A Simplified Blocking Probability Calculation in the Retry Loss Models for Finite Sources

1. A Simplified Blocking Probability Calculation in the Retry Loss Models for Finite Sources I. Moscholios M. Logothetis G. Kokkinakis Wire Communications Laboratory, Department of Electrical & Computer Engineering, University of Patras, 265 00 Patras, Greece. E-mails: moscholios, m-logo, gkokkin@wcl.ee.upatras.gr

2. STRUCTURE • Review: • The Erlang Multirate Loss Model (EMLM) for Finite Sources (Engset Multirate Loss Model-EnMLM) • The Analytical Model - Equivalent System • The Glabowski – Stasiak Simplified Method for the Blocking Probability Calculation based on the Kaufman – Roberts formula • The Single and Multi-Retry Models for Finite Sources • The Analytical Models – Equivalent System • A Simplified Method for the Blocking Probability Calculation in the Retry Models • Numerical Results – Evaluation • Comparison between the two methods based on simulation results. • Conclusion

3. Review: The Engset Multirate Loss Model (EnMLM) Background • Link of capacity,C. • K service classes. • Nk number of sources of service class k • nk number of in-service sources of service-class k • λkarrival rate: λk= (Nk – nk) vk (vk mean arrival rate per idle source) • μk service rate, or hk=μk –1 holding time (exponentially distributed) • bk bandwidth requirements

4. Review: The EnMLM - The Analytical Model Link Occupancy Distribution - State Probabilities Stamatelos-Hayes, 1994 Kaufman-Roberts, 1981 αk = vk hk offered traffic load per idle source Blocking Probability Bk of service-class k where Example: C = 5 bandwidth units (b.u.) K = 3, N1=N2=N3= 6 sources b1= 3 b.u., b2= 2 b.u., b3= 1 b.u. α1=α2=α3= 0.01 erl (per idle source) • The problem • Calculating G(j)’s

5. n1 n2 n3 j B1 B2 B3 jeq 0 0 0 0 0 0 0 1 1 5 0 0 2 2 10 0 0 3 3 * 15 0 0 4 4 * * 20 0 0 5 5 * * * 25 0 1 0 2 12 0 1 1 3 * 17 0 1 2 4 * * 22 0 1 3 5 * * * 27 0 2 0 4 * * 24 0 2 1 5 * * * 29 1 0 0 3 * 16 1 0 1 4 * * 21 1 0 2 5 * * * 26 1 1 0 5 * * * 28 State Space Blocking States Review: EnMLM – Equivalent System It has been proved (Stamatelos – Hayes, 1994): Two stochastic systems with the same state space and the same parameters K, Nk, αk are equivalent – they give the same CBP By choosing b1=16, b2=12, b3=5, C=29 an equivalent system results with unique link occupancy per state, jeq B1=6.08%,B2=0.75%, B3=0.32%

6. Review: The Glabowski – Stasiak Simplified Method for the Blocking Probability Calculation based on the Kaufman – Roberts formula Kaufman-Roberts, 1981 Average number of service-class k calls in state j Glabowski-Stasiak, 2004

7. Review: The Single-Retry Model for finite sources (f-SRM) • Non Product Form Solution • Assumptions: • Local Balance between adjacent states • Migration Approximation: The occupied link bandwidth from the retry calls is negligible when the link occupancy is at most equal to the retry boundary State Probabilities – Link Occupancy Distribution (Stamatelos-Koukoulidis, 1997) EnMLM Migration Approximation the retry calls are negligible when j  C-bk+bkr i.e. γk(j) = 0 when j  C-bk+bkr

8. Review: The Multi-Retry Model for finite sources (f-MRM) State Probabilities – Link Occupancy Distribution (Moscholios, Logothetis, Nikolaropoulos 2005) EnMLM When Nk approaches infinity for k =1,…,K : State Probabilities – Link Occupancy Distribution (Kaufman 1992) Kaufman/Roberts formula

9. n1 n2 n1r j B1 B2 B1r jeq 0 0 0 0 0 0 1 0 2 2000 0 2 0 4 * * 4000 0 2 1 5 * * * 5001 1 0 0 3 * 3000 1 0 1 4 * * 4001 1 0 2 5 * * * 5002 1 1 0 5 * * * 5000 Review: Determination of the State Space – Equivalent System in the f-SRM Example: C = 5 K = 2 service-classes N1=N2= 6 sources a1=a2= 0.01 erl b1= 3, b2= 2, b1r= 1, a1r = 0.03 erl State Space Blocking States (C:b1:b2:b1r)new ~(C:b1:b2:b1r)init.e.g. 5002:3000:2000:1001~5:3:2:1 B1r=0.33%,B2=0.66%

10. The proposed Simplified Method for the Blocking Probability Calculation based on the Glabowski-Stasiak Method

11. Numerical Results – Evaluation Example: C = 50, K = 2, N1=N2= 12 α1= 0.06 erl, α2= 0.2 erl b1= 10, b2= 7, b1r1= 8, b1r2=6, α1r1=0.075 erl, α1r2=0.1 erl b2r1=4, α2r1 = 0.35 erl Equivalent system (30 states): Ceq=50006, b1eq=10000, b2eq=7001, b1r1eq=8000, b1r2eq=6000,b2r1eq=4000. The corresponding infinite system used for the y( )’s and CBP calculation is: C=50 b.u., b1=10 b.u., α1=0.72 erl, b1r1=8 b.u., α1r1=0.9 erl, b1r2=6 b.u., α1r2=1.2 erl and b2=7 b.u., α2=2.4 erl, b2r1=4 b.u., α2r1=4.2 erl.

12. Numerical Results – Evaluation (cont.) At each point (P) the values of α1, α1r1, α1r2 are constant while those of α2, α2r1 are increased by 0.4/12 and 0.7/12 respectively, (for P=1: α2=0.2 and α2r1=0.35 erl).

13. Conclusion - Summary • Review: • The Engset Multirate Loss Model-EnMLM • The Analytical Model - Equivalent System • The Glabowski – Stasiak Simplified Method for the Blocking Probability Calculation based on the Kaufman – Roberts formula • The Single and Multi-Retry Models for Finite Sources • The Analytical Models – Equivalent System • A Simplified Method for the Blocking Probability Calculation in the Retry Models • Numerical Results – Evaluation • The comparison between the two methods shows the superiority (in terms of simplicity) of the Glabowski – Stasiak method.