1. Model under consideration: Loss system Collection of resources to which calls with holding time  (c) and class c arrive at random instances. An arriving call is either admitted into the system or is blocked and lost. If the call is admitted, it remains in the system for the duration of its holding time. The admittance decision is based on the call’s class and the system’s state. University of Twente - Stochastic Operations Research 27 blocked call C

2. Birth-death process Poisson arrivals rate  call length L mean  =1/  number of channels C State n = # calls in queue State space S = {0,1,…,C} Markov chain X = {X(t), t  0} birth rate q(n,n+1)=  death rate q(n,n-1)=n  Equilibrium distribution University of Twente - Stochastic Operations Research 26    n n+1 n  (n+1) 

3. Proof: (exponential case)    n n+1 n  (n+1)  equilibrium distribution solution global balance rate out of state n = rate into state n detailed balance Erlang loss probability University of Twente - Stochastic Operations Research 25

4. Exercise: University of Twente - Stochastic Operations Research 2 Consider the Erlang loss queue, but now assume that the call length L has an Erlang(2,2  ) distribution (consisting of 2 exponential phases with rate 2 . Construct the Markov chain recording the number of calls in phase 1 and phase 2, and compute the probability that an arriving call is blocked. Now consider L with Hyperexponential distribution, parameters and rates, such that Construct the Markov chain recording the number of calls in each phase, and compute the blocking probability for arriving calls. Compare the two results obtained above with the result for exponentially distributed call lengths with rate.

5. Blocking probabilities: general case Poisson arrivals rate  call length L mean  =1/  Erlang Hyperexponential Blocking probability University of Twente - Stochastic Operations Research 24

6. General distribution (vD) Erlang( k,  ) mean EL = k/  CV=1/  k  1 Hyperexponential mean CV  1 University of Twente - Stochastic Operations Research 23  

7. General distribution: phase type distribution With probability Erlang( k,  ) phase type distribution mean University of Twente - Stochastic Operations Research 22    

8. General distribution: phase type distribution With probability Erlang( k,  ) phase 1 phase 2 phase 1 phase type distribution dense in class of distributions with non-negative support University of Twente - Stochastic Operations Research 21    

9. General distribution: phase type distribution Markov chain that records the remaining number of phases and that restarts in phase k wp each time phase 1 is completed state k records number of remaining phases of renewal process state space S={1,2,…} transition rates q(k,k-1) =  q(1,k) =  Let H(k) denote equilibrium distribution, then H(k) satisfies global balance: H(k)  = H(1)  + H(k+1) , k=1,2,… or discrete renewal equation (TK VII-6) H(k) = H(1) + H(k+1), k=1,2,… solution where University of Twente - Stochastic Operations Research 20

10. General distribution: phase type distribution is distribution that satisfies discrete renewal equation H(k) = H(1) + H(k+1), k=1,2,… Proof insert H(k) into equation: show that H(k) is distribution: University of Twente - Stochastic Operations Research 19

11. Erlang loss queue: phase type call length Poisson arrivals rate  call length L mean  =1/  number of channels C State call i has remaining phases; State space Markov chain X = {X(t), t  0} Transition rates Equilibrium distribution H(k) is distribution of the remaining number of phases = remaining call length University of Twente - Stochastic Operations Research 18

12. Erlang loss queue: phase type call length Equilibrium distribution Proof global balance University of Twente - Stochastic Operations Research 17 H(1)=   and use discrete renewal equation

13. Erlang loss queue: phase type call length Theorem 1 Equilibrium distribution where moreover, equilibrium distribution of number of calls depends on call length distribution only through its mean (insensitivity property): Proof sum distribution over all possible configurations of phases University of Twente - Stochastic Operations Research 16