I Introduction
Many emerging applications depend on the collection and delivery of status updates between a set of ground terminals and a central terminal using mobile agents. Examples include: measuring traffic at road intersections [1], temperature, and pollution in cities [2], ocean monitoring using underwater autonomous vehicles [3], and surveillance using UAVs [4]. All of these applications depend upon regular status updates, that are communicated in a timely manner, so as to keep the central terminal and the ground terminals updated with fresh information.
Age of Information (AoI) is a newly proposed metric that captures timeliness of the received information [5, 6, 7]. Unlike packet delay, AoI measures the lag in obtaining information at the destination node, and is therefore suited for applications involving gathering or dissemination of time sensitive updates. Age of information, at a destination, is defined as the time that elapsed since the last received information update was generated at the source. AoI, upon reception of a new update packet, drops to the time elapsed since generation of the packet, and grows linearly otherwise.
We consider the problem of AoI minimization in gathering and dissemination of information updates, between a set of ground terminals and a central terminal. The information updates can be as small as a single packet containing temperature information or a high fidelity image or a video file. The ground terminals are equipped with low power transmitters, and a mobile agent is used to gather and disseminate information.
The age or freshness of information gathered and disseminated depends on the trajectory of the mobile agent, whose mobility is constrained to a mobility graph . The mobile agent can move from ground terminal to ground terminal only if . This model can be used to capture the fact that the agent may not be able to move between any arbitrary locations due to topological limitations.
The problem of persistent monitoring in dynamic environments has been considered in [8, 9, 10] using tools from optimal control. These works focus on minimizing uncertainty when source locations are time varying, rather than timely monitoring over a fixed set of locations. Minimizing delay in a similar setting with packets arriving randomly in space and time has been considered in [11]. There has also been work on trajectory control of a mobile agent for minimizing transmission energy in sensor networks [12].
Closer to our work are [13] and [14], in which some approximation trajectories to minimize maximum latency on metric graphs were proposed. In [15], the authors consider trajectory planning for a mobile agent to minimize AoI. They obtain the best permutation of nodes for the mobile agent to visit in sequence, given Euclidian distances between the nodes. In our work, mobility is constrained by a general graph , and we seek the optimal trajectory over the space of all trajectories allowed on this graph , not just permutations of nodes. To the best our knowledge, this is the first work to consider the AoI minimization on general mobility graphs , and provide polynomial time approximation algorithms.
In the information gathering problem, we consider the design of trajectories for the mobile agent to minimizes peak and average age, two popular metrics of AoI. We first consider the space of randomized trajectories, in which the mobile agent traverses edges according to a random walk on the mobility graph
. We show that a randomized trajectory is in fact peak age optimal, and that it can be obtained in polynomial time using the MetropolisHastings algorithm. We then prove that solving for the average age optimal trajectory is NPhard, in a symmetric setting, and propose a heuristic randomized trajectory that is simultaneously peak age optimal and factor
average age optimal, where is the mixing time of the randomized trajectory on . The factor can scale with the graph size, especially if the graph is not well connected. Thus, we propose an agebased trajectory, in which the mobile agent uses the current AoI to determine its motion, and show that it is factor optimal in a symmetric setting.In the information dissemination problem, the central terminal sends updates for each ground terminal via the mobile agent. The mobile agent queues these update packets in a firstcomefirstserve (FCFS) queue, and delivers them to the respective ground terminal when the mobile agent reaches it. The FCFS queue assumption is motivated by uncontrollable MAC layer queues, where the generated updates get queued for transmission [16, 7]. We, now, not only have to design the trajectory of the mobile agent, but also determine the optimal rate at which the central terminal generates information updates for each ground terminal. We show that the peak age optimal randomized trajectory of the information gathering problem, along with a simple update generation rate, is at most a factor optimal, in both peak and average age. Also derived is an explicit formula for peak age of the discrete time Ber/G/1 queue with vacations, which may be of independent interest.
Ii System Model
We consider a central terminal that needs to communicate with a set of ground terminals . The ground terminals are equipped with low power, low range radio communication devices, and cannot directly communicate with the central terminal, or with each other. An autonomous mobile agent , is used as a relay between the central terminal and the ground terminals, while moving across the geographical region where the ground terminals are spread.
The mobility of the agent is constrained by a mobility graph , where can travel from ground terminal to ground terminal only if . The graph , thus, constraints the set of allowable moves. We consider a timeslotted system, with slot duration normalized to unity. In the duration of a timeslot, the mobile agent stays at a ground terminal to gather or disseminate information, and moves to any of its neighbours in for the next timeslot. The mobility graph can be constructed from the limitations of a slot duration, distances between ground terminals, and speed of the mobile agent.
