulates to the dispatcher the first time, its value of c will be 1. Since c ≥ h
does not hold, the dispatcher will consider this job as a high priority job on this occasion.
The second time that this job re-circulates to the dispatcher, its value of c will be 2. Since
c ≥ h holds, the dispatcher will consider this job as a low priority job on this occasion.
We have now explained how the dispatcher classifies an arriving re-circulated job into either a
high or low priority job. We have yet to explain the detailed working of the dispatcher. We will
do that together with the description of how departures are handled. This is because the arrival
of a re-circulated job at a dispatcher follows the job’s earlier departure from a server, see Figure
1. The following steps describe how a job, which has completed a server visit, will be handled.
For ease of referral, we will use the term tagged job to refer to this job that has just completed
its server visit.
The tagged job is considered to be a permanent departure if the number of complete visits
that it has already made is equal to the total number of visits that this job requires. If the
tagged job is not a permanent departure, then it will be re-circulated to the dispatcher. The
server that was working on the tagged job would send a message to the dispatcher to inform
it that it is available to serve another job.
If the tagged job is a re-circulated job, then it will be sent to the dispatcher which will
classify it into either a high or low priority job using the values of c and h as described
earlier. The dispatcher will then place the tagged job at the end of the appropriate queue.
The dispatcher is aware that a server has just completed a visit of a job and is available to
process another job. The dispatcher executes the following:
– If the high priority queue is non-empty, then the job at the head of the high priority
queue will be sent to the available server for processing.
– If the high priority queue is empty and the low priority queue is non-empty, then the job
at the head of the low priority queue will be sent to the available server for processing.
– If both high and low priority queues are empty, then the dispatcher does not need to
do anything. The server that has just been made available will go idle.
We remark that the above description means that the dispatcher uses the non-preemptive queue-
ing discipline. We will be discussing queueing disciplines in Week 7 and you can read about it on
p. 500 of [1]. However, the above description should be enough for you to get your project going
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now even before we discuss priority queues in Week 7.
We make the following assumptions on the system in Figure 1. First, it takes the dispatcher
negligible time to process a job, to classify a job and to send a job to an available server. Second,
it takes a negligible time for a server to send a re-circulated job to the dispatcher and to inform
the dispatcher on its availability. As a consequence of these assumptions, it means that: (1) If a
job arriving at the dispatcher is to be sent to an available server right away, then its arrival time at
the dispatcher is the same as its arrival time at the chosen server; (2) The departure time of a job
from the dispatcher is the same as its arrival time at the chosen server; and (3) The departure time
of a re-circulated job from a server is the same as its arrival time at the dispatcher. Ultimately,
these assumptions imply that the response time of the system depends only on the queues and
the servers.
We have now completed our description of the operation of the system in Figure 1. We will
provide a number of numerical examples to further explain its operation in Section 4.
You will see from the numerical examples in Section 4 that the threshold h can be used to
influence the system’s mean response time. So, a design problem that you will consider in this
project is to determine the value of the threshold h to minimise the mean response time of the
system. You can read in [1] how priority queueing can be used to reduce the mean response time
of computer systems.
Remark 1 This project is inspired by a recent work [2] which studies how priority queueing can be
used to improve the performance of a multi-server system that provide service to multi-phase jobs.
A multi-phase job also requires multiple visit to the servers in order to get its work done. However,
the multi-phase job in [2] will sometimes require only the service of a server but sometimes it may
require a number of servers in parallel. In order to make this project more do-able, we have
simplified many of the settings in [2]. For example, we do not use preemptive queueing, processor
sharing and parallel servers.
4 Examples
We will now present two examples to illustrate the operation of the system that you will simulate
in this project. In all these examples, we assume that the system is initially empty.
4.1 Example 1: number of servers n = 2 and threshold h = 1
In this example, we assume the there are n = 2 servers in the system and the threshold h for
determining whether a re-circulated job is of low or high priority is 1.
In this example, each job requires one or two visits to the servers before it permanently departs
from the system. Table 1 shows, for each job, its arrival time and the service times for its visits. If
there is only one service time in the third column in Table 1, then it means the job only requires
one server visit. If there are two service times, then the job requires two server visits. For example,
Job 1 in Table 1 requires two visits where the first and second visits require, respectively, 3 and
10 time units of service times. As another example, Job 3 requires only one visit and the service
time required for that visit is 6 time units.
