Route Selection Based on Connectivity-Delay-Trust in Public Safety Networks

Recently, the mission-critical push-to-talk (MCPTT) system has emerged as the new broadband technology for public safety networks. In this paper, we propose a route selection method based on the connectivity, delay, and trust in the MCPTT system to offer the best connectivity-delay-trust performance. To begin with, the connection probability between a pair of user equipment (UE) has been analyzed with the interference from other MCPTT groups. Then, the timeliness probability is studied to capture the level of confidence for the delay requirement. Moreover, the trust probability is calculated by a beta reputation system to prevent potential attacks from malicious nodes and ensure secure interactions among the forwarding UE. By combining three probabilistic results, we investigate the optimal path route selection method based on the Dijkstra's algorithm to find the path, which can meet the requirement of the MCPTT system. The simulation results validate our models and theoretical analysis. The results provide a useful insight into designing the MCPTT system for 5G public safety and disaster relief networks.


A. Background and Motivation
A S INTRODUCED in the 3rd Generation Partnership Project (3GPP) Release 13, mission-critical push-to-talk (MCPTT) provides an enhanced arbitrated method by which two or more users can engage in the mission-critical group communications based upon 3GPP evolved packet system services [1].Floor control is provided in the mission-critical group communications to regulate the access to shared channels and arbitrate transmission request contentions.When multiple requests occur, one user equipment (UE) is guaranteed to talk (i.e., take the floor), while others are either rejected or queued based on the priorities of the users in contention.Overriding the current talker by a higher priority user and limitation of talk time promote the chances for other users to use the service.Further development of mission-critical services such as mission-critical video and mission-critical data created an opportunity to reuse base func-tionality documented in the stage one requirements for MCPTT [2].Although mission-critical service on long-term evolution (LTE) will be primarily targeting public service agencies, i.e., police, fire brigade, ambulance, it is still capable of general commercial applications for other industries and service departments such as utility companies and railway.
The users from different mission-critical organizations communicate in separate groups for different tasks by using their MCPTT UE.In many practical off-network scenarios, responders are dispatched to investigate an area that is far away from their affiliated group.To report the situations back to this group in the formats of text messages, recorded voice or video clips, the leading UE from nearby groups can cooperate to forward the messages in an ad hoc fashion [3]- [5].To yield a reliable, fast, and secure path, the models related to connectivity, delay, and trust need to be addressed for route selection.

