Delay Analysis of Multichannel Opportunistic Spectrum Access MAC Protocols
We provide in this paper a comprehensive delay and queueing analysis for two baseline medium access control (MAC) protocols for multi-user cognitive radio (CR) networks and investigate the impact of different network parameters, such as packet size, Aloha-type medium access probability and number of channels on the system performance. In addition to an accurate Markov chain, which follows the queue status of all users, several lower complexity queueing theory approximations are provided. Accuracy and performance of the proposed analytical approximations are verified with extensive simulations. It is observed that for CR networks using an Aloha-type access to the control channel, a buffering MAC protocol, where in case of interruption the CR user waits for the primary user to vacate the channel before resuming the transmission, outperforms a switching MAC protocol, where the CR user vacates the channel in case of appearance of primary users and then compete again to gain access to a new channel. The reason is that the delay bottleneck for both protocols is the time required to successfully access the control channel, which occurs more frequently for the switching MAC protocol. We also propose a user clustering approach, where users are divided into clusters with a separate control channel per cluster, and observe that it can significantly improve the performance by reducing the number of competing users per control channel.
Opportunistic spectrum access (OSA) communication models , implemented by cognitive radios (CR), offer both the capacity to decrease the communication infrastructure expenses by using the vacant portions of the spectrum and to improve the quality of wireless communications by permitting wireless CR nodes111Node and user are used interchangeably. to switch to a better channel when the channel quality is not satisfactory. These notable features have made cognitive-radio based wireless networks a technology of choice for incoming wireless technologies. Frequent spectrum handovers along with new requirements such as spectrum sensing and spectrum decision distinguish a cognitive radio network from its predecessors. New medium access control (MAC) protocols which take into account the inherent nature of cognitive radios are thus indispensable. For this aim, several medium access protocols have been proposed in the literature [2, 3, 4]. Different protocols may differ in details such as the existence of a common control channel, number of radios required for the nodes and sensing algorithm (please see [5, 6] and references therein for a detailed survey of CR MAC protocols). Since most of the proposed MAC schemes assume a dedicated control channel, and given that being equipped with multiple radios increases the hardware and complexity cost, we investigate in this paper a realistic cognitive radio network with a common control channel, distributed sensing appropriate for ad-hoc networks, and a single radio per node.
In the literature, most of the attention on the analysis of the different medium access protocols and their variations has been focused on the throughput analysis in the presence of saturated traffic (e.g., [6, 7, 2, 8, 3, 4]), while little work has been done on delay and jitter analysis. Throughput analysis provides an upper bound for the performance of the network, which can be used for evaluation and comparison purposes of different protocols. However, delay analysis and packet level performance evaluation are required to investigate how a protocol, along with its parameter settings, behaves for delay-sensitive applications, such as multimedia communications [9, 10].
In the few papers available on delay analysis (e.g., [11, 12, 13, 14, 15]), detailed system parameters are not considered or the CR network under consideration is simple, such that full insights into the impact of different parameters on the network performance have not been provided. The target has also been specific networks; e.g., sensor networks, with specific requirements , such that the results can not be easily extended to general models. In [17, 18], we proposed a general M/G/1 queueing model  for a cognitive radio link where the server is subject to interruptions . The channel (i.e., the server of the queue) is subject to interruptions because the CR node has to suspend its communication session for the period of spectrum handover (switching) or primary users’ activity (buffering). The proposed models in those papers were for a single node with continuous-time operating and interruption periods. However, those models didn’t address the complex problem of multi-node operation in a multi-channel environment.
Our objective in this paper is thus to fill this MAC protocols analysis gap by providing a comprehensive delay analysis for two baseline medium access control protocols discussed in [6, 7]. Note that the models described in those papers are not detailed MAC models for practical purposes. The intent is rather to have models with enough details, yet general enough, to use as baseline reference models for families of MAC protocols and investigate the effect of different parameters on the throughput of a cognitive radio network. In this paper, we have the same objective for the delay analysis and thus only relax the saturated traffic assumption and keep the same other assumptions and modeling details to analyze the average delay. In the first part, a buffering MAC model [6, 7], in which a node stays on the channel in case of interruptions until the transmission of the packet is finished, is investigated. In the second part, a switching MAC model , where the node leaves the interrupted channel and participates in a new reservation competition every time that an interruption occurs, is investigated.