We consider two problems: information gathering and information dissemination. In the information gathering problem, every time the mobile agent reaches a ground terminal , the ground terminal sends a fresh update to the mobile agent, which is immediately relayed to the central terminal. The age , at the central terminal, for the ground terminal drops to . When the mobile agent is not at the ground terminal , the age increases linearly. See Figure 1. The evolution of in the information gathering problem can be written as:
(1) 
where denotes the location of the mobile agent at time . Note that the age evolution depends on the trajectory that the mobile agent follows on the mobility graph .
In the information dissemination problem, the central terminal generates updates for each ground terminal. The generated updates are then transmitted to the mobile agent. The mobile agent queues updates received from the central terminal in a set of FCFS queues, one for each ground terminal. The mobile agent delivers the headofline update in queue , to ground terminal , when it reaches . The central terminal has no control over the FCFS queues on the mobile agent, however, it can control the update generation rate , for each ground terminal .
The age , at the ground terminal , increases by every time the mobile agent is not at , or when it is at but has no updates to deliver. Otherwise, a successful delivery of the headofline update occurs in time slot , and the age drops to the age of the headofline update in queue . See Figure 2. This evolution of age can be written as:
(2) 
where is the time of generation of the head of line packet in queue , at time , and denotes the set of packets in the mobile agent’s queue at time .
Iia Age Metrics
AoI is an evolving function of time. We consider two time average metrics of AoI. Average age, for ground terminal , is defined as the time averaged area under the age curve:
(3) 
In Figures 1 and 2, we see that the age peaks before a fresh update is delivered. In the information gathering case, a fresh update is delivered every time the mobile agent visits , i.e. . Whereas, in the information dissemination case, a fresh update is delivered whenever and the queue . The peak age , for ground terminal , defined as an average of all the peaks in the age evolution curve , can be written as
(4) 
in the information gathering case and
(5) 
in the information dissemination case.
We define the network peak and average age to be
(6) 
where are weights representing the relative importance of a ground terminal . Our goal is to minimize network peak and average age.
IiB Trajectory Space
We use to denote a reasonably large space of trajectories:
where denotes the fraction of timeslots, the trajectory , is at ground terminal :
(7) 
For a trajectory , the limit (7) exists and is positive for all . This requirement is to ensure that the peak and average age are both finite and well defined.
Peak and average age depend on the trajectory . We use and to denote network peak and average age, respectively, for .
Iii Information Gathering
In this section, we consider the problem of information gathering. We define optimal peak and average age to be
(8) 
where denotes the space of all trajectories for the mobile agent.
We first consider randomized trajectories, where the mobile agent moves according to a random walk on the mobility graph. We shall show that for peak age optimality, such randomized trajectories suffices. We then show that the average age optimization is NPhard, and propose a heuristic randomized trajectory. In Section IIID, we propose an agebased trajectory for better average age performance.
Iiia Randomized Trajectories
We start by defining the class of randomized trajectories:

A trajectory , on mobility graph , is said to be a randomized trajectory if
is an irreducible Markov chain defined by a transition probability matrix
:(9) for all and , where for .
For convenience, we shall refer to , defined above, as the randomized trajectory , where to denote the matrix with entries . Note that is the probability that the mobile agent, when at ground terminal , moves to ground terminal for the next time slot. The constraint: for , ensures that the randomized trajectory adheres to the mobility constraints defined by .
We assume in the definition of a randomized trajectory , that is an irreducible Markov chain over the state space . This is desired, since the mobile agent has to traverse through all the nodes, repeatedly, for a positive fraction of time, or otherwise the resulting peak and average age would be unbounded.
For any randomized trajectory , we obtain explicit expressions for network peak and average age. We use the notation and to show explicit dependence of peak and average age on the randomized trajectory .
Theorem 1
The network peak and average age for a randomized trajectory is given by
(10) 
where is the unique stationary distribution obtained by solving and are diagonal elements of the matrix , where is an matrix with entries .
IiiB Peak Age Minimization
We first formulate the peak age minimization problem over the space of randomized trajectories. We shall see that a peak age optimal randomized trajectory suffices for optimality over the space of all trajectories.
Using the results in Theorem 1, we can write the peak age minimization problem over the space of randomized trajectories as:
(11) 
Note that characterizes a randomized trajectory, while is the unique stationary distribution associated with it.
This problem is difficult to solve because the irreducibility constraint cannot be expressed in a simple, solvable manner. Further, relaxing the irreducibility constraint can yield a trivial solution like , which are neither irreducible nor anywhere close to optimal.
