I Introduction
Timely delivery of information updates is gaining increasing relevance with the advent of technologies such as cyberphysical systems, internet of things, and futuristic unmanned aerial vehicular networks. In unmanned aerial vehicular networks, timely delivery of status updates, such as vehicle position and velocity, may be critical to network safety [1, 2]. In internet of things or cyberphysical systems, timely delivery of sensor information can significantly improve the overall system performance [3].
Age of information (AoI) is a recently proposed metric that measures the time elapsed since the last received update was generated by the source [4, 5]. Figure 1 shows typical evolution of AoI at a destination node, as a function of time. AoI, upon reception of a new update packet drops to the time elapsed since generation of the packet, and grows linearly otherwise. Therefore, AoI is a destination node centric measure, unlike packet delay, and is more suited for applications involving dissemination of time sensitive information.
In [4], a simulation study considered a network of vehicles exchanging status updates, with an 802.11 based communication infrastructure, and showed that the AoI is minimized at a certain optimal packet generation rate. It further showed that AoI can be improved by changing the queue discipline of the MAC layer FIFO queue to lastinfirstout (LIFO). This observation was theoretically proved under a general network setting in [6]. Motivated by [4], AoI was analyzed for several queueing models [5, 7, 8, 9, 10, 11, 12].
However, age minimization for a network under general interference constraints and channel uncertainty has received very little attention. A problem of scheduling finitely many update packets under physical interference constraints was shown to be a NPhard problem in [13]. Age for a broadcast network, where only a single link can be activated at any time, was studied in [14, 15]. Preliminary analysis of age for a slotted ALOHA like random access was done in [16], and a distributed algorithm for age optimal ALOHA was only recently proposed in [17]. Age in multihope interference networks has been studied in [18].
We considered the problem of age minimization for a wireless network under general interference constraints, and time varying channel, in [19]. We considered two types of sources: active sources, which generate fresh information in every slot, and buffered sources
, which cannot generate fresh information in every slot. We showed that for a network with active sources, a stationary scheduling policy, which schedule links according to a stationary probability distribution, is peak age optimal and factor
average age optimal. We also showed that the same scheduling policy, with a certain packet generation rate control, is nearly optimal in the buffered case.In [19], however, the space of policies was limited to not using the channel state information. In this paper, we relax this assumption and consider scheduling policies which have perfect channel state information at every time slot . We limit ourselves to the active sources case, and propose two policies: virtualqueue based policy and agebased policy, which uses the current channel state information to make scheduling decisions. We show, via numerical simulations, that availability of channel state information can significantly improve the AoI performance of the network.
We prove that the virtualqueue based policy is nearly peak age optimal, up to an additive factor, while the agebased policy is at most a factor away from the optimal peak and average age. Similar result has been recently derived for another agebased policy proposed for broadcast network, in which only a single link can be activated, in [20]. Numerical simulations suggest that this bound is pessimistic, and that the proposed scheme performs much better.
In numerical simulations, we observe the benefit/utility of using channel state information in scheduling to minimize age, especially when the network has ‘high’ level of interference or ‘bad’ channel quality. We demonstrate by considering a specific network example that the gap in age performance between known channel state and unknown channel state can be as large as fold. Even though channel state information may not be perfectly available in certain network settings, this work establishes the utility of acquiring such channel state information for scheduling to minimize age. To the best of our knowledge, this is the first work to make this observation.
Ii System Model
Consider a wireless network , where denotes the set of nodes and the set of directed links. Not all links can be activated simultaneously. Thus, we call a set that can be activated simultaneously without interference as a feasible activation set. We use to denote the collection of all feasible activation sets. We consider a slotted time system, where the slot duration is normalized to unity.
We use to denote the channel process, where if the channel is in ON state at time and if the channel is in OFF state at time . The space of all channel states is given by . We consider to be independent and identically distributed (i.i.d.) across time , with , for all . We call this the i.i.d. channel process. Note that the channel process is not identically distributed across links, and that can be different for different links .