In this example, a job will be identified with using the tuple (i, c/r) where i is the job’s index
(see the first column of Table 1), c is the number of complete servers visits made by the job and
r is the total number of server visits required by the job. For example,
The job (1, 0/2) refers to the job with index 1. We know from Table 1 that Job 1 requires
2 visits to the servers and this is indicated by “/2”. The notation “0/2” says that this job
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Job index Arrival time Service times of the job’s server visits
1 0.9 3, 10
2 1.5 2, 1
3 2.2 6
4 3.3 2
5 8.0 1, 4
Table 1: Data for Example 1.
has done zero complete visits to the servers. When Job 1 re-circulates to the dispatcher for
the first time, its tuple becomes (1, 1/2).
The job (5, 1/2) refers to Job 5 which requires altogether 2 visits to the servers. The notation
“1/2” says that this job has done one complete visit to the servers out of the two required
visits.
Remark 2 We remark that the job indices are not necessary for carrying out the discrete event
simulation. We have included the job index to make it easier to refer to a job in our description
below.
The events in the system in Figure 1 are the arrival of a new job to the dispatcher and the
completion of a visit at a server. Note that we have not included the arrival of a re-circulated job
to the dispatcher as an event. This is because the arrival of a re-circulated job at the dispatcher
is immediately after the completion of a server visit. So the simulation will handle the arrival of
re-circulated job at the dispatcher and its associated server completion together.
We will illustrate how the simulation of the system works using “on-paper simulation”. The
quantities that you need to keep track of are:
Next arrival time is the time that the next new job will arrive
For each server, we keep track its server status, which can be busy or idle.
We also keep track of the following information on the job that is being processed in the
server:
– Next completion time is the time at which the job will complete its current server
visit. If the server is idle, the next completion time is set to ∞. Note that there is a
next completion time for each server.
– The time that this job arrived at the system. This is needed for calculating the response
time of the job when it permanently departs from the system.
– A list of the service times for the future server visits of this job. Note that we enclose
the list of service times within a pair of square brackets [ ].
– The job’s tuple.
For example, the job information “3.5, 1.5, [1], (2,0/2)” indicates that current visit will be
completed at time 3.5 and this job arrived at the system at time 1.5. The “0/2” indicates
that the job has not completed any server visits so the current visit is the job’s first visit to
the server. The “[1]” indicates that the job needs one more visit in the future and this visit
will require a service time of 1. Note that if the job has no more future visits to make, then
we will use [ ] to indicate that.
The contents of the high and low priority queues. Each job in the queue is identified by 3
fields: the job’s tuple, the job’s arrival time to the system, a list of the job’s service times
for its future server visits. For example, we write a job in a queue as
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[(1,1/2), 0.9, [10] ]
which means the job (1,1/2) arrived at the system at time 0.9, has 1 visit completed and its
future visit to the server will require a service time of 10.
The “on-paper simulation” is shown in Table 2. The notes in the last column explain what
updates you need to do for each event. Recall that the two event types in this simulation are the
arrival of a new job to the dispatcher and the completion of a visit at a server, we will simply refer
to these two events as Arrival and Completion in the “Event type” column (i.e., second column)
in Table 2.able 6: The completion times for the server visits.
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5 Project description
This project consists of two main parts. The first part is to develop a simulation program for the
system in Fig. 1. The system has already been described in Section 3 and illustrated in Section 4.
In the second part, you will use the simulation program that you have developed to solve a design
problem.
5.1 Simulation program
You must write your simulation program in one (or a combination) of the following languages:
Python 3 (note: version 3 only), C, C++, or Java. All these languages are available on the CSE
system.
We will test your program on the CSE system so your submitted program must be able to
run on a CSE computer. Note that it is possible that due to version and/or operating system
differences, code that runs on your own computer may not work on the CSE system. It is your
responsibility to ensure that your code works on the CSE system.
Note that our description uses the following variable names:
1. A variable mode of string type. This variable is to control whether your program will run
simulation using randomly generated arrival times and service times; or in trace driven mode.
The value that the parameter mode can take is either random or trace.
2. A variable time_end which stops the simulation if the master clock exceeds this value. This
variable is only relevant when mode is random. This variable is a positive floating point
number.
