Tamtam Proceedings - lamsin

402 El Kamili et al.of a capacity configuration c to that of some configuration ĉ. "Hat" values will refer toconfiguration ĉ, while "nohat" values refer to c. For example, ˆT l and T l are the equilibriumdelay of link l under ĉ and c, respectively. Now, we introduce some performancemeasures , by which the designer can evaluate the efficiency of a capacity configuration.We shall concentrate on measures that are expressed by means of the user prices.Definition 4.1 Consider two capacity configurations c and ĉ, and let η i and ˆη i be theprice of user i at the respective equilibrium. Then, configuration ĉ is said to be user priceefficient relative to configuration c, if ˆη i ≤ η i , for all i ∈ I.Let us start by proving some technical results that will play a key role in the proof ofthe fact that the demands increase when some capacity is added.Lemma 4.1 We have the following relations:(i) { ˆT l ≤ T l ; ˆλ i ≥ λ i ; ˆr i < r i } ⇒ ˆf i l ≥ f i l(ii) { ˆT l > T l ; ˆλ i ≤ λ i ; ˆr i ≥ r i } ⇒ ˆf i l ≤ f i l(iii) { ˆT l ≤ T l ; ˆλ i > λ i ; ˆr i ≤ r i } ⇒ ˆf i l ≥ f i lThe following Theorem is fundamental in the study of the capacity addition problemwhen user demands are variables, it shows that user demands increase in the case of anaddition of the capacity.Theorem 4.1 If a capacity configuration ĉ is an augmentation of configuration c, thenthe demand of each user i is higher under configuration ĉ, i.e., ˆr i ≥ r i , ∀i ∈ I.Lemma 4.2 If a capacity configuration ĉ is an augmentation of configuration c, then theequilibrium delay of each link l is lower under configuration ĉ, i.e., ˆT l ≤ T l for all l ∈ L.The following proposition shows that an addition of capacity is always price efficient.Proposition 4.1 (PRICE EFFICIENCY )If a capacity configuration ĉ is an augmentation of configuration c, then ĉ is user priceefficient relative to c , i.e., ˆη i ≤ η i for all i ∈ I.5. References[ElA2004] R. EL AZOUZI, M. EL KAMILI, E. ALTMAN, M. ABBAD AND T. BAŞAR, “CombinedCompetitive Flow Control and Routing in Multi-User Communication Networks withHard Side-Constraints.”, 2004 Chapter of book Analysis Control and Optimization of ComplexDynamic Systems, Editors E. K. Boukas and R. P. Malhamé[Alt2002] E. ALTMAN, R. EL AZOUZI AND V. VYACHESLAV, “Non-cooperative routing in lossnetworks. ”, Performance Evaluation, Vol 49, Issue 1-4, pp. 257-272, 2002.TAMTAM –Tunis– 2005