We use to denote transmission decision on link at time . if link is scheduled to transmit at time . Not all transmissions succeed even if the set of activated links is a feasible activation set due to channel uncertainties. A successful transmission occurs over link , at time , if and only if .
We consider active nodes, which transmit fresh information at every transmission opportunity. We define age , of a link at time , to be the time elapsed since the last successful activation of link . Figure 2 shows evolution of age for a link . Age reduces to upon a successful activation of link , while it increases by in every slot in which there is no successful activation of link , i.e.,
(1) 
This age evolution equation can more compactly be written as
(2) 
for all and .
We consider two popular age measures, namely, average age and peak age. Average age is the area under the age curve in Figure 2, while peak age is the average of all the peaks of the age curve. More precisely, we define average age of a link as
(3) 
and the average age of the network to be the weighted sum
(4) 
Note that the sum of all the peaks, until time , in the age curve can be expressed as . This is because only at times when age peaks. We, therefore, define the peak age to be
(5) 
for every link , and the peak age of the network to be the weighted sum
(6) 
We are interested in designing policies that minimize peak and average age.
Since both peak and average age are time average measures, performance of a policy does not depend on the initial age at time . We, therefore, assume that the system starts with for all , unless stated otherwise.
Iia Unknown Channel Case
In [19], we considered age minimization under the unknown channel case. Specifically, we considered all policies which scheduled feasible activation set at time as a function of the history
(7) 
We showed in [19] that stationary policies, which schedule links according to a probability distribution that is independent of , is in fact peak age optimal and factor average age optimal.
In stationary scheduling policies, every feasible activation set is assigned a fixed probability with which it is activated in slot , independent across slots. The probability that a link is activated in a slot is given by
(8) 
for all . These set of equations can be compactly written as . Note that an activated link may fail in successfully transmitting the packet due to channel errors. The probability of successful activation of a link in any slot is , since the scheduling decision is independent of the current channel state.
Further, notice that, if a link is successfully activated with probability in each slot, independent across slots, then the time since last transmission, i.e. age
, is geometrically distributed with rate
. In [19], we showed that this is indeed equal to the peak age of link , under stationary policy. As a result, the peak age for the stationary policy, determined by distribution , is given by , and thus, the optimal peak age is given by(9) 
The peak age optimal stationary policy is obtained by solving (9).
In the next subsection, we discuss the space of policies considered in this paper, and show how knowing the channel state affects age minimization. We argue that in the case when channel state information is available for scheduling, age better than what is given by (9) can be achieved.
IiB Scheduling Policies
A scheduling policy determines the set of links that will be activated at each time , i.e., . The policy can make use of the past history of link activations and observed channel states to make this decision, i.e., at each time , the policy will determine as a function of the set
(10) 
We consider centralized scheduling policies, in which this information is centrally available to a scheduler, which is also able to implement its scheduling decision. This assumption is consistent with that in network scheduling literature [21, 22].
To see the difference between age minimization under known and unknown channel process consider the two link example shown in Figure 3. In this example, only one link can be activated at a time. Let the weights for the two links, and the channel success probabilities be . When the channel state is unavailable the peak age minimization problem is given by (from (9)):
(11) 
Here, denotes the fraction of times link is scheduled and denotes the fraction of times link is scheduled. Since , the optimal solution to (11) is given by , i.e. with probability each link gets scheduled in each slot, and as a result the optimal peak age is .
However, if we can observe the channel state in every slot before making scheduling decision, we can achieve even smaller age than . Consider the following policy: schedule link whenever , else schedule link . The successful link activation frequency on link is then , while on link it is . The peak age is given by . This happens primarily because the set of achievable successful link activation frequencies, namely , is larger in the case when the channel can be observed before deciding on the schedule in each slot. In Figure 3, we show these regions in the observed and unobserved channel state case for the two link example.
This shows that when the channel state is available for making scheduling decisions, the network age performance can be improved upon. In the next subsection we define a subclass of policies that make scheduling decision based only on the current channel state , and not the entire history . We will see later that these policies can be peak age optimal.