Note that your simulation program must be a general program which allows different param-
eter values to be used. When we test your program, we will vary the parameter values. You can
assume that we will only use valid inputs for testing.
For the simulation, you can always assume that the system is empty initially.
5.1.1 The random mode
When your simulation is working in the random mode, it will generate the inter-arrival times
and the workload of a job in the following manner.
1. We use {a1, a2, . . . , ak, . . . , ...} to denote the inter-arrival times of the jobs arriving at the
dispatcher. These inter-arrival times have the following properties:
(a) Each ak is the product of two random numbers a1k and a2k, i.e ak = a1ka2k ?k = 1, 2, ...
(b) The sequence a1k is exponentially distributed with a mean arrival rate λ requests/s.
(c) The sequence a2k is uniformly distributed in the interval [a2l, a2u].
Note: The easiest way to generate the inter-arrival times is to multiply an exponentially
distributed random number with the given rate and a uniformly distributed random number
in the given range. It would be more difficult to use the inverse transform method in this
case, though it is doable.
2. The workload of a job is characterised by; (i) the number of server visits that the job re-
quires; and (ii) the service times of all these server visits.
The first step to generate the workload of a job is to generate a random positive integer to
use as the number of server visits that this job requires. You will be given a sequence of J
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non-negative real numbers p1, p2, ..., pk, ... and pJ with the property
∑J
k=1 pk = 1. Given
these numbers, we want the probability that a job needs k server visits to be equal to pk,
for k = 1, ..., J .
For example, if you are given the sequence 0.5, 0.2, 0.3, then the jobs generated has the
following properties:
(a) Prob[a job requires 1 server visit] = 0.5
(b) Prob[a job requires 2 server visits] = 0.2
(c) Prob[a job requires 3 server visits] = 0.3
Note that you may interpret J as the maximum number of server visits that a generated job
requires. In the example above, we have J = 3, which implies that all generated jobs need
at most 3 server visits.
3. If a job requires k server visits, then you will need to generate k random service times for
each of the k server visits. These k service times are independent and they all come from
the same probability distribution.
The service time per server visit is generated by the probability density function (PDF) g(t)
where:
g(t) =
{
0 for 0 ≤ t ≤ α
γ
tβ
for α < t
(1)
where
γ =
β ? 1
α1?β
Note that this probability density function has two parameters: α and β. You can assume
that α > 0 and β > 3.
As an example, if a job requires 3 server visits, then you will need to generate 3 random
numbers which come from the probability distribution whose PDF is given by g(t).
5.1.2 The trace mode
When your simulation is working in the trace mode, it will read the list of inter-arrival times
and the list of service times of the server visits from two separate ASCII files. We will explain the
format of these files in Sections 6.1.3 and 6.1.4 .
An important requirement for the trace mode is that your program is required to simulate
until all jobs have departed from the system. You can refer to Table 2 for an illustration.
Hint: Do not write two separate programs for the random and trace modes because they share
a lot in common. A few if–else statements at the right places are what you need to have both
modes in one program.
5.2 Determining the threshold h that minimises the mean response time
After writing your simulation program, your next step is to use your simulation program to de-
termine the threshold h that can minimise the mean response time.
For this design problem, you will assume the following parameter values:
Number of servers: n = 6
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For inter-arrival times: λ = 3.9, a2` = 0.91, a2u = 1.27
For the number of server visits required for each job: the sequence p1, p2, p3, p4, p5 is 0.52,
0.21, 0.15, 0.08, 0.04.
For the service time per server visit: β = 3.4, α = 0.3.
In solving this design problem, you need to ensure that you use statistically sound methods
to compare systems. You will need to consider simulation controls such as length of simulation,
number of replications, transient removals and so on. You will need to justify in your report on
how you determine the value of the threshold h.
6 Testing your simulation program
In order for us to test the correctness of your simulation program, we will run your program using
a number of test cases. The aim of this section is to describe the expected input/output file format
and how the testing will be performed.
Each test is specified by 4 configurations files. We will index the tests from 1. If 12 tests are
used, then the indices for the tests are 1, 2, ...., 12. The names of the configuration files are:
For Test 1, the configuration files are mode_1.txt, para_1.txt, interarrival_1.txt and
service_1.txt. The files are similarly named for indices 2, 3, .., 9.