However, the problem (11) can be transformed to finding an irreducible , with a given stationary distribution. This is a simpler problem and can be solved using the MetropolisHastings algorithm.
Lemma 1
Let , for all , to be a distribution on , and a randomized trajectory satisfy . Then, solves (11).
Proof:
See Appendix B.
Lemma 1 implies that a randomized trajectory , that satisfies , is a peak age optimal, over the space of all randomized trajectories. We now construct one such randomized trajectory: for given in Lemma 1, define a MetropolisHastings randomized trajectory :
(12) 
where
(13) 
and equals the out degree of terminal in the mobility graph . It is known that such a randomized trajectory satisfies [17]. We, therefore, have the following result.
Theorem 2
The MetropolisHastings randomized trajectory solves (11), i.e. it is peak age optimal over the space of all randomized trajectories.
We considered randomized trajectories, where the mobile agent moves from terminal to with probability . We now show that, for peak age optimality, such a randomization suffices.
Theorem 3
The MetropolisHastings randomized trajectory is peak age optimal over the space of all trajectories , namely .
Proof:
We establish a more general result. Namely, any randomized trajectory which satisfies , where , is peak age optimal over the space of all trajectories:
To prove this, it suffices to argue that the peak age for any trajectory is lower bounded by , where is as given in Theorem 2. We show this in Appendix C.
Thus, we are able to obtain a peak age optimal trajectory, namely . Further, the matrix can be computed in polynomial time; in time. Therefore, the peak age minimization problem is solved in polynomial time.
IiiC Average Age Minimization
We now consider the average age minimization problem. We first argue that in the symmetric setting, namely ,^{1}^{1}1The weights only measure relative significance of ground terminals. Thus, setting is equivalent to setting . the average age minimization problem is NPhard
Theorem 4
The problem of finding an average age optimal trajectory is NPhard in the symmetric setting of .
Proof:
See Appendix D.
Since solving the average age minimization problem is hard, we derive a lower bound on average age. Intuitively, if the mobility graph is better connected then it should yield a lower age. This is because a better connected mobility graph imposes fewer restrictions on mobility. The following result obtains a lower bound on network average age by comparing it with the network average age of a complete graph.
Theorem 5
For any trajectory , the network average age is lower bounded by
(14) 
where for all .

See Appendix E.
Note that the term is nothing but the optimal peak age ; see Theorem 3. Furthermore, the lower bound in Theorem 5 is independent of the trajectory . Therefore, we get
(15) 
where is the space of all trajectories. It must be noted that a similar result was derived in the case of link scheduling for age minimization in [7]. The similarity of the result is rooted in the fact that the information gathering problem in the complete graph case is equivalent to the link scheduling problem in [7], in which at most one link can be activated simultaneously.
IiiC1 A Heuristic Randomized Trajectory
Motivated by the peak age optimality results of the previous section, we restrict ourselves to the space of randomized trajectories, and propose a heuristic, called the fastestmixing randomized trajectory, and prove an average age performance bound for it.
Using the results in Theorem 1, the average age minimization problem over the space of randomized trajectories can be written as
(16) 
Here, is the randomized trajectory and the unique stationary distribution corresponding to . Solving (16) can be computationally complex. Not only do we have the irreducibility constraint, but also a nonlinear constraint in .
We next upper bound the network average age, for any randomized trajectory of the mobile agent. We first define mixing time for a randomized trajectory.
To do this, we first discuss the notion of stopping rules and stopping times in a Markov chain. A stopping rule is a rule that observes the walk on a Markov chain and, at each step, decides whether or not to stop the walk based on the walk so far. Stopping rules can make probabilistic decisions and therefore the time at which the walk stops, called the stopping time, is a random variable.

The hitting time from state distribution to on a Markov chain is the minimum expected stopping time over all stopping rules that, beginning at , stop in the exact distribution of . In other words, it is the expected number of steps that the optimal stopping rule takes to move from to . This is denoted by . The mixing time of a Markov chain is then defined as
(17) where is the collection of all distributions on and is the stationary distribution of . In other words, it is the expected time taken to reach stationarity using the optimal stopping rule and starting at the worst initial distribution.
Lemma 2
The network average age for a randomized trajectory is upper bounded by
(18) 
where denotes the mixing time of the randomized trajectory .
Proof:
First, we define the quantity , called the discrepancy of the randomized trajectory . This definition implies that Thus, we get the following upper bound:
(19) 
However, from [19] we know that , where is the mixing time of the randomized trajectory . Thus, we have the required result
where the last equality follows from Theorem 1.
We use this relation and suggest the following heuristic for minimizing age: Find the fastest mixing randomized trajectory on the mobility graph that minimizes peak age.
