Problem 2.1

1. Problem 2.1 Consider K independent sources of customers where the interarrival time between customers for each source is exponentially distributed with parameter lk (i.e., each source is a Poisson process). Now consider the arrival stream, which is formed by merging the input from each of the K sources defined above. Prove that this merged stream is also Poisson with parameter l= l1+ l2+…+ lk.

2. Solution Let Xi be the Poisson counting process for the ith source and let X=X1+…+XK. The z-transform for X is which is the z-transform for a Poisson process with parameter l1+…+lk. (since the Xi are independent)

3. 1-p 1 2 1 a p 3 1-a Problem 2.5 Consider the homogeneous Markov chain whose state diagram is

4. Problem 2.5 (cont.) • Find P, the probability transition matrix. • Under what conditions (if any) will the chain be irreducible and aperiodic? • Solve for the equilibrium probability vector p. • What is the mean recurrence time for state E2? • For which values of a and p will we have p1= p2= p3? (Give a physical interpretation of this case.)

5. Irreducible and periodic for all 0<p≤1 and 0<a≤1 except a=p=1.

6. From p=pP [p=(p1,p2,p3)] we obtain only two independent equations, namely p1=p2 p2=(1-p)p1+ap3 Using the conservation of probability, we also have p1+p2+p3=1. Thus

7. We need a=p Interpretation: Since each visit to E1 is followed by exactly one visit to E2 and vice versa for all p, a we have p1=p2 always. Also p (=a) of the time we go from E1 to E3 and the average number of steps (or mean time) spent in E3 per visit is . Thus is the average number of visits to E3 per visit to E1.

8. Problem 2.11 Consider a birth-death process with coefficients which correspond to an M/M/1 queueing system where there is no room for waiting customers. • Give the differential-difference equations for Pk(t) (k=0,1). • Solve these equations and express the answers in terms of P0(0) and P1(0).

9. Solution

10. Since P0(t)+P1(t)=1, we may rewrite the first equation from part (a) as As in section I.4, we recognize that the homogeneous solution must be of the form Ae-(l+m)t; we find that the particular solution is a constant (say B) and so P0(t)=B+Ae-(l+m)t.

11. Substituting into our differential equation, we have . Evaluating the solution at t=0, we have . Hence P1(t) may be found from P1(t)=1-P0(t) or by symmetry of the defining equations from part (a) to yield

12. Problem 3.2 Consider a Markovian queueing system in which • Find the equilibrium probability pk of having k customers in the system. Express your answer in terms of p0. • Give an expression for p0.

13. Solution

14. So Note for 0≤a<1, this system is always stable.

15. Problem 3.5 Consider a birth-death system with the following birth and death coefficients: • Solve for pk. Be sure to express your answers explicitly in terms of l, k, and m only. • Find the average number of customers in the system.

16. Solution Here we demonstrate the “differentiation trick” for summing series (similar to that on page 69).

17. Since we have

18. Thus and so

19. (b)