Consider a tram at a tourist attraction. The tram is on a fixed loop stopping at
ID: 3134836 • Letter: C
Question
Consider a tram at a tourist attraction. The tram is on a fixed loop stopping at a number of sites. At each site passengers queue up to board the tram and passengers on the tram may elect to get off. Assume the capacity of the tram is 5. The probability that any passenger on the tram will get off at any site is 0.8. The number of passengers waiting to board at a site has the following probability mass function. P[0 customers waiting to board ] = 0.4 P[ 1customer waiting to board] = 0.4 P[2 customers waiting to board] = 0.2.Each site is assumed to be mathematically equivalent. Assume at each stop any passengers who wish to get off will do so before anyone waiting to board. Also, anyone who does not find a scat will simply walk to their desired destination. Formulate the problem as a Markov process Assume the state of the system will be the number of passengers on the bus (Just after boarding). Compute the one-step transition probability matrix P Assume at the beginning of the day, the tram is empty. Compute the probability of the tram being full after making each of the following stops: 1,2, 3, and 4Explanation / Answer
A process is said to be markov process, if Xt+1 depends only on Xt, and not on X0, X1 ,...,Xt-1.
In Markov process the future depends only on the present, not on the past.
A Markov Process consists of two things:
1. Finite number of states: the number of passengers in the tram is finite
2. Transition probabilites for moving between these states
Given : state of system = number of passengers.
Hence, S = {1,2,3,4,5}
Thus in mathematical notation, We can present the markov property as:
P(Xt+1=s/Xt = st, Xt-1 = st-1,...,X0=s0) = P(Xt+1=s/Xt=st)
Where t=number of passengers