From the proof of Theorem 3, we know that for a randomized trajectory to be peak age optimal all we need is , where is the stationary distribution of . It, therefore, suffices to find that satisfies , and simultaneously minimizes the mixing time . We call this the fastestmixing randomized trajectory, and use to denote it. The following result provides a way to obtain by solving a convex program.
Theorem 6
The fastest mixing randomized trajectory can be found by solving the following convex optimization problem:
(20) 
Here denotes the spectral norm of matrix and .
Proof:
See Appendix F.
This convex program (20) finds a randomized trajectory on that is closest to the stationary randomized walk , in the spectral norm sense. Also, is peak age optimal on graph , since it satisfies Note that, the problem (20) can be solved in polynomial time by converting it to a semidefinite program [20].
We now bound the average age performance of the fastestmixing randomized trajectory.
Theorem 7
The network average age of the fastestmixing randomized trajectory is at most factor away from the optimal average age:
(21) 
where is the mixing time of .

See Appendix G.
To see the usefulness of the fastestmixing randomized trajectory, and Theorem 7, consider a random geometric graph . The graph consists of nodes spread over a unit square with a link between every two nodes that are within a distance . If is the physical speed of the mobile agent, then must equal , where is the slot duration. We know that mixing time of is , and therefore, the fastestmixing randomized trajectory would be at most factor optimal. For highly connected graphs, such as Dirac graphs in which the degree of each node is at least , we have constant factor of optimality; since the mixing times are [21] establishes a connection between the existence of long paths in graphs and their mixing times.
IiiD Agebased Trajectories
In the last two subsections, we proposed two randomized trajectories, namely and . Both were peak age optimal, while the latter was also factor average age optimal. We also noted that solving the average age problem is generally hard. We now propose an agebased trajectory which can be constant factor age optimal.
In the symmetric setting, where , we observe that the agebased trajectory is a repeated depthfirst traversal of the mobility graph . This can be verified easily when the mobility graph is a tree. Consider the tree in Figure 3, and assume that we start at the root node 1. The trajectory of the agent following the rule described above would be This is precisely the depthfirst traversal of the tree graph.
In the symmetric setting, where , we now prove that the agebased trajectory is factor2 optimal.
Theorem 8
In the symmetric setting , the network average age for the agebased trajectory is bounded by
(23) 
for any increasing function .
Proof:
See Appendix H
This agebased policy can be implemented in an online fashion if the mobile agent has access to age of the neighboring terminals. The complexity of implementing this trajectory is then at most linear in the timehorizon and .
Iv Information Dissemination
We now consider the information dissemination problem. The central terminal generates updates for every ground terminal , at rate , according to a Bernoulli process. The generated updates for the ground terminal are sent to the mobile agent, which get queued in the th FCFS queue. The mobile agent follows a trajectory , and delivers the headofline update in queue to terminal , when it reaches it.
Our objective is to minimize the network peak age and average age over the space of update generation rates and all trajectories :
(24) 
where denotes peak age and denotes the average age of terminal . Their evolution is given by (2). For convenience, we have omitted their explicit dependence on and .
Motivated by the results for the information gathering problem, we consider randomized trajectories. Note that an arriving update in queue has service time equal to the intervisit times to ground terminal , provided the update arrived when the queue was notempty; . However, when an update arrives to an empty queue , the time to delivery is not the intervisit time, and depends on the location of the mobile agent at the time of arrival.
Since the analysis of age for such a queueing system may be difficult, we provide an upper bound, by comparing the the th queue with a discrete time Ber/G/1 queue with vacations: whenever the th queue is empty pretend that it goes on a vacation, with vacation times having the same distribution as intervisit time; otherwise the service times for the queue are just intervisit times. Clearly, the age process of such a FCFS queue is an upper bound for the age process . Thus, we upper bound the peak age and average age , by the peak and average age of this Ber/G/1 queue with vacations. We first analyze peak and average age of a Ber/G/1 queue with vacations.
Iva Age for Ber/G/1 Queue with Vacations
Consider a discrete time FCFS Ber/G/1 queue with vacations, where an arrival occurs with probability , the service times are generally distributed with mean , and the vacation times are also generally distributed.
We obtain an expression for the peak age of a discrete time Ber/G/1 queue with vacations, and a bound on average age.
Lemma 3
The peak age for a discrete time FCFS Ber/G/1 queue with vacations is given by
(25) 
where , while the average age is upperbounded by peak age, namely .
