EE555: Queuing Model Project

ept for one group of 2 students). It is preferred to be in Excel form with embedded
Macros (any language will do. Best and easiest is to use Visual Basic)). Your user Interface
must be VERY user friendly!!!! A drop box will be created for your submissions in the
Extra Credit Assignment Folder of our web site. A single Excel Attachment (Sheet 1, Part
1, Sheet 2, Part 2, Sheet 3, Part 3, Sheet 4: Part 4 and Sheet 5, Part 5)). After submission,
each group is required to visit me in my office (after the deadline) to demonstrate with
arbitrary chosen test cases. I have 3 DEN students. You can form a group, or you can
work individually. A zoom meeting will be set for DEN students to demonstrate their
work. You can also use MATLAB if you prefer to do so.
Part 1:
It is desired to design an Erlang B calculator for the M/M/c/c “lost calls queuing model”
that we discussed in Discussion 1 session. The input parameters are the average arrival
rate () in packets/min, the average service rate () in packets/min and the probability of
blocking, PB. The output parameter should be the number of servers “c” required to
satisfy the PB requirement. Remember that the number of servers must be an integer.
So, if the answer from your “calculation is not an integer, you need to take the “next”
higher integer.
Part 2:
It is desired to design an Erlang C calculator for the M/M/c “delayed calls queuing model”
that we discussed in Discussion 1 session. The input parameters are the average arrival
rate () in packets/min, the average service rate () in packets/min. In addition, you are
given the following two “input constraints”:
a. The probability that an arriving Packet will find all servers busy (i.e. P (W > 0)) should
not exceed  where  is an input parameter.
b. Given that an arriving packet must wait, the average waiting time should not exceed
 minutes where  is an input parameter. Hint: This is conditional expectations.
The outputs of your calculator should be the number of servers required to satisfy the
above requirements/constraints. In addition, your calculator should enable me to find
the following averages: The average number of busy servers and the average number
of packets in the system (both waiting and being served)
Part 3 (Reference: Textbook of EE503 (Posted), section 12.5)
You have a cyclic system with a web server serving N clients (N is an input parameter).
Each client can be in two states. In the first state, the client is “preparing a request for
service”. In the second state, the client has generated the request that is either waiting in
the Queue or being served. Each source spends an exponentially distributed amount of
time  preparing each server request ( is an input parameter). The server serves one
client at a time with an exponentially distributed service rate of  requests/sec ( is an
input parameter). What percentage of the time is the web server busy? What is the
throughput of the system? What is the average time spent in the system for each
request? What proportion of the time that each client spends waiting for the
completion of his request?
Part 4:
Suppose we have a single server. Packets arrive according to a Poisson Process with an
average arriving rate of  packets/sec (an input Parameter). The capacity of the system is
K (an input parameter). The time required to serve each packet is an exponentially
distributed with mean service time of 1/ ( which is the average service rate is an input
parameter. If there are n packets in the system, the probability that an arriving packet will
“balk” (Balk means refuse to enter) is n/K for n = 0, 1, 2, 3, … K (So for example if n = 0,
the probability that an arriving packet will balk from entering the system is 0. If n = 1, the
probability that an arriving packet will balk from entering the system is 1/K and so on.).
It is desired to calculate the server utilization and the average number of Packets in the
system (Both output parameters)
Part 5:
Suppose we have a system with two servers, S1 and S2. S1 is faster than S2. Both servers
provide exponential service with average service rates of 1 and 2 respectively (Both
input parameters and of 1  2). Packets arrive according to a Poisson Process with an
average arrival rate of  (input parameter). Server S1 has infinite buffer. Server S2 has no
buffer. Define the state of the system as (n1, n2) where n1 is the number of packets in the
first system (Both waiting and being served) and n2 is the number of packets being served
by the second server for example (3, 1) means there are 1 packet being served by S1 and
two packets waiting for s1 and one packet being served by S2. Another example (0, 1)
means S1 is idle and one packet is being served by S2 and so on (Note that state (2, 0) is
NOT a possible state). Arriving packets that see the system is empty will join the faster
Server. It is desired to find (i.e. output parameters), the probability that is the system
is idle and the server utilization
Hint: In all above questions you need to sketch (on paper) the state rate transition
diagram like we did in the discussion session #1 and get the steady state probabilities
(PMF).

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