The main contribution of this work is thus providing a comprehensive analytical delay analysis for a multichannel cognitive radio network with both buffering and switching recovery policies where the impact of different network parameters (number of users, number of channels, control channel access probability, channel availability, and arrival rate) on the performance is considered. The results can thus be used in network optimization and design problems. For both MAC models, we derive an exact Markov chain model which can be solved to obtain the system time distribution and moments. Since the number of states in the Markov chains grows exponentially with the number of nodes, we also propose lower complexity approximations based on discrete-time M/G/1 queueing theory results. In both MAC models, the service time of a packet is composed of two parts: the time spent to compete with other users to reserve a channel (for the switching MAC model, the reservation period may occur multiple times during the service of a single packet), and the transmission time. We use Markov chain models, combined with Renewal theory results, to derive approximate distributions of both parts of the packet service time. The service time moments are then used to find the approximate average system time of packets in the OSA network for both MAC protocols. We also provide numerical results to validate the analysis and provide insight on the delay performance of the MAC protocols for different network parameters.
The paper proceeds as follows. In Section II, the system model is presented and related parameters are introduced. The next two sections discuss the buffering model. In Section III, an accurate Markov chain for queue occupancy is proposed from which different parameters of interest can be derived. As the queue occupancy Markov chain is not scalable, a service cycle analysis is provided in Section IV. Section V discusses the switching model where the same approach is followed: first an accurate Markov chain for queue occupancy is proposed and then a delay cycle analysis and related approximations are provided. Numerical and simulation results for both buffering and switching models are discussed in Section VI. Finally Section VII concludes the paper with some remarks on future research direction. The notation used in this paper are listed in Table I.
|Number of CR nodes|
|Total number of channels|
|Number of data channels ()|
|Poisson packet arrival rate|
|Geom/G/1 service time|
|Transmission time (buffering)|
|Interrupted transmission time (switching)|
|Enlarged packet length (switching)|
|State variable, number of competitors|
|Probability of unavailability due to PUs|
|Control channel availability per timeslot|
|Packet capture probability|
|Data channel availability|
|Maximum No. of links equal to|
|Probability of being equal to|
|Probability of success in Aloha-type competition|
|Probability of success in state (6)|
|Probability of success in state for a marked node|
|Probability of channel access attempt in Aloha-type competition|
|Maximum Queue Size|
|Probability of transmission among|
|Probability of successful reservation|
|Defined in (8)|
|Steady-state probability of state|
|Updated steady-state probabilities, eliminating states with|
|Inter-arrival time of PUs (switching model)|
Ii System Model
In this section we provide a summary of the two generic multichannel OSA MAC protocols for which we derive the delay analysis under unsaturated traffic in later sections. More details on those protocols, as well as their throughput performance under saturated traffic, can be found in [6, 7]. It is assumed that there are nodes in an ad-hoc cognitive radio network and OSA channels where one channel is the dedicated control channel and the remaining channels are used for data transmission. Without loss of generality and for notation simplicity, we assume that the nodes have intended receivers that are not part of those nodes. A maximum of links might thus simultaneously exist.
As illustrated in Fig. 1, operations in the OSA network is time-slotted. Primary and secondary users are fully synchronized. In the beginning of each timeslot, there is a short quiet period during which the whole cognitive network stops operation to sense the channels. Primary users (PU) activity in any channel, including the control channel, is modeled by a Bernoulli random variable: in each timeslot and independent of the other timeslots and channels, a channel is unavailable for CR users (occupied by primary users or due to false alarms) with a probability . We also use a packet capture model whereby a transmission during an available timeslot is successful, without considering collisions and interference, with probability for the data channels and probability for the control channel. Channel availability and packet captures are assumed to be independent.
A geometric (Bernoulli) arrival process is assumed for all nodes with a probability of packet arrival in a timeslot . The packet length is given in the number of timeslots required to transmit the packet and has a geometric distribution with parameter .