For Test 10, the configuration files are mode_10.txt, para_10.txt, interarrival_10.txt
and service_10.txt. The files are similarly named if the test index is a 2-digit number.
We will refer to these files using the generic names mode *.txt, para *.txt etc. We will describe
the format of the configuration files in Section 6.1
Each test should produce 2 output files whose format will be described in Section 6.2. We will
explain how testing will be conducted in Sections 6.3 and 6.5.
6.1 Configuration file format
Note that Test 1 is the same as Example 1 discussed in Section 4.1. We will use that test to
illustrate the file format.
6.1.1 mode *.txt
This file is to indicate whether the simulation should run in the random or trace mode. The file
contains one string, which can either be random or trace.
6.1.2 para *.txt
If the simulation mode is trace, then this file has two lines. The first line is the value of n (=
number of servers) and the second line has the value of h (= threshold for priority queueing). If
the test is Example 1 in Section 4.1, then the contents of this file are:
2
1
These values are in the sample file para_1.txt.
If the simulation mode is random, then the file has three lines. The meaning of the first two
lines are the same as above. The last line contains the value of time_end, which is the end time of
the simulation. The contents of the sample file para_7.txt are shown below where the last line
indicates that the simulation should run until 1000.
15
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1000
You can assume that we will only give you valid values. You can expect n to be a positive
integer greater than 1. You can expect h to be a positive integer. For time_end, it is a strictly
positive integer or floating point number.
6.1.3 interarrival *.txt
The contents of the file interarrival *.txt depend on the mode of the test. If mode is trace,
then the file interarrival *.txt contains the interarrival times of the jobs with one interarrival
time occupying one line. You can assume that the list of interarrival times is always positive. For
Example 1 in Section 4.1, the arrival times are [0.9, 1.5, 2.2, 3.3, 8.0] which means the inter-arrival
times are [0.9, 0.6, 0.7, 1.1, 4.7]. For this example, the inter-arrival times will be specified by a file
(see sample file interarrival 1.txt) whose contents are:
0.9000
0.6000
0.7000
1.1000
4.7000
If the mode is random, then the file interarrival *.txt contain 2 lines. The first line contains
three values corresponding to the parameters λ, a2` and a2u. The second line contains the the
values for the sequence p1, ..., pJ . As an example, the contents of interarrival 8.txt are:
1.0 0.95 1.2
0.5 0.3 0.15 0.05
For this example, the values of λ, a2` and a2u are respectively 1.0, 0.95 and 1.2. The values of
p1, p2, p3, p4 are 0.5, 0.3, 0.15, 0.05. This means that you can infer the value of J by counting the
number of values found in the second line of the file. For interarrival 8.txt, J = 4. Note that
you can assume that we will only give you valid pk, i.e. all pk’s are non-negative and the sum of
all pk’s is 1.
6.1.4 service *.txt
For trace mode, the file service *.txt contains the service times of the server visits. As an
illustration, the service times of the server visits for Example 1 in Section 4.1 will be specified by
a file (see sample file service 1.txt) whose contents are:
3.0000 10.0000
2.0000 1.0000
6.0000 NaN
2.0000 NaN
1.0000 4.0000
where you will find the service times of the server visits of each job in a line of the file.
Note that the symbol NaN is a Python floating point number to denote not a number and is
often used to indicate an absence of numbers. In this example, if there are two numbers on the
line, the job requires two server visits; if there is a number and an NaN, the job is requires only
one server visit.
The following shows the contents of service 3.txt for trace mode simulation:
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2.1000 3.2000 1.9000 NaN
4.0000 3.0000 4.9000 6.1000
5.1000 2.3000 1.2000 NaN
7.2000 1.8000 NaN NaN
4.6000 NaN NaN NaN
Note that there are 4 entries in each line where the number 4 corresponds to the maximum number
of server visits among all the jobs. You can conveniently load the contents of this file by using the
function numpy.loadtxt() into a numpy array. You may also find the function numpy.isnan()
useful.
In general, if the maximum number of server visits among all jobs is V , then there are V entries
in each line of service *.txt.
For random mode, the file service *.txt contains one line, corresponding to the values of β
and α.
You can assume that the data we provide for trace mode are consistent in the following way:
The number of inter-arrival times and the number of lines of service times are equal.