B. Related Work
Network connectivity is an essential requirement to guarantee reliable communications among UEs.Due to the underlying dynamics of fading and interference [6], the link quality between two wireless nods has been studied statistically.The connection probability or the outage probability has been used to study a link quality under the fading channel statistically.The outage probability for cooperative relay network in the presence of intercell interference was analyzed in [7].The probability of full connectivity of high density random networks in confined geometries was developed in [8].The use of device-to-device (D2D) relays as a disaster relief solution and their network performance were studied in [9].In [10], a systematic cooperative routing scheme named Constructive-Relay-based CooPerative Routing (CRCPR) was proposed to provide greater robustness against node mobility-induced link breaks.Meanwhile, geographical constraints [11] and some task properties such as rescue missions inside houses on fire could make the group communications exhibit clustering behavior.The interference and outage in clustered wireless ad hoc networks modeled by Matérn cluster and modified Thomas cluster processes were studied in [11].The performance of modified Thomas clusters in D2D networks based on the distance probability density function was analyzed in [12].The performance of Matérn clusters based on the newly discovered distance probability density function was addressed in [13].In [14], a communication-aware route selection strategy was proposed to yield high throughput for ongoing data trans-missions.However, it did not consider the clustering behavior of group communications.Meanwhile, the quality of intercluster communications in MCPTT networks depends on the call activities of UE within each cluster, and the call activities are related to the call delay modeling of MCPTT networks.Therefore, the connectivity analysis should be linked to the delay modeling to achieve more accurate results.
During route selection, choosing the forwarding UE with low latency is desired.The queuing for utilizing the shared resource blocks in the MCPTT group communications decides that the service for forwarding the messages is not always immediately available.Consequently, the guaranteed deadline requirement from classic real-time systems is infeasible and a timeliness probabilistic model is proposed to capture the level of confidence for the delay requirement [15].The timeliness probability is defined to be the probability that a transmission request needs to wait for at most a given time threshold τ until the resources blocks are available.To study the transmission request delay in each cluster head for message forwarding, the trunking theory is used.Two major classes of trunked radio systems named lost call cleared and lost call delayed (LCD) systems were studied in [16].The former does not provide a queue for requests.The latter, however, allows a transmission request to the queue, in a buffer, and waits until a server is available, which fits the flow control for MCPPT service.Meanwhile, a probabilistic metric was proposed in [15], where the Gaussian distribution was used to approximate the distribution of end-to-end delay.The interflow and intraflow interferences have been considered in [17] to reduce packet losses and retransmissions for shorter delay.
Secure message delivery is very crucial to the mission-critical service since nodes may not cooperate or act maliciously toward the message forwarding, which can cause the received message to be confused or beyond comprehension.Secure sockets layer (SSL) and some other traditional cryptographic tools attach a message authentication code to the transmitted data so that the UE at the destination can detect the malicious behaviors of forwarding UE such as data tampering [18].However, by using SSL alone, it is still very challenging for the UE at the destination to locate the malicious forwarding UE along a path and report them back to the source via a different path where the behaviors of the UE along the new path also need to be confirmed.Hence, to determine whether a neighboring node will assist to forward the messages in a timely manner, trust establishment schemes have been proposed [18], [19].Trust exploits the fact that a source UE can monitor the forwarding operation of the neighboring UE due to the broadcast nature of radio.Based on an accumulated number of observations on whether each time the forwarding is successful, the source UE can choose to trust one neighboring UE in the future forwarding operations.Given the number of observations on the forwarding operations over a link, the trust probability of a link can be formulated based on the beta probability density functions which have been actively used in Bayesian inference [20].An early study on the trust establishment in a mobile ad hoc network was conducted in [21], where the forwarded packet was buffered and checked with the original message for a match.A passive monitoring of forwarded data traffic to estimate the behavior of a node without buffer was proposed in [19].The proximity-based trust and experience-based trust modelings in random wireless networks were proposed in [22].The selective packet drop behavior in the selfish relay has been studied in [23] for route selection, which ignored the effects of connectivity and delay.

C. Contributions and Paper Structure
To the best of our knowledge, this is the first paper to propose a route selection method based on connection, timeliness, and trust probabilities for mission-critical group communications.The contributions of the paper are as follows.
1) We characterized the connection probability among the UE in an MCPTT system in the presence of the interference coming from other MCPTT clusters.2) We proposed a robust trust probability which considered the confidence of the trust as a result of the number of iterations in a beta reputation system.3) We first proposed a route selection method which simultaneously considered the connection, timeliness, and trust probabilities for route selection in an MCPTT public safety network.As a result, the proposed route selection method could provide a useful guideline to design the MCPTT system for 5G public safety and disaster relief networks.
The rest of the paper is structured as follows.Table I gives the notation and symbols that we use throughout the paper.Section II begins with a description of the system model and addresses the probabilistic models for connection, delay, and trust of a given link.Section III focuses on the route selection based on the three probabilistic models mentioned above.Section IV gives the simulation results and discussion, and Section V concludes the paper.

