A computer router receives data packets that need to be transmitted to their destinations. The router operates in small time-slots that are such that on a given time-slot only two things can happen: either a packet is received or no packets are received. The probability that a packet is received during a given time-slot isp, independently of any other time-slots. Packets are stored in a waiting queue at the router until they are successfully transmitted to their destination; there is no limit to how many packets can be stored. On any given time-slot, the router will also attempt to send out one packet from the waiting queue to its destination. The probability that a packet is successfully transmitted during a time-slot isq, independently of any other attempts. A packet that arrives and finds no other packets in the waiting queue will wait until the next time-slot to be transmitted. LetX_ndenote the number of packets waiting to be transmitted at the beginning of then^thtime-slot. Note that the state space ofX_nisS= {0, 1, 2, 3,...}.

A.Computep_0,0= P(X_n+1= 0 | X_n= 0).

B.Computep_i,i= P(X_n+1= i | X_n= i)for i = 1, 2, 3,....

C.Indicate which of the following corresponds to the detailed balance equations for {X_n:n>=0}.

a.₀ =₀p_0,0 +₁p_1,0and _i =_i1p_i1,i+_ip_i,i +_i+1p_i+1,i , i=1,2,3,...

b._ip(1q) =_i+1(1p)q, i=0,1,2,3,...

c._i =j=0 _jp_j,i, i=0,1,2,3,...

d._iq(1p) =_i+1(1q)p, i=0,1,2,3,...

e.None of the above

D.Explain in detailwhy isp < qa necessary and sufficient condition for {X_n: n>=0}to be positive recurrent.

E.Supposep= 0.5andq= 0.8. Compute _ifori= 0, 1, 2,...., where= (₀, ₁, ₂,...)is the stationary distribution of {X_n: n >=0}.