6.2 Output file format
In order to test your simulation program, we need two output files per test. One file containing
the mean response time. The other file contains the completion times of the server visits from the
servers.
We want to start by clarifying what we mean by mean response time. You can calculate the
response time of a job by subtracting the time that this job arrives at the system as a new job
from the time it permanently departs from the system. Tables 1 and 5 illustrate this concept.
For trace mode, the mean response time will be calculated using all the jobs provided in the
interarrival *.txt and service *.txt. This is because, as mentioned in Section 5.1.2, a trace
mode simulation is required to simulate until all jobs have permanently departed from the system.
For random mode, the mean response time should be calculated using all those jobs that have
permanently departed the system by time_end. In other words, for those jobs which are still in
the queue or are being processed in the server at time_end, you do not include these jobs when
calculating the mean response time.
Note that you do not have to consider transient removal for the mean response before you
write the result to the output file. However, you should consider transient removal when you do
your design.
The mean response time should be written to a file whose filename has the form mrt_*.txt.
For Example 1 in Section 4.1, the expected contents of this file are:
7.5600
The other file dep_*.txt contains the completion times of of the server visits from the servers.
For Example 1 in Section 4.1, the expected contents of this file are:
1.5000 3.5000 1 2
0.9000 3.9000 1 2
3.3000 5.9000 1 1
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1.5000 6.9000 2 2
2.2000 9.5000 1 1
8.0000 10.5000 1 2
8.0000 14.5000 2 2
0.9000 16.9000 2 2
Note the following requirements for the file containing the completion times:
1. Each line contains 4 entries.
2. Each line provides the information on the completion time of a server visit.
3. For each line, the first entry is the arrival time of the job to the system (i.e., as a new job),
the fourth entry is the total number of server visits required by this job, the third entry is
the number of complete server visits that this job has made at the time given by the second
entry. Let us take the first line 1.5000 3.5000 1 2. It says that the job that arrives at the
system at time 1.5 requires a total of 2 server visits, and at time 3.5, this job has completed
1 server visit. You should be able to reconcile the contents of the above file with Example 1
in Section 4.1.
4. The server visits must be ordered according to ascending completion times.
5. If the simulation is in the trace mode, we expect the simulation to finish after all jobs have
been processed. Therefore, the number of lines in dep_*.txt should be equal to the total
number of server visits of all jobs.
6. If the simulation is in the random mode, the file should contain all the server visits that have
been completed by time_end.
All mean response times, arrival times and completion times in mrt_*.txt and dep_*.txt
should be printed as floating point numbers to exactly 4 decimal places. Note that your simulation
should be performed in full floating point precision and you should only do the rounding when
you are writing the output files.
6.3 The testing framework
When you submit your project, you must include a Linux bash shell script with the name
run_test.sh so that we can run your program on the CSE system. This shell script is required
because you are allowed to use a computer language of your choice.
Let us first recall that each test is specified by a four configuration files and should produce
two output files. For example, test number 1 is specified by the configuration files mode_1.txt,
interarrival_1.txt, service_1.txt and para_1.txt; and test number 1 is expected to produce
the output files mrt_1.txt and dep_1.txt.
We will use the following directory structure when we do testing.
the directory containing run test.sh
config/
output/
We will put all the configuration files for all the tests in the sub-directory config/. You should
write all the output files to the sub-directory output/.
To run test number 1, we use the shell command:
./run_test.sh 1
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The expected behaviour is that your simulation program will read in the configuration files for
test number 1 from config/, carry out the simulation and create the output files in output/.
Similarly, to run test number 2, we use the shell command:
./run_test.sh 2
This means that the shell script run_test.sh has one input argument which is the test number
to be used.
Let us for the time being assume that you use Python (Version 3) to write your simulation
program and you call your simulation program main.py. If the file main.py is in the same directory
as run_test.sh, then run_test.sh can be the following one-line shell script:
python3 main.py $1
The shell script will pass the test number (which is in the input argument $1) to your simula-
tion program main.py. This also implies that your simulation program should accept one input
argument which is the test number.
Just in case you are not familiar with shell script, we have provided two sample files: run_test.sh
and main.py to illustrate the interaction between a shell script and a Python (Version 3) file. You
need to make sure run_test.sh is executable. If you run the command ./run_test.sh 2, it will
produce a file with the name dummy_2.txt in the directory output/. You can also try