A. Network Layout
As shown in Fig. 1, K number of mission-critical groups are randomly distributed on the plane.We focus on choosing a multihop to forward messages from one communication group to another in a mission-critical public safety network working in the off-network mode.We assume that each group forms a cluster with radius R, and there are m i members and one group leader within cluster i.The group leader is fixed at the cluster center, and the group members are uniformly distributed within the cluster.The group leaders' UE work as cluster heads and are automatically authorized as the member of other groups.Thus, the cluster head is capable of forwarding messages from one group to another.The spatial distribution of UE during each hop is assumed to be mutually independent.According to the standard in [24], the talker is in charge of the flow arbitration, and the floor request queue will be transferred from the former talking UE to the new talking UE in the off-network mode.The flow control module for each cluster is assumed to use M/M/1 queuing system, where two M s, in turn, denote the memoryless Poisson distribution of the call requests process and exponential distribution of the call holding time [16], [25], and the number 1 indicates that a single server exists for the queue because, at most, one UE within a group is allowed to transmit at a time.During the busy time, the call requests from a user in cluster i are assumed to be Poisson distributed with an average rate of λ u i calls per minute, and a user in any cluster holds a call for τ hold seconds in average.The holding time is assumed to be exponentially distributed [25].The entire number of users in cluster i is m i + 1.Therefore, (m i + 1)λ u i is the total mean call arrival rate.The transmit power of each UE is P d .The gain due to coding, transmitting antenna, and receiving antenna is G e .We assume that all channels experience block Rayleigh fading and the channels remain constant over one block but vary independently from one block to another.The corresponding channel gains are independently and exponentially distributed within a unit mean.

B. Connectivity Modeling
The connectivity is a study of the link performance in the presence of path loss, multipath fading, interference, and noise.By taking connectivity into consideration, a path with qualified link performance will be used for route selection.

1) Signal to Interference-Plus-Noise Ratio (SINR):
The receiver of a UE captures not only the signal but also the interference from the UE who are transmitting simultaneously [26].The total interference power measured at a receiving UE j is given by where L denotes the set of interferers; P kj is the received interference power at UE j from the UE k ∈ L; (d kj ) = (λ/4π) 2 d −η kj , where λ represents the carrier wavelength, d kj is the pairwise distance between UE k and j, and η is the path loss exponent; |h kj | 2 is the instant power of Rayleigh fading.The SINR of receiving signal from UE i at UE j is given by where N 0 is the additive white Gaussian noise power at each UE.Next, we characterize an average interference based on the spatial distributions of the simultaneous transmitting UE within the clusters.
2) Interference From One Cluster: According to the floor control in the MCPTT group communications, the UE of a cluster work in a delay and served manner so that, at most, one UE is transmitting on the shared resource blocks.For simplicity of notation, we let a cluster and the cluster head inside have the same index.Given that a cluster k is active due to a UE is transmitting, let E k denote the event that cluster head k is transmitting and E k denote the event that a cluster member of k is transmitting.The Laplace transform of the interference from cluster heads k to j given E k is written as where a kj is the distance between cluster heads k and j as shown in Fig. 2. When the members of cluster k need to transmit, at most, one UE is active.Hence, the Laplace transform of the interference from a member, which is uniformly distributed in Fig. 2. Interference is coming from the head or member of cluster k when cluster head i is transmitting a message to cluster head j.
cluster k, to the head of cluster j given E k is written as where is the distance between a member of cluster k and the head of cluster j by using the Law of Cosines as shown in Fig. 2, and b k is the distance between cluster head k and a possible location of a member of cluster k; G = G e P d (λ/4π) 2 .
A closed-form expression for (3) exists as η = 2, which is given by ( For η > 2, instead of using (4), L m I (s | a kj ) can also be achieved by numerically evaluating a single integral, which is written as where and c 1 (r d ) is defined by ( 9), shown at the bottom of this page.
The Laplace transform of the interference from cluster k to cluster head j is written as where P {•} denotes the probability of an event.Assuming that the head and members of cluster k statistically have equal opportunities to access the resource blocks, P {E k } = 1/(1 + m k ).P {Q k } is the blocking probability that the resource blocks have already been occupied by another UE when a UE requests its transmission.A detail discussion about blocking probability is given in Section II-C2.Therefore, the Laplace transform of the interference in (10) considers both the spatial distributions and transmission activities of UE within clusters, and it is used to characterize the connection probability as follows.
3) Connection Probability: The connection probability of a link is the likelihood that the SINR of the link is higher than a predetermined threshold ζ = 2 t − 1 with link spectrum efficiency t.The connection probability from cluster heads i to j, can be written as a kj (11) where C denotes the indices of all clusters.
According to (10), the number of UE and their transmission requests within a cluster will affect the amount of interference to other links; consequently, it will affect the connection probability given by (11).Meanwhile, the request to transmit a message will be queued if the resource blocks are not immediately available.Therefore, it is of great importance to study the delay in an MCPTT network for route selection.

