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8 GREENBERG et al.: RESOURCE SHARING FOR BA & IR CALLS 17 TABLE III PERFORMANCE OF CHTA FOR s = 100, b =10, I =60, B =2, I = B =1 Fig. 4. Plots of the interrupt probability p I versus blocking probability P I for various values of the upper limit parameter r in Example 3. The parameters are s = 100, b =10, I =60, B =2, and B = I =1. Fig. 3. Plots of the interrupt probability p I versus blocking probability P I in Example 2. The parameters are s =40, b =5, I =24, I =1, B =16, B =4. TABLE IV P B AS A FUNCTION OF r WITH s = 100, b =10, I =60, B =2, I = B =1 probabilities. In particular, the parameters are,,,,, and. Plots of versus are shown in Fig. 3. Each curve is based on ten different values of. Again, IPA has the best performance, but in this case the FTA curve is quite close to the IPA curve. Example 3: In this example, we show how the upper limit of against BA calls can be effectively used to improve the rate of revenue. Consider the same set of system parameters as in Example 1, except for a new control variable. The values of as a function of are shown in Table IV. Plots of versus for various values of using the IPA policy are shown in Fig. 4. As in Example 1, the curve is based on ten different values for, and for each value of, the simulation run length was after deleting 25 time units to get rid of transients. As expected for larger values of the reservation parameter, both the blocking and interrupt performance of IR calls are better, at the expense of increased BA call blocking. We also plot versus the average revenue in Fig. 5 for several values of, assuming that we use (2.2) with and. The best results,, are achieved first by and then. In contrast, the best possible revenue with is 76.0 and with link partitioning is 74.3 (with capacity 30 dedicated to BA). Since, the revenue in this example corresponds to the carried load. Since the offered load is 80, the lost revenue has been reduced from 5.7 with link partitioning to 3.7 with resource sharing using, a decrease of 35%. However, the effect of alone on revenue is not great; the upper limit is most useful for making desired tradeoffs between IR and BA performance. We also point out that, from Fig. 4, the blocking probability for IR calls is less than when and the blocking probability for BA calls is when. Comparing this to Table II, we see that our admission control algorithm can achieve high link utilization while keeping and reasonable whereas under link partitioning, to achieve maximum revenue, becomes 0.211, which is very high. From a design point of view, one more relationship has to be specified to run the network at a given operating point on the versus curve. This is the relationship between interrupt probability threshold and the realized interrupt probability. For the optimal reservation parameter, this relationship is shown in Fig. 6. In several examples we have found the relation between and to be very nearly linear. Moreover, tends to be about two orders of magnitude smaller than (by a factor of about ). Example 4: As mentioned in Section I, a natural application for a BA service is one where most BA calls are conference calls. Thus, it is natural to consider an example where BA calls tend to have much longer holding times than regular voice calls. Let us reconsider Example 3 with and. We keep the traffic intensity the same but have increased the holding times of BA calls ten times. We have plotted the revenue as function of for various values of the reservation parameter in Fig. 7.

9 18 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 7, NO. 1, FEBRUARY 1999 Fig. 5. Plots of interrupt probability p I versus revenue R in Example 3. The parameters are s = 100, b =10, I =60, B =2, I = B =1, and various values for r. Fig. 6. P T versus p I for Example 3 with r =50. Comparing this to Fig. 5, we see that revenue increases when BA calls have longer holding times. Thus, the potential gain from the IPA algorithms as compared to strict partitioning will be more when the BA holding times are longer. Further, since BA arrivals and departures are less frequent when the arrival and departure rates decrease simultaneously, we expect the computational complexity of IPA to decrease due to the fact that IR calls will see fewer peaks in the BA profile. Since the average BA holding time is ten times the average holding time for IR calls, it is reasonable to expect the near decomposability property explained in Section IV to hold. A comparison of the revenue in Fig. 7 to the results in the nearly decomposable case presented in Table I shows that the simulation results are close to those predicted by the Markov chain computations. As in Table I, the revenue is relatively Fig. 7. Example 4: p I versus R for I = B =10. insensitive to since the traffic intensity of BA calls is small relative to the capacity of the link. In all our examples, the revenue computed assuming a nearly decomposable structure was an upper bound on the revenue achievable through any of our admission control schemes for the actual model. We conjecture that this property holds more generally, so that the analytic results in Section IV should serve as a useful reference point. VII. NONEXPONENTIAL HOLDING-TIME DISTRIBUTIONS On dropping the exponential holding-time assumption for IR calls, we lose the memoryless property of the IR holding times. This makes the computation of the interrupt probability complex. Given the holding-time cumulative distribution function (cdf) and the elapsed holding time for any call in progress,

10 GREENBERG et al.: RESOURCE SHARING FOR BA & IR CALLS 19 the remaining holding time of this call has complementary cdf (8) where. Hence, given the elapsed holding times for calls in progress, the remaining holding times are independent random variables with cdf s defined as in (8). Since these cdf s are different, the exact computation of future events is a difficult combinatorial problem. Fortunately, the following simplification appears to be quite effective. We ignore the elapsed holding times, which reduces the amount of information that we need to store. Ignoring the elapsed holding times, we assume a Poisson arrival process and make an infinite-server approximation. Then, conditioned on there being calls in progress at some time in equilibrium, the remaining holding times of these calls are distributed as independent random variables with cdf, the stationaryexcess cdf associated with, given by This property holds because the arrival-time and holding-time pairs are distributed according to a Poisson random measure on the plane [4]. This result appears on [21, p. 161]. An alternative proof can be found in [5]. The infinite-server approximation ignoring elapsed holding times makes it possible to directly extend the FTA, OPA, and IPA admission control algorithms to nonexponential holdingtime distributions, without increasing the computational complexity. For example, instead of (7), the interrupt probability for IPA becomes where for (9) (10) and (11) i.e., the old calls have cdf, while the new call has cdf. Expressions for the OPA and FTA algorithms can be obtained in a similar fashion. Example 5: To illustrate, we consider the parameters of Example 3 with and let the holding-time distribution be hyperexponential with balanced means, overall mean and squared coefficient of variation. With the hyperexponential distribution, when a call arrives, with probability it chooses a holding time from an exponential distribution with mean and, with probability, it chooses a holding time from another exponential distribution with mean. This model is natural to represent two subclasses of IR calls with different exponential holding times. The balanced means assumption specifies one parameter by requiring that. Each simulation run was for time units, after deleting 25 time units to get rid of transients. Each curve is plotted based on ten simulation runs. The simulation runs are longer Fig. 8. Example 5: p I versus R for ExpIPA and Hyperx. than in Example 2 to get suitably small confidence intervals and reasonably smooth curves. This is due to the increased variability of the holding-time distributions [19]. We compared IPA based on the hyperexponential distribution (denoted by Hyperx) with IPA based on the exponential distribution (denoted by ExpIPA); i.e., ExpIPA uses the IPA algorithm in Section V-B assuming the exponential distribution when the actual holding times are hyperexponential. The purpose of this comparison is to study the performance of Hyperx as well as to check whether or not the ExpIPA is sensitive to holding-time distributions. For, Hyperx and ExpIPA are compared in Fig. 8. The performance of Hyperx is clearly superior to that of ExpIPA. This example shows that knowledge of the holding-time distribution of IR calls can be exploited to improve the revenue without sacrificing computational complexity. Example 6: We now consider the case in which the holding times of IR calls are deterministic. We consider this case because we can compare the results to CHTA which is clearly optimal for a fixed value of. Therefore, consider the same parameters as in Example 5 with the only difference being that the holding times of IR calls are deterministic. Note that the for and is equal to 1 for. Each simulation run was for time units, after deleting 25 time units to get rid of transients. Each curve is plotted based on ten simulation runs. DetIPA and CHTA are compared in Fig. 9, where the CHTA curve is the constant, allowing no interruptions. The gap between the CHTA and DetIPA is what is lost by not keeping track of and exploiting the ages. However, the curves in Fig. 9 indicate that this gap is small. Thus, the infinite-server approximation is indeed good and is nearly optimal in the only case for which we know the optimal solution. VIII. WHEN BA CALLS NEED NOT BOOK FAR AHEAD So far, we have required BA calls to book far head relative to IR holding times. That clearly is an important case, but it is also of interest to consider what happens when that assumption is relaxed, which we now do.

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