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matlabstatisticsmarkov-chainsmarkov
v4lerO
v4lerO

Reputation: 5

Transition probabilities in Continuous Time Markov Chain following Poisson Processes

Let's imagine we have the following states corresponding to the number of users in a system in a Continuous Time Markov Chain:

---- [10] ------ [11] ------ [12] ........... [17]

The arrival of a new users in the system follows a Poisson distribution at rate \lambda, and service time (users leave the system) at rate \mu. I want to compute the probability that staying at state [10] jumps into state [17]. That is, the probability that having 10 users in the system, 7 more users arrive in the system. It is possible to compute?

I tried the s-transform method that use first derivative but I'm unable to compute it.

lambda_ = 1

# Service rate (Poisson parameter)
mu = 1/60 

# Define the rate matrix Q
Q = np.array([[-lambda_, lambda_],
              [mu, -mu]])

# Compute the stationary distribution π
pi = np.linalg.solve(Q.T.dot(np.eye(2)), np.ones(2))
pi /= np.sum(pi)

# Initial state distribution
p0 = np.array([0.6, 0.4])  # Example initial state distribution

# Time points
t = 1.0  # Example time point

# Compute transient probabilities
pt = p0.dot(np.linalg.matrix_power(np.exp(Q * t), 1))

print("Transient probabilities at time t:", pt)

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