I need assignment help in queuing theory
4.1 Assuming one-step behavior, flow balance, homogeneous arrivals, and homogeneous service at a single-server station i, which is a part of a network with population N, show that the mean queue length can be expressed as Use equations and What simplifications can you make to this equation as N → ∞, assuming that station i is not a bottleneck? 4.2 Show that the proof of Little’s formula holds for last-come first-served and processor sharing disciplines by verifying that area still represents the total backlog 4.3. Consider a station where the arrivals occur in packets of size 1 or 2, but the services occur singly. The packet size is randomly chosen, and the probability of it being of size 1 is q. Both arrivals and services are homogeneous and the system is flow-balanced. The interarrival and service times are constant and given by a and b respectively. Show by a counterexample that the arriver’s distribution is different from departure’s distribution, which, in turn, is different from a random observer’s distribution. 4.13. Suppose that we wish to determine the service rate function µi(n) for station i in a closed model with population N. Suppose that the station is load-independent except for the overhead, which increases slowly with load. So, we assume that the station capacity decreases linearly with n with rate δ. The system reports utilization, denoted U ∗ i (N), as the fraction of time the station is busy. Starting with the throughput theorem, show that where S, U ∗ i (N) and Qi(N) are the measured values of service time, utilization and queue length respectively. That is, S = U ∗ i (N)/λ(N), U ∗ i (N) = 1 − P(0|N), and Q(N) = PN n=1 nP(n|N). Devise a procedure for estimating δ. 5.6. Draw the state transition diagram and classify all the states of a discrete time Markov chain with the following transition probability matrix Suppose that the process is in state 1 initially. Show by explicit calculations of limiting and stationary distributions that they are identical. 5.7. Consider a M/M/c/SI/24/∞ queuing system that represents the behavior of service windows in a bank. There are a total of six windows, but not all of them are always manned. All customers form a single queue. The mean interarrival time is one minute, and the mean service time per customer is four minutes. Let n denote the number of customers in the system. If 0 ≤ n ≤ 4, only one window provides service. Whenever n > 4, another window is opened up, and whenever n ≤ 4, the extra window is closed down. Thus, for 5 ≤ n ≤ 8, two windows are operational. The same happens for higher values of n; that is, for 9 ≤ n ≤ 12, three windows will be open, and so on. Solve this system and determine the average queue length and the average number of open windows. 5.11. Consider a M/E2/1/FCFS/3/3 queuing system with mean arrival rate of 2/sec and the mean service rate of 4/sec. After serving one customer, the server takes a vacation with exponentially distributed time if and only if there are no waiting customers. The mean vacation time is 0.5 seconds. (a) Choose a suitable definition of system “state” and draw the complete state transition diagram. (b) Write down global balance equations considering boundary in a way that yields the simplest set of equations. (c) Solve the equations and obtain expressions for the probability of finding n customers in the system. (d) Using the throughput law, compute the system throughput. Also compute the average queue length and average response time of the station. What would be an appropriate definition of utilization in this problem?