A node is in the busy state when it has a reserved data channel for transmission, otherwise the node is in the idle state (with an empty or non-empty packet queue). After the sensing period, if the control channel is available, the idle nodes with a non-empty packet queue attempt to reserve a channel using an Aloha-type competition over the control channel. That is, each of the nodes will transmit in the timeslot a reservation request with a probability over the control channel. The competition is successful if only one reservation request is transmitted and it is correctly received. The probability of success in competition is given by :
where is the probability of a successful transmission on the control channel without considering collisions. The probability of success for a marked node among nodes is . It is assumed that the list of channels not being used by CR nodes is given. So, a success in competition in this timeslot is followed by reserving the channel which is on the top of the list by the competition winner, to be used from the beginning of the next timeslot. An exception is when all channels are occupied in this timeslot. If a channel is released at the end of this timeslot, it will be assigned to the successful node in competition, otherwise the successful node in competition is unsuccessful in reserving a channel and returns for a new competition in the next timeslot. It is therefore important to distinguish the event of success in competition, which depends solely on the number of competing users, from the event of success in reservation, which depends also on the status of channels. In case that a node is successful in reserving a channel, it starts transmitting one packet in the reserved channel from the beginning of the next timeslot. Other nodes can not make any interference on the reserved channel, but fading, sensing errors and PU activities may interrupt the transmission.
Two MAC protocols are considered in this paper. A packet transmission cycle example for each protocol is illustrated in Fig. 1. In the buffering MAC protocol , a node stays on its reserved channel, even when it becomes occupied by PUs, until the packet is entirely transmitted. In each timeslot, a successful transmission therefore occurs with probability . For the buffering MAC protocol, the (enlarged) packet service time therefore consists of a reservation period of length followed by a transmission time which consists of successful transmission, unsuccessful transmission and unavailable timeslots, until the entire packet is transmitted.
In the switching MAC protocol , a node that senses it channel occupied by PUs leaves the channel and returns to the idle state to participate in the competition to reserve a new channel in this timeslot. For the switching MAC protocol, the (enlarged) packet service time therefore consists of several reservation periods and the successful transmission and unsuccessful transmission timeslots required to transmit the entire packet.
For both MAC protocols, once the packet transmission is terminated, the node releases the channel and returns to the idle state. It will enter the competition at the next timeslot if it has another packet in its queue. A service-resume transmission model is assumed for both MAC protocols where the remaining part of the packet is transmitted after each interruption. Note that due to the packet length geometric distribution model, the delay analysis is also the same for a service-repeat model where the entire packet must be transmitted after each interruption.
Iii Buffering MAC Protocol: Queue Occupancy Markov Chain Analysis
The discrete and memoryless nature of the system model and MAC protocols described in Section II enables us to propose a general discrete-time Markov chain (MC) which tracks all arrivals, departures and interruption events for all nodes. The different queue performance metrics can then be directly computed using the steady-state MC solution. The state variable is given in the form where is the current number of packets in the system (waiting in the queue or being transmitted) of node and is a status flag indicating whether node has a reserved channel () or is idle (). For each state , we also define the variables and , where is an indication function with , , which are, respectively, the total number of busy nodes and the total number of idle nodes with a non-empty queue. An MC example with is illustrated in Fig. 2.
Because of the geometric arrival process and since at maximum one packet can be served in one timeslot over each channel, the queue status changes at most by one packet in two consecutive slots. That is , where is the node queue occupancy at time . However, as a function of the status flag, the following refinements can be made:
If then , and only an arrival event can occur, in which case , otherwise . Furthermore, the next status state is also , irrespectively of an arrival event occurrence.
If and , two events of reservation success and an arrival can occur at the same timeslot and the events are independent. We have depending on arrival or no arrival. Reservation success only changes the value of to 1.
If and , two events of packet transmission termination and an arrival can occur at the same timeslot and the events are independent. We have . The case occurs with an arrival and no transmission termination, and the case occurs with no arrival and transmission termination. Meanwhile, a node will have the same number of packets in the queue in the next timeslot for two events: transmission termination and no arrival, or no transmission termination and no arrival. A transmission termination results in changing the value of to 0, irrespectively of an arrival event occurrence.
Furthermore, all busy nodes and idle nodes with an empty queue operate independently, so the joint probability of different events can easily be found by multiplication.
For two states and , , the set of impossible transitions, can then be defined as follows:
where ( is the number of positions for which the difference between the elements of and is 1) is the number of nodes which started transmitting in the new timeslot from state to and is the number of busy nodes in state A whose transmission is finished in state .
