Modeling and Performance Analysis of Telephony Gateway REgistration Protocol

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5 .7.6 TABLE II PARAMETERS FOR DIFFERENT SCENARIOS Parameter equal Value for skewed T T update interval 3 sec 3 sec simulation model logical gateway Fig. 5. Blocking Probability vs. Load using MUBS Algorithm (Gateways having Equal Capacity) If a set M of m gateways have the same utilization i/t and i/t < i k /T k for all k G and k / M, then the incoming calls are split equally between the m gateways. λ i /m if j M λ j = otherwise If k out of the m gateways have actual number of ongoing calls equal to the total capacity, then the reward assigned to that state is k/m and reward is assigned to the other states. 1) Experimental Results: We have run experiments using the CTMC model given in Figure 4 (using PRISM) to solve the steady state probabilities with two gateways. The parameters used for these scenarios are summarized in Table II. Gateway capacities used are small for quick computation of blocking probability using the Markov chain. We also have developed a TGREP simulator and implemented the gateway selection algorithms in it. We provide the results based on the model as well as using the simulator. All simulation results are presented with a 95% confidence interval of less than.1. Finally, we also run the same experiment on logical gateway, which has capacity equal to the sum of capacities of the two individual gateways. Two scenarios are considered here for the experimental results: equal with the two gateways having equal capacities. skewed with one of the gateways having much higher capacity than the other. As is clear from Figures 5 and 6, the blocking probability when using the CTMC model closely follows the blocking probability obtained from the simulator as the load is varied on the system. This validates that our CTMC model is correct simulation model logical gateway Fig. 6. Blocking Probability vs. Load using MUBS Algorithm (Gateways having Skewed Capacity) Also, as expected, the performance of the system with one logical gateway is always better in terms of blocking probability in both the scenarios. B. Probabilistic Selection Based on Utilization (PSBU) A problem with minimum utilization based selection is that between two updates, it will always pick the same gateway. Thus, if the update rate is low, that particular gateway can quickly reach its capacity and then it will start dropping calls. This will adversely affect the performance of the system. Hence, a better way to select a gateway is to assign probabilities to each gateway based on their current state in terms of capacity and select gateways based on those probabilities. There are two ways that the probability can be assigned. The first method is to have the probability of choosing a gateway be inversely proportional to the utilization of the gateway. In this case, λ 1 = λ i(i 2 /T 2 ) i 1 /T 1 + i 2 /T 2 λ 2 = λ i(i 1 /T 1 ) i 1 /T 1 + i 2 /T 2 The second method is to have the probability directly proportional to the remaining capacities of the gateways, i.e., λ 1 = λ i(1 i 1 /T 1 ) 2 i 1 /T 1 i 2 /T 2 λ 2 = λ i(1 i 2 /T 2 ) 2 i 1 /T 1 i 2 /T 2 Evaluation of the two methods using the CTMC model and simulation suggests that the second method results in more balanced load and lower blocking probability. As an example, consider a case where two gateways have utilizations.7 and.9 respectively. The values of λ 1 and λ 2 as calculated the two methods are given in Table III. The first method results in almost the same rate for both gateways, leading to unbalanced load. The second method assigns much lower probability for the more utilized gateway, which leads to lower system blocking probability. Hence, we use the second method for the probabilistic gateway selection algorithm and disregard the first method. We finesse it a little bit to handle the case when the denominator

6 TABLE III AN EXAMPLE SHOWING THE TWO METHODS OF THE PROBABILISTIC SELECTION ALGORITHM Gateway 1 Gateway 2 utilizations.7.9 First method (probabilities in inverse proportion of utilization) Second method (probabilities in direct proportion of remaining capacity).75 5 becomes zero, which happens if both gateways are fully utilized. With this change, the rates λ 1 and λ 2 are given by λi 2 if i 1 = T 1 and i 2 = T 2 λ 1 = otherwise and r = λ 2 = λ i(1 i 1/T 1) 2 i 1/T 1 i 2/T 2 λi 2 if i 1 = T 1 and i 2 = T 2 otherwise λ i(1 i 2/T 2) 2 i 1/T 1 i 2/T 2 Reward assignment for the states is 1 if a 1 = T 1 and a 2 = T 2 1 i 1/T 1 2 i 1/T 1 i 2/T 2 if a 1 = T 1 and (i 1 T 1 or i 2 T 2 ) 1 i 2/T 2 2 i 1/T 1 i 2/T 2 if a 2 = T 2 and (i 1 T 1 or i 2 T 2 ) if i 1 = T 1 and i 2 = T 2 and a 1 = T 1 if i 1 = T 1 and i 2 = T 2 and a 2 = T 2 otherwise The above analysis can be generalized to n gateways. Then the call rate is given by λ i /n ( ) if i 1 /T 1 = i 2 /T 2 = = i n /T n = 1 λ j = λ i 1 i j/t j n P n k=1 i k/t k otherwise The reward assignment for a state in which i 1 /T 1 = i 2 /T 2 = = i n /T n = 1 is k/n where k is the number of gateways for which a = T. For other states, the rewards are assigned as follows. Let B, B φ, be the set of k gateways for which actual number of ongoing calls is equal to the total capacity. Then the reward assigned is k g B i g/t g n n k=1 i k/t k When none of the gateway has reached its capacity in terms of the actual number of ongoing calls (i.e. when B = φ), the reward assigned is. 1) Experimental Results: We again ran experiments using the CTMC model and also using our simulator. The blocking probabilities vs. load for this algorithm are shown in Figures 7 and 8 using the parameters in Table II. Here again we see that results using our simulator very closely follow the CTMC model, again validating our model. Comparing these simulation model logical gateway Fig. 7. Blocking Probability vs. Load using PSBU Algorithm (Two Gateways with Equal Capacity) simulation model logical gateway Fig. 8. Blocking Probability vs. Load using PSBU Algorithm (Two Gateways with Skewed Capacity) graphs with those of minimum utilization based algorithm (i.e., comparing Figure 7 with Figure 5 and Figure 8 with Figure 6), it can easily be seen that the PSBU performs better than the MUBS algorithm in terms of blocking probability. We ran some experiments with different number of gateways keeping the total capacity of the system constant. Figures 9 and 1 show the effect of increasing the number of gateways on blocking probability (when the system capacity is constant). The parameters for these simulation runs are summarized in Table IV. The difference in blocking probabilities between the two algorithms is much higher here because the gateway capacities are higher. Thus, relative performance of MUBS becomes worse than PSBU as the total capacity of the system increases. Note that the two algorithms have been plotted to different scales. This is because the blocking probability attained by the MUBS algorithm is an order of magnitude worse than that attained by the PSBU algorithm. V. EFFECTS OF UPDATE RATE Another parameter that we have not discussed so far is the update rate. Intuitively, decreasing update rate results in the information at the LS getting increasingly stale, which should result in increase of blocking probability. The problem that a gateway provider might face here is how to choose the proper update rate to be able to provide the appropriate quality of service in terms of blocking probability.

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