• Question 1
We can regard the queue length of an M/M/∞ queue as a Birth-and-Death process, e.g., the customer is arriving according Poisson Process with parameter λ, and the service rate of each server is μ with exponentially distributed service time. Answer the following questions,
1. Build a Birth-and-Death model of this kind of queue model and draw a state transition rate diagram.
2. Derive the steady state probability, if they are existing.
3. What is the utilization and throughput of this queueing model.
4. What is average queue length E[X], Average System time E[S].
• Question 2
The M/M/m/m queue is an extension of the M/M/1/K queue (see the Figure 1), which can be also modeled by using Birth and Death process.
Figure 1: The M/M/m/m queueing model
1. Build the Birth-and-Death model for this queue and draw a state transition rate diagram.
2. Compute the steady state probability.
3. Recall the definition of the Blocking Probability PB for the M/M/1/K queue
and compute PB for this model.
• Question 3
A manufacturing workstation receives a continuous supply of parts and processes them at a rate of 12 parts per hour with exponentially distributed processing times.
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Figure 2: The 5-nodes queueing network
However, each part needs to be placed on a special fixture before it can be processed. There are N such fixtures available, and the time it takes to remove a processed part from its fixture and load a new part on it is exponentially distributed with mean 15 min. Determine the smallest number of fixtures such that the average waiting time of a part at the workstation queue is less than 2 min. Does this number of fixtures satisfy the additional requirement that the utilization of the workstation be at least 80%?
• Question 4
A packet-switched communication network is viewed as an open queueing network with each link modeled as an M/M/1 queueing system. Packets arrive at various nodes as independent Poisson processes, and the length of every packet is assumed to be exponentially distributed. Let λij be the rate of packets to be transmitted over link (i,j), and let μij be the transmission rate (packets/s) of link (i,j).
1. Show that the average delay of packets (measured from arrival at some node to departure at some other node) is given by
E[S] = 1 ∑ λij
Λ ∀ (i,j) μij − λij
,
where Λ is the total arrival rate(packets/second) into the network.
2. Consider the 5-node 7-link network shown in Fig 2, where the transmission rate of all links is μ = 3 packets/s. Packets arrive at nodes 1 and 2 with rates r1 = 1 packet/s and r2 = 3 packets/s respectively, and have to be transmitted to nodes 3,4,5. The following routing probabilities are used: p12 = p14 = 1/2, p23= 3/5, p24 = p25 = 1/5, p35 = p45 = 1/2 . Find the average delay of packets in the network, and all the link utilizations.
3. Suppose that all packets arriving at node 1 are destined for node 5. Find the average delay of these packets.
• Question 5 Simulation:
1. Writeaprogram(matlaborpython)tosimulatetheM/M/1,M/M/m,M/M/1/K
QS. Verify the Little Law, verify the expression of the π0, the average queue 2
length and utilization. (λ = 2, μ = 3(all servers have the identical service rates), K = 5, m = 3)
2. (Bonus Question) Write a program to verify the results you got in Question 4.
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