The transition probabilities of going from state to a new state can then be computed using Alg. (1). For , the transition probability is zero. In Alg. (1), is the set of nodes participating in competition in state whose size is given by . If is empty, the last part of the algorithm is not run. represents the case where all channels are busy and no termination occurs.
This exact Markov chain model is general and can address several system model extensions, such as non-homogeneous nodes with different arrival rates or different packet length. However, as there is no buffer limit, the number of states is infinite and grows exponentially with the number of nodes. To be able to solve the Markov chain numerically a buffer limit should be set and the truncated Markov chain is analyzed instead. If the transition probabilities are written for a truncated version of the Markov chain with the buffer size , the states with should be treated separately because for cases , or and no termination, the queue occupancy in the next state with arrival or without any arrival (with probability equal to one) will still be .
Even for the truncated MC, the queue occupancy MC model suffers from exponential growth of the number states and is thus not scalable. For example, for a buffer limit of , the number of states will be (e.g., 484 states for and ). In the next section, the homogeneity of the nodes is taken into account to propose an approximate but scalable combined Markov chain which will be used for a service cycle analysis.
Iv Buffering MAC Protocol: Service Cycle Analysis
Since a geometric arrival process is assumed, we have for any node a discrete-time M/G/1 queue with geometric arrivals called Geom/G/1. Continuous-time M/G/1 and Geom/G/1 become equivalent when the timeslot is short with respect to the arrival rate. Furthermore, since the arrival process and packet length distribution are identical for all nodes, the queues are homogeneous. To solve a tagged node queue, we should therefore only find the service time for the buffering MAC protocol. and are independent, so the first and second moments of the service time can easily be found from the moments of and . Discrete version of the Pollaczekâ-Khinchine formula  can then be used to find the queue performance metrics of interest. The exact distribution of and approximations for the distribution of are derived in Section IV-A and IV-B, respectively.
Iv-a Transmission Time
For the buffering MAC protocol, the transmission time is started after the timeslot in which a channel has been reserved with success and lasts until the end of the packet transmission (see Fig. 1). Since the node is using a reserved channel, there is no interference from other CR nodes and in each timeslot the probability of a successful transmission is given by (see Section II). For a packet with length timeslots, successful transmissions are required to complete the packet transmission. For timeslots, the last slot should necessarily be a successful transmission and among the remaining timeslots, exactly of them should be successful transmissions. Therefore, conditioned on the packet length , the transmission time of the packet has a negative binomial distribution given by:
The packet length has a geometric distribution such that the unconditional distribution of is given by:
The first and second moment of can then be found as:
Iv-B Reservation Periods
At the beginning of every timeslot, nodes can be divided into three groups: busy nodes with a reserved channel, idle nodes with an empty queue and idle nodes with a non-empty queue. The probability of reservation success at the end of the timeslot depends on the number of nodes in the last group, except for the case where all channels are occupied. Due to packet arrival and packet transmission termination events, the number of idle nodes with a non-empty queue changes from one timeslot to the next one such that the probability of reservation success in a timeslot is time-varying. We will proceed as follows to find the reservation period distribution. First, we will propose a combined Markov chain to find the distribution and transition probabilities of the number of nodes competing in the reservation procedure. We will then find the reservation period distribution by first conditioning it on the number of competing nodes in the first reservation timeslot for the tagged node, and then using the steady-state probabilities to obtain the unconditioned distribution.
Iv-B1 Combined Markov Chain
Since the sum of nodes at every timeslot is fixed to , the state variable of the proposed combined Markov chain is given by a 2-tuple where is the number of busy nodes and is the number of idle nodes with a non-empty queue. The number of idle nodes with an empty queue is then given by . We have the following facts about the system:
Every idle node with an empty queue may independently become a node with a non-empty queue in the next timeslot with the probability , which is the probability of one arrival during one timeslot.
Among the idle nodes with a non-empty queue, at most one node may become busy with the probability of reservation success and all others stay in the same group. is the probability of getting a channel when nodes participate in competition and channels are already reserved. It is given by:
where is the probability of no transmission termination among the ongoing communication sessions.
Every busy node may independently finish in a timeslot its packet transmission with probability and then, with probability , the node joins the idle nodes with an empty queue. converges to the steady-state queue occupancy of a node queue and is the same for all busy nodes because nodes are homogeneous and have a geometric memoryless packet length.