C. Delay Modeling 1) LCD System:
The flow control in MCPTT group communications can be modeled as an LCD system where the incoming transmission request is held in a queue until an in-progress transmission terminates and the resources blocks become free [16].The blocking probability is used to characterize the likelihood that the resources blocks are already in use when a new call request happens.If no resource blocks are immediately available after a call request to a cluster, it will be helpful to know the level of confidence that a call can be served within τ seconds.
2) Blocking Probability: Let Q i denote the event that a call is delayed in the queue of cluster head i.The blocking probability of cluster head i is given by the Erlang C formula as follows: where A i = (m i + 1)λ u i /μ is the traffic intensity, μ = 1/τ hold is the mean service rate, and τ hold is the average duration of a call; and N is the number of servers.0 < A i < 1 is required so that the system is stable; otherwise, it would result in a queue which grows unboundedly.For the group communications of MCPTT, the resource blocks are utilized by a UE per unit time.
Therefore, the number of servers N for a queue is 1.Consequently, P {Q i } = A i , and ( 10) is expanded as 3) Timeliness Probability: Meanwhile, if no resource blocks are immediately available, the call will be delayed.In a steady state, the delay time for a call request in the queue obeys exponential distribution [25].Let the random variable W i denote the delay of a user in cluster i, the probability of a call waiting for more than τ s in the queue is given by where τ is the delay time of interest.Hence, the timeliness probability that a caller needs to wait for at most τ s to be served is given by Thus, we define the timeliness probability from cluster heads i to j as D ij = D j .Apparently, if the selected threshold τ is 0, the timeliness probability will be solely decided by 1 − A i , which is a linear function of m i .Fig. 3 gives the timeliness probabilities in cluster i under different values of delay threshold (τ ) and the average number of calls per unit time (λ u i ).It is clear that as the number of users increases, the timeliness probability drops heavily.The timeliness probability can be grown by either increasing τ or decreasing λ u i .

D. Trust Modeling
Trust is an expectation of technically competent role performance.It is expected to play a major role for secure message forwarding.As we seen in Fig. 4, an attacker (or a misbehaving node) can modify the messages while it is forwarding.This would create confusion and misjudgment at the receiver.To rate the trustworthiness of a link, we adopted the beta reputation system to the binary rating on whether each forwarding operation (interaction) is redeemed to be successful or not.

1) Beta Reputation System and Mean Trust Probability:
The expression for the posterior probability estimation of binary events can be represented as beta distribution functions [20].Let α ij − 1 denote the number of previously successful direct forwarding operations from cluster heads i to j, while β ij − 1 denote the number of previously failed operations.The probability density function of observing the successful forwarding operations in the future is a beta function given by where θ ij ∈ [0, 1] denotes the probability of observing successful forwarding operations in the future, and shape parameters α ij , β ij > 0. The mean trust probability from cluster heads i to j is the probability expectation value of the reputation function, which is written as where O denotes the information about the observed interactions between cluster heads i and j until now.When the number of interactions is very large, T mean ij is the accurate mean trust probability.
2) Confidence of Trust and Robust Trust Probability: In an MCPTT network, two cluster heads may have very few interactions due to lack of forwarding interest, early stage of a network running, and network topology changes.Thus, adequate observations are missing to calculate the accurate mean trust probability.The confidence intervals of trust can be used to infer the robustness of the trust from a limited number of observations.
A confidence interval c of θ ij makes sure that θ ij will lie in a probability range with c × 100% certainty.According to [27], the Bayesian-based lower confidence bound for θ ij is given by where Beta −1 (c; a, b) is the cth quantile function of a beta distribution, i.e., a beta inverse cumulative distribution function, with shape parameters a and b.Compared with traditional confidence interval estimators such as the normal approximation and the Clopper and Pearson approach, Bayesian confidence technique is more capable of providing a mean level of coverage close to the nominal level even for a small-to-intermediate number of interactions [27]. 1 Since the accurate mean trust probability can reach as low as T lower ij for a given confidence interval c, we define this lower bound as the robust trust probability T robust ij .As a result of the above discussion, we have formulated two trust probabilities, namely, mean and robust trust probabilities.The latter has considered the uncertainty existed in the proportion of success interactions in a small number of observations.Fig. 5 gives the confidence of the trust under a different number of forwarding operations.It is clear that although the mean trust probabilities are the same, the robust trust probabilities vary.Thus, the robust trust probabilities can help to differentiate the links with the same mean trust probabilities.Meanwhile, robust trust can also give different ranking results when the mean trust probabilities are not the same.= 0.5774, which implies that the link from cluster heads 1 to 6 is more robust than the link from cluster heads 1 to 5 in terms of trust even though the latter has a better mean trust probability.

