Discrete-time stochastic processes

The map rule is to choose h1 or h0 depending on

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Unformatted text preview: ially distributed service times with rate µ2 . Each departure from system 1 independently 1 − Q1 . System 2 has an additional Poisson input of rate ∏2 , independent of inputs and outputs from the first system. Each departure from the second system independently leaves the combined system with probability Q2 and re-enters system 2 with probability 1 − Q2 . For parts a, b, c assume that Q2 = 1 (i.e., there is no feedback). ∏1 ✲ System 1 µ1 Q1 ❄Q 1− 1 ∏2 ✲ ✲ ✲ System 2 µ2 Q2 ✲ 1−Q2 (part d) a) Characterize the process of departures from system 1 that enter system 2 and characterize the overall process of arrivals to system 2. b) Assuming FCFS service in each system, find the steady state distribution of time that a customer spends in each system. 6.9. EXERCISES 273 c) For a customer that goes through both systems, show why the time in each system is independent of that in the other; find the distribution of the combined system time for such a customer. d) Now assume that Q2 < 1. Is the departure process from the combined system Poisson? Which of the three input processes to system 2 are Poisson? Which of the input processes are independent? Explain your reasoning, but do not attempt formal proofs. Exercise 6.16. Suppose a Markov chain with transition probabilities {Pij } is reversible. 0 Suppose we change the transition probabilities out of state 0 from {P0j ; j ≥ 0} to {P0j ; j ≥ 0}. Assuming that all Pij for i 6= 0 are unchanged, what is the most general way in which 0 we can choose {P0j ; j ≥ 0} so as to maintain reversibility? Your answer should be explicit about how the steady state probabilities {πi ; i ≥ 0} are changed. Your answer should also indicate what this problem has to do with uniformization of reversible Markov processes, if anything. Hint: Given Pij for all i, j , a single additional parameter will suffice to specify 0 P0j for all j . Exercise 6.17. Consider the closed queueing network in the figure below. There are three customers who are doomed forever to cycle between node 1 and node 2. Both nodes use FCFS service and have exponentially distributed IID service times. The service times at one node are also independent of those at the other node and are independent of the customer being served. The server at node i has mean service time 1/µi , i = 1, 2. Assume to be specific that µ2 < µ1 . Node 1 µ1 ✲ Node 2 µ2 ✛ a) The system can be represented by a four state Markov process. Draw its graphical representation and label it with the individual states and the transition rates between them. b) Find the steady state probability of each state. c) Find the time-average rate at which customers leave node 1. d) Find the time-average rate at which a given customer cycles through the system. e) Is the Markov process reversible? Suppose that the backward Markov process is interpreted as a closed queueing network. What does a departure from node 1 in the forward process correspond to in the backward process? Can the transitions of a single customer in the forward process be associated with transitions of a single customer in the backward process? 274 CHAPTER 6. MARKOV PROCESSES WITH COUNTABLE STATE SPACES Exercise 6.18. Consider an M/G/1 queueing system with last come first serve (LCFS) pre-emptive resume service. That is, customers arrive according to a Poisson process of rate ∏. A newly arriving customer interrupts the customer in service and enters service itself. When a customer is finished, it leaves the system and the customer that had been interrupted by the departing customer resumes service from where it had left off. For example, if customer 1 arrives at time 0 and requires 2 units of service, and customer 2 arrives at time 1 and requires 1 unit of service, then customer 1 is served from time 0 to 1; customer 2 is served from time 1 to 2 and leaves the system, and then customer 1 completes service from time 2 to 3. Let Xi be the service time required by the ith customer; the Xi are IID random variables with expected value E [X ]; they are independent of customer arrival times. Assume ∏ E [X ] < 1. a) Find the mean time between busy periods (i.e., the time until a new arrival occurs after the system becomes empty). b) Find the time-average fraction of time that the system is busy. c) Find the mean duration, E [B ], of a busy period. Hint: use a) and b). d) Explain briefly why the customer that starts a busy period remains in the system for the entire busy period; use this to find the expected system time of a customer given that that customer arrives when the system is empty. e) Is there any statistical dependence between the system time of a given customer (i.e., the time from the customer’s arrival until departure) and the number of customers in the system when the given customer arrives? f ) Show that a customer’s expected system time is equal to E [B ]. Hint: Look carefully at your answers to d) and e). g) Let C be the expected system tim...
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