The number of participants in the competition in the next slot compared to the previous slot may thus decrease at most by one, but may increase to any larger value up to depending on the number of idle nodes with an empty queue with an arrival and the number of busy nodes that finish their transmission and have a non-empty queue.
Using these facts and the results in [6, Eq. (16)], the transition probabilities of the combined Markov chain can be given by:
and are respectively defined in [6, Eqs. (14),(15)] as the probability of transmission terminations among ongoing communication sessions and the probability of success in competition among competitors (note that ).
is the probability that new nodes join the group of idle nodes with a non-empty queue given that there are busy nodes that terminate their transmission and idle nodes with an empty queue. This happens if nodes out of the busy nodes that terminate their transmission have a non-empty queue at the end of the timeslot and nodes out of the idle nodes with an empty queue have a packet arrival in the timeslot, where . is thus given by:
Assuming that is known, the steady-state probabilities of the combined Markov chain can be found. We explain in Section IV-B3 how can be found.
Iv-B2 Reservation Period Distribution
We now derive the distribution of for a tagged node belonging to the group of idle nodes with a non-empty queue. Let denote the reservation length given that the system is in the state at the beginning of the reservation period. The probability of a successful reservation for the marked node in the first timeslot, and thus have is equal to . However, if the reservation is unsuccessful, the system transits to a new state with the probability . In this state, the probability of successful reservation and thus have is now . Otherwise, the system transits to a new state and the process repeats. In general, can be found in a recursive manner from the probability of transition to other states and from other states, given that the node was not successful in reservation.
It is important to note that for this analysis, the transition probabilities of the combined MC discussed in the previous section can not be directly used because it is implicitly assumed that the marked node was not successful. We therefore need a different MC with transition probabilities denoted by conditioned on the fact that at each transition, the marked node was not successful, and that if the reservation competition was a success, it was one of the other competitors who was successful. The transition probabilities are given by (7) where given in (1) is replaced by .
The distribution of is then given by:
can then be unconditioned to obtain as follows:
Since when we mark a node to find , we implicitly assume that the node is idle with a non-empty queue, the states in which the number of competing users is zero () should therefore be eliminated to obtain the steady-states probabilities for the states as .
Iv-B3 Solution Procedure
To find the combined MC transition and steady-state probabilities, , the steady-state probability of a node’s queue being empty, is required. can be computed from the Geom/G/1 queue relation . However, depends on the combined MC solution. We therefore proposed an iterative solution where an initial value is set for . The combined MC chain can then be solved and computed. A new value of is then obtained and used for the next iteration. The iteration continues until the value of converges. To compute the initial value of , we propose to use the lower bound case for where only the tagged node participates in the competition. The distribution of is then geometric and .
Note that the number of competing nodes in one state is not independent of the other states. However by using the steady-state probabilities in (10) combined with the iterative solution procedure, this dependency is ignored. Furthermore, the different functions involve in this system are not necessarily convex. The combined-MC approach is therefore inherently an approximate solution. For example, we have observed that in some cases the combined-MC solution converges to a non-zero value of while simulations indicated that the system is unstable. However, all studies that we have performed showed that in the region where the system is stable, the combined-MC approximation is very accurate (see results presented in Sec. VI).
Iv-B4 Other Approximations
The proposed approach based on the combined Markov chain is scalable. However, the number of slots until reservation success is infinite and the distribution of should be truncated which results in an approximation. Furthermore, even when the distribution of is truncated, the computations are intensive. We therefore propose in this section three different approaches to reduce the computational complexity required to calculate the moments of .
Reduced Markov Chain
The number of states in the Markov chain can be reduced by assuming that idle nodes will independently have a non-empty queue at the beginning of a timeslot, and therefore participate in the competition, with probability . This assumption therefore ignores the dependency between timeslots of the status of an idle node and we don’t need to track in the MC state variable the number of idle nodes with a non-empty queue. That is, the combined MC chain presented in Section IV-B1 can be replaced with the single dimension Markov chain proposed in  which uses the single state variable , the number of busy nodes. In each timeslot, nodes then participate in the competition with an Aloha access probability given by . The distribution of can then be computed as explained in Section IV-B2 where given in (1) is replaced by , the summations in (9) and (10) are over a single variable, and the modified transition probabilities in (9) and steady state probabilities in (10) are obtained from the MC in . The solution procedure is then the same as the one described in Section IV-B3. This approximate model, denoted by ’Pawelczak-MC’ in the numerical results, reduces the computational complexity since the state space is smaller.