III. ROUTE SELECTION
In this section, we consider a route selection method where the selection process explicitly takes into account the connectivity, delay, and trust of an MCPTT network.We assume that the connection, timeliness, and trust probabilities of each link are exchanged among the cluster heads.The cryptographic primitives are used to authenticate and protect the integrity of the data [21].In many multicriteria route selection algorithms, the criteria are normally weighted and mapped to the costs of the links.Although the measurements of the connectivity, delay, and trust are all probability values, the distributions of them are different.Hence, it is challenging to find proper weights to the probabilities.To make progress, we propose the following steps to achieve the costs of links for route selection.
Step 1: Based on the exchanged probability values regarding the connectivity, delay, and trust among the cluster heads, the source cluster head forms three adjacency matrices, which are given by where each element of the matrices represents the relationships between two links regarding connection, timeliness, and trust probabilities; the elements on the diagonal of the matrices in ( 20) are set to be 1.
Step 2: Define H t , D t , and T t as the thresholds of the connection, timeliness, and trust probabilities, respectively.To define the thresholds that are appropriate to the environment, a general guideline is given as follows for reference: Threshold H t can be calculated based on the pairwise connection probability in (13) with the exchanged information on clusters and their locations.Among the calculated connection probabilities, the threshold is chosen such that a predefined percentage of the nodes scores above that of threshold.Threshold D t can be obtained by (15), where the average duration of a call τ hold is achieved by previous experience.Threshold T t can be given by ( 19) with the anticipated interactive results on α and β that is controlled by risk assessment.A criterion from the connectivity, delay, and trust is specified as the basic criterion for route selection, and the route selection is also constrained by the thresholds of other two criteria.
Step 3: For the route selection strategy based on connectivity, the following requirements must be satisfied: If T ij ≤ T t or D ij ≤ D t , the cost of the directional link i, j will be 0; otherwise, the cost of the same link will be H ij .The resultant adjacency matrix for the cost of links is given by Algorithm 2: The interface of Dijkstra's algorithm.1:function J = Dijkstra (C, l 1 , l n ); Input : C: adjacency cost matrix; l 1 : index of source; l n : index of destination.Output : J: the path with minimum end to end costs. where The similar method will apply to delay or trust if either of them is the basic criterion for route selection.
We propose to find a path with the maximum end-to-end probabilities.For a given path J = {l 1 , l 2 , . . ., l n }, l 1 is the index of source forwarding UE, and l n is the index of final (destination) forwarding UE; n is the number of forwarding UE involved, and n ≥ 2. The end-to-end probability of a route is the product of every independent probability of the links consisting of the path.The end-to-end connection, timeliness, and trust probabilities of a path are, respectively, defined as Fig. 6.Route selection results based on connectivity and constrained by T t and D t for a snapshot of a randomly deployed MCPTT clusters.The trust is calculated based on the mean trust probability.
Many route algorithms have been proposed to find the shortest path [25].Among these algorithms, the Dijkstra's algorithm is designed to find a path with the minimum sum of link costs.To find a path with the maximum multiplication of probabilities with the Dijkstra's algorithm, we input − log R with the indices of source and destination cluster heads to the Dijkstra's algorithm.The steps of the route selection method and the interface of the Dijkstra's algorithm are summarized in Algorithms 1 and 2, respectively.