Proof:
The peak age for a FCFS queue is given by
(26) 
where denotes the time an update spends in the queue and is the interarrival time between two updates. Given that vacation times are distributed i.i.d according to random variable , we have
(27) 
where denotes the service time distribution. Substituting this and in (26), we obtain the expression for peak age. For the derivation of average system time , see Appendix I.
The upperbound on average age directly follows from the observation that total time spent in the queue is negatively correlated with interarrival times. For details, see Appendix I.
IvB Age Minimization Problem
Using Lemma 3, we now obtain an upperbound on both, network peak and average age, for a given randomized trajectory and update generation rates .
Lemma 4
For a randomized trajectory and packet generation rates , the peak and average age for a ground terminal is upperbounded by
(28) 
for all , where is the unique stationary distribution of , ,
is a matrix with all rows equal to the stationary distribution vector
, and .Proof:
See Appendix J.
We propose a policy, i.e. a randomized trajectory and update generation rate , that minimizes the age upperbound :

Separation Principle Policy

Mobile agent follows the randomized trajectory obtained by solving (20).

Generate updates for the ground terminal at rate
(29) where and are diagonal elements of the matrix .

We call it the separation principle policy for two reasons. Firstly, is the fastestmixing randomized trajectory, which we proposed for minimizing average age in the information gathering problem. Secondly, the update generation rate for the ground terminal , depends only on and , which are functions of the first and second moments of the return times to terminal under trajectory :
where denotes the return time to terminal , starting from , under the fastest mixing randomized trajectory . We now bound the performance of this separation principle policy.
Theorem 9
The peak and average age of the separation principle policy is bounded by
where is the mixing time of the randomized trajectory .
Proof:
See Appendix K.
V Simulation Results
We test the performance of our proposed trajectories on three different kinds of mobility graphs: random geometric graphs ,^{2}^{2}2Setting for random geometric graphs ensures connectivity w.h.p. grid graphs with diagonal edges, and 3connected ring or cycle graphs; see Figure 4. We use to denote the number of ground terminals, namely . For the agebased policy, we set the function , inspired by the index based policies in [7]. Link weights are picked uniformly at random from the interval in an independent manner. We run our simulations for a total of
timeslots, to get a good estimate of the peak and average age.
We first consider the information gathering problem, and plot peak and average age for all the proposed trajectories of the mobile agent: the MetropolisHastings randomized trajectory , fastest mixing randomized trajectory , and agebased trajectory. Figure 5 plots peak age as a function of network size for the random geometric graph . We observe that the peak age for all the three proposed trajectories match. We know from Theorems 3 and 6 that that the two randomized trajectories, namely, the MetropolisHastings randomized trajectory and the fastest mixing randomized trajectory , are both peak age optimal. Figure 5, therefore, suggests that even the agebased trajectory for the mobile agent is peak age optimal.
In Figure 6 we plot the average age performance of the proposed trajectories, as a function of network size . Also plotted is the lower bound for average age derived in Theorem 5. We see that the agebased policy is nearly average age optimal, while the fastest mixing randomized trajectory performs slightly better than the MetropolisHastings randomized trajectory .
Theorem 7 proved that the fastest mixing randomized trajectory is at least factor optimal. Figure 6 validates this conclusion: for example, for ground terminals, the average age for the fastest mixing randomized trajectory is approximately a factor away from the lower bound.
In Figures 7 and 8 we plot the average age performance for several proposed trajectories, as a function of the network size. The agebased policy, again outperforms the two randomized trajectories, and is nearly optimal. We observe that the average age for the fastest mixing randomized trajectory , namely , is much worse in the ring graph than in the grid graph. This is because the mixing time for the ring graph is much larger than for the grid graph. Similar observation holds in comparing and the grid graph.
In Figure 9, we simulate the performance of the separation principle policy for the information dissemination problem, for graph , and compare its age performance with the information gathering problem. We observe a significant deterioration of age, as a function of network size , in the information dissemination case in comparison to the information gathering case. This, we note, is the cost of uncontrollable queues in the system on age performance.
Vi Conclusion
We considered the trajectory planning problem for a mobile agent, that traverses through a mobility graph , to help timely exchange of information updates between a central terminal and a set of ground terminals . In the information gathering problem, we showed that a randomized trajectory, namely the fastestmixing randomized trajectory, is peak age optimal and factor average age optimal. We showed that obtaining an average age optimal trajectory can be NPhard, while we constricted the peak age optimal trajectory in polynomial time. To improve the average age, we proposed an agebased policy, and showed it to be factor average age optimal, in a symmetric setting. In the information dissemination problem, we proposed a separation principle policy, in which the mobile agent follows the fastest mixing randomized trajectory with a simple rate control. We proved that the separation principle policy is factor optimal, in both peak and average age.
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