We now propose an approximate approach which significantly reduces the complexity associated with finding the distribution of so that it can be used in scenarios with limited computation capacity. The complexity in (9) arises from the fact that the reservation success probability is time-varying and has memory. As an approximation, we propose to simply assume that the number of participating nodes during a reservation period is always . With this assumption, , the reservation time given that the number of participating nodes is at each timeslot, has a geometric distribution with probability , where is the probability of being equal to and is defined as follows:
From the combined MC chain presented in Sec. IV-B1, we can compute (), the distribution of the number of nodes participating in the competition excluding the cases where there is no competitors. We can then approximate the distribution of as follows:
Given that is geometric, it is then easy to compute the moments of . Since is not known, the same solution procedure as the one described in Section IV-B3 should be used. This approximate model is denoted by Combined-MC-Dist in the numerical results.
This third approach further simplifies the computations by calculating the reservation time with the average number of idle nodes in competition with a non-empty queue. Let define
where the steady-state probabilities are obtained from the combined MC chain presented in Section IV-B1. Then is approximated with a geometric distribution with probability . Since is not known, the same solution procedure as the one described in Section IV-B3 should be used. This approximate model is denoted by Combined-MC-Avg in the numerical results.
V Switching MAC Protocol Delay Analysis
In this section, we provide the delay analysis for the multichannel OSA switching MAC protocol. The same approach as for the delay analysis for the buffering MAC protocol is followed: we first provide a system occupancy Markov chain in Section V-A and then we derive an Geom/G/1 queue analysis in Section V-B. The only difference between the buffering and switching MAC protocols is that in the switching protocol, if a SU senses that its reserved channel is occupied by a PU, it immediately returns to the group of idle nodes with non-empty queue and participates in the competition, while in the buffering MAC protocol the node would remain on its reserved channel. To facilitate the analysis of the switching MAC protocol, the system occupancy MC and combined MC for service time analysis are embedded after the sensing period. Finally, for sake of brevity, we omit details in the analysis that are similar to the buffering MAC protocol analysis.
V-a Occupancy Markov Chain
The states of the queue occupancy Markov chain are similar to the buffering case and are given by . However, the observation time where the Markov chain has been imbedded is immediately after the initial sensing period when the status of the channels has already been determined. We therefore have that when the reserved channel for node is available in this timeslot for transmission (the transmission may however be unsuccessful due to the packet capture model).
Similar to the buffering case, for a state , we can define as the total number of busy nodes who have a channel and whose channels are available in this timeslot and as the total number of idle nodes with non-empty queue. The switching model facts are very similar to the buffering model with the difference that a node who transmitted on its assigned channel in this timeslot may, if it does not finish its packet transmission, come back to the competition with probability if a PU is detected on its assigned channel. For , the Markov chain is illustrated in Fig. 3 where the transitions after states and are not shown. It can be seen that compared to Fig. 2 for the buffering policy, there are only two changes: a new transition may occur from state (1,1) to (1,0) if no arrival occurs, transmission is not finished, but the channel is missed in the next timeslot, and to state (2,0) with the same conditions and an arrival occurs.
Transition probabilities for this Markov chain can be found using Alg. 2 with the same set of impossible states (IS). Similar to the buffering model, when a node is busy (), its transitions are independent of other nodes. For and , the node participates in competition. Compared to the buffering model, it can be seen that the main change is for the case where there is a transition between and because this transition may occur in the switching model due to both a service termination or a PU positive sensing. Moreover, we can see that the probability of success is updated by because a success in competition should also be followed by the availability of the reserved channel to be a reservation success.
For the case where we have , the success depends on the nodes who had a transition from to . Any of them may have a transition due to missing the channel and PU arrival (kept in vector ) or a termination (kept in vector ). keeps the total probability of both events. Then, is defined to be the probability that all transitions from to have been due to PU arrival. In this special case, no reservation success is possible. If at least one node among them has finished its transmission, with the probability , then a competition success may result into a reservation success with the probability .