IV. SIMULATION AND DISCUSSIONS
In this section, we provide simulations to validate the route selection method detailed above.The parameters used in the simulation are given by Table II.The given link spectrum efficiency and bandwidth offer a target rate of 1.2 M/ps for group communications, which aims to fulfill the speed requirement for forwarding recorded messages and even holding a highdefinition video call (720 p, 30 frames/s by H.264 profile).The average holding time of 6 s is long enough for sending approximately three pictures of the scenes taken by an 8-megapixel mobile phone each time at 1.2 Mb/s.There are 16 MCPTT clusters deployed on the plane.The cluster 1 at (0, 0) needs to send the messages to cluster 16 at (1000, 1000).The rest of the clusters are randomly deployed and could cooperate to forward the messages.For a given snapshot of the randomly deployed clusters, the route selection method is run to get a series of cluster heads for forwarding messages.The connection, timeliness, and trust probabilities of the links are exchanged among the cluster heads by cryptographic tools [21].The simulations results are obtained by averaging over 10 4 independent Monte Carlo trials.Fig. 6 shows the route selection results for a snapshot of a randomly deployed MCPTT clusters.The route selection is based on the link connection probabilities and constrained by T t and D t .Table III gives the coordinates of the cluster heads and the number of cluster members.The selected paths and their end-to-end connection probabilities are listed as follows.
2) Path J 2 = {1, 4, 14, 3, 6, 16}, and P H J 2 = 0.8975.3) Path J 3 = {1, 5, 14, 10, 9, 7, 16}, and P H J 3 = 0.9512.4) Path J 4 = {1, 4, 14, 3, 6, 16}, and P H J 4 = 0.8926.Comparing J 1 with J 2 , the higher the threshold D t , the smaller the end-to-end connection probability.This is because, as D t increases, the cluster heads that can form paths with better end-to-end connection probabilities may not be chosen if the consisting links by the heads could not satisfy a higher timeliness probability threshold.The similar discussion could be applied to J 3 and J 4 .J 2 and J 4 which are forking from cluster 1 to clusters 2 and 4 separately imply that cluster head 4 could not be chosen as a forwarding head as T t = 0.9, because P H J 2 > P H J 4 and the rest links of the two paths are the same.The same applies to the forking from cluster head 10 to cluster heads 3 and 9 for J 1 and J 3 .Fig. 7 gives the route selection results for a snapshot of a randomly deployed MCPTT clusters.The route selection is based on link timeliness probabilities and constrained by T t and H t .H 1,16 is set to be smaller than H t so that there is no direct route from source to destination.Thus, the delay and trust among the forwarding UE need to be considered.According to (15), the delay of the links to cluster i is relying on m i .The paths and their end-to-end timeliness probabilities are listed as follows.From J 1 to J 3 , one can easily see that the end-to-end timeliness probability is decreasing as H t is increasing.It further implies that if there are several clusters which have the same least number of cluster members and also meet the thresholds requirements for route selection, one route formed by them with the least number of hops will be selected.Besides, the reason that J 1 is selected when T t = 0.5, H t = 0.75 instead of J 4 , although they have the same number of users within, is due to the implementation of Dijkstra's algorithm.
Fig. 8 gives the route selection results based on link trust probabilities and constrained by D t and H t .Among the given paths, J 1 and J 2 are selected based on mean trust probabilities and J 3 and J 4 are based on robust trust probabilities with c = 0.95.J 1 and J 3 shows that two-hop routes are available for route selection, and the route via cluster 14 gives the best end-to-end trust probabilities for both the mean and robust trust probability formulations.J 2 and J 4 show that the routes selected given by the mean and robust trust probabilities could also give different results even though their values of D t and H t are the same.
Fig. 9 gives the averaged end-to-end connection and timeliness probabilities after route selections.The proposed route selections are based on connection probabilities and constrained by T t only (i.e., D t = 0).The average number of users within each cluster is 2. For a snapshot of the random clusters, an increasing number of cluster heads coming consecutively from clusters 2 to 15 are labeled as malicious, and the route selections are performed.A head k is regarded as malicious if both T j k and T kj are less than T t for any j = k.To compare, a benchmark of route selection solely based on connectivity [14] is given.When there is only one malicious node on the plane, the end-to-end connection probability for both the benchmark and the proposed route selection method are around 97.5%.As all cluster heads in clusters 2 to 15 become malicious nodes, there is a 5% performance drop in the end-to-end connection probability for the proposed method.Although, the benchmark achieves the best end-to-end connection probability, it does not consider the malicious node which makes the path susceptible to attacks.Fig. 9 also shows that the end-to-end connection and timeliness probabilities decreases and increases, respectively, as the number of malicious cluster heads increases.This is because, as the number of malicious cluster heads increases, the cluster heads that can form paths with better end-to-end connection probabilities have a high chance to become malicious.Hence, they are not available for route selection.The increasing number of malicious cluster heads impose the reduction of the average number of hops for a selected path.For a given m i , the delay for each cluster is the same.Therefore, as the average number of hops decreases, a higher end-to-end timeliness probability is achieved.

V. CONCLUSION
In this paper, we have proposed a novel route selection method based on connectivity, delay, and trust for a mission-critical public safety network.We have characterized the connection probability, where the amount of interference considered the statistics of call activities within each cluster.The robust trust probability based on the Bayesian-based lower confidence bound has been formulated.The simulations have shown that compared with single criterion route selection, the proposed route selection method offers versatile solutions with different basic criteria and thresholds on connection, timeliness, and trust probabilities.The academic implications, limitations, and future studies are given as follows.
Academic implications: First, this research has demonstrated that connectivity, delay, and trust are all crucial to the missioncritical system since the system is susceptible to connection outages, delays, and attacks.Second, by incorporating the delay model into interference and connection probability analysis, the effect of the activity of the group communications on the connectivity has been captured.Third, the trust probability based on the lower confidence bound could provide a more robust result than the mean trust probability especially when the number of interactions is small.
Limitations: This paper considers three crucial criteria for route selection, although there might be other uncovered but important criteria for certain applications.If they can be formulated in a probabilistic form, then they can be fitted easily into the proposed route selection method in this paper.Moreover, the trust evaluation may not always be available since the observing UE could be busy while the next forwarding UE is forwarding its messages.
Future Studies: In this paper, the group members are uniformly distributed within the clusters, but it would be interesting to study other types of clusters such as clusters with Gaussian distributed members.Meanwhile, it is worthwhile to investigate how to carry out the trust rating when the observing UE is busy and cannot witness the forwarding behavior of the next forwarding UE.
Thus, the proposed route selection method would be well suited to 5G public safety and disaster relief communication networks.

Fig. 3 .
Fig. 3. Timeliness probability of cluster i under different parameters, and τ h old = 6 s.

Fig. 5 .
Fig. 5. Distribution of observing successful forwarding operations in the future after the observations so far given by (α ij , β ij ).

Fig. 7 .
Fig. 7. Route selection results based on link timeliness probabilities and constrained by T t and H t for a snapshot of a randomly deployed MCPTT clusters.The trust probability is the same as mean trust probability.The trust is calculated based on the mean trust probability, and H 1 , 16 < H t .

Fig. 8 .Fig. 9 .
Fig. 8. Route selection is based on link trust probabilities and constrained by D t and H t for a snapshot of a randomly deployed MCPTT clusters, and H 1 , 16 < H t .
| a k j ) Laplace transform of the interference from heads k to j given E k ij Timeliness probability from cluster heads i to j α ij − 1 Number of successful interactions from heads i to j β ij − 1 Number of failed interactions from heads i to j θ ij Probability of successful interactions T m ean ij

TABLE III COORDINATES
OF CLUSTER HEADS AND NUMBER OF CLUSTER MEMBERS