Transition probabilities of markov chains

Reference no: EM131005829

Suppose that the times between successive arrivals of customers at a single-server station are independent random variables having a common distribution F. Suppose that when a customer arrives, he or she either immediately enters service if the server is free or else joins the end of the waiting line if the server is busy with another customer. When the server completes work on a customer, that customer leaves the system and the next waiting customer, if there are any, enters service. Let Xn denote the number of customers in the system immediately before the nth arrival, and let Yn denote the number of customers that remain in the system when the nth customer departs. The successive service times of customers are independent random variables (which are also independent of the interarrival times) having a common distribution G.

(a) If F is the exponential distribution with rate λ, which, if any, of the processes {Xn}, {Yn} is a Markov chain?

(b) If G is the exponential distribution with rate μ, which, if any, of the processes {Xn}, {Yn} is a Markov chain?

Reference no: EM131005829

Questions Cloud

What is the composition of the two phases and the pressure : The laboratory division of your company has sent you the following data for the system n-pentane (1) propionaldehyde (2) at 35 °C: The activity coefficients of the two components at infinite dilution are 4.0 and the saturation pressures of the pur..

Independent of the arrival process : The service times of the arrivals are independent exponential random variables with rate μ, and are independent of the arrival process. Customers depart the system when their service ends. Let N be the number of arrivals before the ?rst departure.

Verify the given velocity vector : Relative to polar coordinates, the position vector of an object moving in the plane can be written as r (t) = r(t)er. Verify that the velocity vector is v(t) = (dr/dt) er + r (dθ/dt)] eθ.

What is bubble pressure of the mixture : Use the data below for the system benzene(1)/acetonitrile(2) at 45 °C to answer the following question: What is the saturation pressure of the pure components at 45 °C?

Transition probabilities of markov chains : (a) If F is the exponential distribution with rate λ, which, if any, of the processes {Xn}, {Yn} is a Markov chain? (b) If G is the exponential distribution with rate μ, which, if any, of the processes {Xn}, {Yn} is a Markov chain? (c) Give the trans..

Find total expected return : (a) Find your total expected return. (b) Find the value of t that maximizes the total expected return.

Calculate the conditional distribution : Find the conditional distribution of T given that N = n. Calculate the conditional distribution of N, given that T = t, and notice that it is distributed as 1 plus a Poisson random variable with mean λ(1 - p)t.

Determining the poisson process with rate : Policyholders of a certain insurance company have accidents at times distributed according to a Poisson process with rate λ. The amount of time from when the accident occurs until a claim is made has distribution G.

Number of customers in the system : 1. Consider an in?nite server queuing system in which customers arrive in accordance with a Poisson process with rate λ, and where the service distribution is exponential with rate μ. Let X(t) denote the number of customers in the system at time t. F..

User Account

All Pages