V-B Service Cycle Analysis
The nodes queue in the multichannel OSA network with the switching MAC protocol can also be modeled with a Geom/G/1 queue. As illustrated in Fig. 4, the Geom/G/1 service time in this queue is composed of alternating renewal processes: uncompleted packet transmissions (a part of the packet however can be transmitted) and competitions for channel reservation alternate until the packet is entirely transmitted. Since the packet length is geometric and the occurrence of an interruption is a Bernoulli event, and both are independent, the uncompleted packet transmission time is thus memoryless and identical. The unsuccessful transmission attempts therefore form a renewal process. The reservation periods during the transmission of a packet are also identical and form the second renewal process.
Let be the service time of the Geom/G/1 queueing model, which can be given by
where is the enlarged packet length (in number of slots), which is geometrically distributed with the probability , and is the number of switching during the transmission of the packet. To find the service time, we must find the distribution of and the number of renewals , which are addressed in Section V-B1 and Section V-B2, respectively.
V-B1 Reservation Period
The approach to find the distribution of for the switching MAC protocol is similar as the one followed for the buffering model in Sec. IV-B. The state variable of the combined Markov chain is still given by a 2-tuple , however, the observation time where the combined Markov chain is embedded for the switching MAC protocol is after the initial sensing period at the beginning of each timeslot such that the status of the channels for the current timeslot is known. is therefore the number of reserved channels on which there will be a transmission attempt in this timeslot and is the number of idle nodes with a nonempty queue which participate in the competition on the control channel for reservation. The number of idle nodes with an empty queue is thus given by .
To find transition probabilities, we use four auxiliary variables , , and . Note that in the following, and refer to and for the current timeslot . Among busy nodes, nodes independently finish the transmission of their packets, each with the probability . We use the notation for the probability of this event. Any of those nodes may join in the next timeslot, with probability , the idle nodes with a nonempty queue. Otherwise, they join the idle nodes with an empty queue. The probability that nodes out of the finishing nodes will be among the nodes in the next timeslot is thus given by . For the remaining nodes which do not finish their transmission, nodes may leave their channels due to PU presence sensing with probability . Those nodes also join the idle nodes with a nonempty queue after the sensing period. The remaining nodes stay on their channels, so they are among the nodes.
Considering participants in competition, we have the same results for competition success as for the buffering case. One node may then be successful and join the nodes with probability . The term is included because the node may be successful in competition and reservation during this timeslot, but the reserved channel might be unavailable in the next timeslot. Therefore, without starting the transmission, the node has to come back to competition. All other nodes remain in the group of idle nodes with a nonempty queue, i.e., they are among nodes. Finally, any of idle nodes with an empty buffer may receive a packet and join the participants in competition in the next timelot. The probability to have such nodes is given by .
We can thus, for the switching MAC protocol, express the transition probabilities of the combined Markov chain in the form as follows:
where is given in (8). Given the transition probabilities for the combined Markov chain for the switching MAC protocol, the distribution of can be obtained as explained in Section IV-B2. The second and third approximations given in Section IV-B4 to find the distribution of can also be used (the approximation based on the model presented in  is not applicable for the switching MAC protocol).
V-B2 Renewal Process
As illustrated in Fig. 4, , any part of the whole service time where the node has a reserved channel, is the inter-arrival time of the channel unavailability events. Let us use the random variable to denote the inter-arrival time of those events. has a geometric distribution with probability . From the Renewal Theory point of view , is the number of renewals from 0 to where the process which is renewed is . The number of renewals can be found from Renewal Theory results .
However, as we have geometric distributions for both and , we can use the following simpler approach. Due to the memoryless packet length, the remaining part of the packet in any operating period is geometrically distributed with parameter . The total number of renewals is thus the number of trials until success, which is a geometric distribution of type-II, with the probability of success in a period being , the probability of sending the whole packet in the given period. However, the number of reservation periods which appear in the calculation of service time is because the first reservation period is not counted as a renewal (see Fig. 4). We thus have:
For any two type-I geometrically distributed random variables and with parameters and , we can show that
We thus have:
To find the second moment of it should be taken into account that we have a random sum and the two parameters and are not independent. Using the following facts: