The probability distribution probability of the unencumbered call interruption time IT is related to the arrival process of PUs and the stochastic processes of the numbers of PUs and SUs with ongoing calls. This proposed model is detailed next. As stated above, the IT of an analyzed SU call due to an arrival of a PU is the elapsed time from the beginning of the call until the arrival of a PU that causes the interruption of the call.
Building on this, let us define P Int as the probability that an arrival of a PU interrupts the analyzed call. However, as it is shown in Appendix 1 , the call interruption process of SUs due the arrival of PUs does not actually follow a Poisson one. In Section 6 , the inaccuracy introduced by this approximation in the numerical results is extensively investigated under a variety of different evaluation scenarios for all the considered call-level performance metrics.
In the following analysis, we focus on the performance of secondary users. In this section, the expressions of both the inter-cell handoff arrival rate and forced termination probability of secondary users are derived.
Cognitive Radio System
For the sake of clarity and completeness, an alternative mathematical expression for calculation of the inter-cellular handoff arrival rate based on the cell departure rate is shown in Section 5 , considering both UST and CDT exponentially distributed. In this section, considering that UST is exponentially distributed and CDT is generally distributed, a closed analytical expression for the forced call termination probability in CRCNs is obtained.
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In mobile communication networks, forced call termination is due to two fundamental features: resource insufficiency and link unreliability [ 9 ]. Without loss of generality, in the following analysis, forced call termination probability due to link unreliability because of the propagation impairments is ignored. Instead, due to the nature of CRNs, a new type of forced call termination is introduced: forced termination due to intra-cellular handoff failure triggered by the arrival of a PU. Finally, it is important to mention that the methodology proposed in this section to derive the forced call termination probability turns mathematically intractable when non-exponential distributions for the UST are considered.
Nonetheless, when phase-type probability distributions are considered to model the UST, the call forced termination probability can be calculated as the ratio of the rate of forced termination calls and the rate of accepted calls as it is done in [ 9 ]. However, in this case, it is necessary to calculate a quantity which is closely related to the steady-state probabilities as in [ 9 , 11 ] and no closed analytical expressions can be obtained.
For the sake of clarity and completeness, an alternative mathematical expression for the forced call termination probability computed as the ratio of the rate of forced termination calls and the rate of accepted calls is presented in the next section, considering both UST and CDT exponentially distributed.
It is important to note that the previous teletraffic analysis represents an approximation approach for the performance evaluation of the considered system. As proven in Appendix 1 , this is not true. However, this approximation renders accurate numerical results for most of the scenarios and performance metrics of interest, as shown in Section 6. The fixed point iteration method is employed to iteratively calculate the inter-cellular handoff rate as explained in [ 40 ]. Notice that 65 is a closed-form approximated expression to compute forced call termination probability.
Equation 65 is an approximated expression in the sense that it was derived in Section 4 considering that call interruption times follow a Poisson process. Moreover, expression 65 depends on P hI and P Int , which are derived in this section considering, also, that call interruption process is a Poisson one. Notice that for the computation of 66 , it is needed to know the steady-state probabilities. However, compared to 65 , the unique source of imprecision of 66 is the fact that steady-state probabilities are derived considering that call interruption process is a Poisson one.
When spectrum handoff is considered, interrupted SUs are allowed to move to other vacant channels if available. The maximum Erlang capacity is computed as the maximum value of the offered traffic for which all the QoS requirements are still met. Maximum Erlang capacity is obtained by optimizing the number of reserved channels to prioritize intra- and inter-cell handoff call attempts over new call requests.
The optimal value of the number of reserved channels is systematically searched by using the fact that new call blocking handoff failure probability is a monotonically increasing decreasing function of the number of reserved channels. The capacity maximization procedure ends when both the new call blocking probability and forced call termination probability achieve their maximum acceptable values. The goal of the numerical evaluations presented in this section is to verify the applicability as well as the accuracy and robustness of our developed mathematical models.
In particular, in this section, numerical results for performance evaluation of mobile CRCN in the presence of both intra-cellular e inter-cellular handoff mechanisms are shown. The analytical numerical results presented in this section were verified by a wide set of discrete-event computer simulation results for a variety of evaluation scenarios. Part of this numerical validation is shown in our previous work [ 23 ], as commented in Section 6.
In Section 6. In this section, numerical results are presented to investigate the extent by which the probability distributions of both CDT and UST as well as the interruption probability P Int affect the statistics of both new call channel holding time CHTn and handoff call channel holding time CHTh. As it is explained in the last paragraph of Section 3 , the statistics of the CHTn and CHTh are obtained using 12 and 35 , respectively.
In this figure, red , blue , yellow , and green lines represent the scenarios where the UST is exponentially, Erlang, hyper-exponentially of order 2, and hyper-Erlang of order 2, 2 distributed, respectively. Several interesting observations can be extracted from Figs. For instance, from Fig. This is an expected behavior that validates our mathematical formulation and can be explained as follows. Also, from Figs. It is an intuitively understandable behavior due to the fact that as the probability of interruption increases, more ongoing secondary calls are prematurely terminated in detrimental of the mean value of both CHTn and CHTh.
However, the most relevant result that can be extracted from the plots presented in Figs. To exemplify this, let us quantify the impact of the distribution type used to model the UST on the statistics of the CHTn the behavior of the CHTn is represented by the solid lines in Figs.
To this end, let us consider the case when both the UST and CDT are exponentially distributed as the reference case this scenario is represented by the red solid-line in Fig. Under the same scenario, Fig. Similarly, Fig.
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Similar results are obtained when the CHTh is considered. On the other hand, Figs. Due to the fact that both distributions i. Analyzing the impact of moments of UST and CDT higher than the third one on channel holding time statistics represents a topic of our current research.
The same observation applies for the CDT. Considering this fact, the following observation can be extracted from Figs. This behavior can be explained as follows. Consequently, the service time cell residence time of calls that end their service in the cell where they were originated is, in general, considerable smaller greater than the mean UST CDT , resulting in a diminution increase on the mean value of the CHTn.
On the other hand, the service time cell residence time of calls that are handed off to another cell is, in general, considerably greater lower than the mean UST CDT , resulting in an augment diminution on the mean value of the CHTh. The combined effects of these facts lead us to the behavior explained above and observed in Figs.
Many others interesting observations can be extracted from Figs. For instance, when the UST is hyper-Erlang distributed and the CDT is exponential distributed notice that this scenario corresponds to the green lines of Fig. Similar behaviors are observed when the CDT is modeled by either an Erlang or a hyper-exponential distribution. Thus, selecting a suitable model that effectively captures the realistic statistics of both CDT and UST is of paramount importance when analyzing the performance of mobile cognitive radio cellular networks.
Both spectrum handoff and non-spectrum handoff cases are considered for scenarios 1 and 3. Optimum number of reserved channels to achieve the Erlang capacity is shown in Fig. Both spectrum handoff and non-spectrum handoff are considered for scenarios 1 and 3. Inter-cell handoff arrival rate for the Erlang capacity is shown in Fig. Erlang capacity shown in Fig. Specifically, the optimal number of reserved channels is computed in such a way that the system capacity is maximized, while the QoS requirements, in terms of new call blocking and forced call termination probabilities, are achieved.
New call blocking and forced call termination probabilities are evaluated using 61 and 66 , respectively. Both of these probabilities are function of the interruption probability, which is evaluated using 62 and 67 for the NSH and SH cases, respectively. From Fig. Let us consider, for example, the scenario S1 for the SH case. This behavior is more evident for the NSH case.
For instance, for the scenario S1, Fig.
In other words, these results reveal that the use of spectrum handoff is essential for maximizing the arrival rate of PUs at which it is still possible to provide the QoS demanded by SUs. On the other hand, for the large relative service time case scenarios S2 and S4 , the critical operational point decreases In general, Fig. For instance, let us consider the scenario S1 for the SH case as the scenario of reference.
This behavior is due to the following two facts. First, the handoff arrival rate increases as the user mobility increases in detrimental of call forced termination probability. Thus, in order to achieve the required forced call termination probability i.
Consequently, in order to achieve the required new call blocking probability i. Second, increasing the relative service time implies that the mean value of the secondary service time relative to the mean value of the primary service time increases. In this sense, as the relative service time increases while the utilization factor of the primary channels remains unchanged , it is more likely that an ongoing secondary call is interrupted due to the arrival of primary sessions in detrimental of forced call termination probability.
Because of its larger duration, each secondary call is exposed to a larger number of interruptions. Thus, as explained above, in order to guarantee the required QoS, in terms of the forced call termination, more channels need to be reserved in detrimental of the maximum achievable system Erlang capacity. Also, for a given value of the utilization factor of the primary channels, as the relative service time increases, the departure rate of successfully terminated calls decreases relative to the arrival rate of primary sessions; thus, the average number of idle channels decreases in detrimental of the new call blocking probability and, consequently, in detrimental of system capacity.
The joint effect of these facts leads us to the behavior explained above and illustrated in Fig. Maximum relative difference between analytical and simulation numerical results for both forced call termination and new call blocking probabilities. In this paper, teletraffic performance and channel holding time characterization in mobile cognitive radio cellular networks CRCNs under fixed-rate traffic with hard-delay constraints was investigated.