IiC only policies
Just as the stationary policies turn out to be peak age optimal in the unknown channel case, we define a subclass of policies that are peak age optimal in the known channel case. These policies do not use any past history, but only the current channel state , defined as follows [22]:

For each observed channel state we assign a probability distribution over the set of feasible activation sets . If channel state is observed then the activation set is activated for slot with probability .
For an only policy, the rate at which a successful transmission occurs over link is given by
(12) 
for all . The space of all such rates will depend on channel success probabilities , and thus, we use to denote this space of all feasible using only policy. For the two link example in Figure 3, is exactly the grey region of successful link activation frequencies . It is known that if is the space of rates achievable under all policies then [22]. This will help us show that an only policy is peak age optimal.
Iii Problem Formulation
In this section, we formulate the peak and average age minimization problems under a general channel process. To do so in a meaningful way, we would like to restrict our search to a certain reasonable policy space. We consider the following policy spaces:
and
Firstly, note that the constraints that the first and second moment of age
should not grow in is natural, because is the time since last successful transmission on link . It growing in time would necessarily mean that the transmissions are becoming less frequent as time goes by.We consider the policy space for peak age minimization, while space for average age minimization. For a ‘good’ policy, we anticipate the process to be ergodic, in which case the policy is in . For ‘good’ average age policy, it stands to reason that ergodicity of would be required. This is because the average age, being the area under the age curve, depends on .
We define optimal peak and average age to be
(13) 
where the minimization is over the space for peak age and over for average age. Note that since by Jensen’s inequality.
Iiia Peak Age Minimization
We first present a lemma that states a conservation law for age. Intuitively, it states that for any policy , the sum of all age peaks is equal to the total time elapsed plus a small insignificant term that goes to as .
Lemma 1
For any policy we have
(14) 
for all .
Proof:
See Appendix A.
A direct consequence of Lemma 1 is that the peak age minimization problem reduces to
(15) 
We prove this equivalence in Appendix B. This result is significant because it shows that the peak age minimization problem is independent of the age evolution equation. For this reason peak age minimization is much simpler than average age minimization. We propose a virtualqueue based algorithm in Section IV to solve this problem.
IiiB Average Age Minimization
In this section, we provide an equivalent formulation for average age minimization under general channel process. By definition, we know that the average age for a link is given by
(16) 
The following result provides a different characterization of the average age in terms of , for all . This result will be useful to get an intuitive grasp over the class of policies proposed in Section V.
Lemma 2
Define for all and , and any given . Then, for , we have
(17) 
for all .
Proof:
See Appendix C.
For an intuitive understanding of Lemma 2, note that average age is essentially the average area of the triangles formed by the age curve in Figure 2. Note that are square of age peaks in Figure 2, because only at the instances when there is a successful transmission on link . The additional term of is due to Lemma 1.
Lemma 2 also implies that average age minimization problem over can be equivalently posed to minimize
(18) 
where , for all , , and any chosen . Since, age reduces to after a link activation it makes intuitive sense to choose such that as
(19) 
in time slot . This, in the least, should minimize age in the next slot. We analyze such policies in Section V, and show that these policies are within a factor of away from the optimal average age . However, in simulations we observe that these policies are very close to optimal.
IiiC Bounds on Peak and Average Age
In this subsection, we provide a characterization of optimal peak age and a lowerbound on average age. We first characterize the optimal peak age by showing that a only policy is peak age optimal.
Theorem 1
The optimal peak age is given by
(20) 
and as a consequence, there exists a only policy that minimizes peak age, and it can be obtain by solving (20).
Proof:
Theorem 1 can be used to obtain a peak age optimal only policy. However, the search space is usually difficult to characterize for general interference constraints. Another issue is that, to solve (20), requires exact knowledge of the channel statistics . We propose two algorithms that attain near optimal peak and average age, even when the channel statistics is not known apriori but learned on the fly by observing channel states .
We now proceed to derive a lowerbound on average age.
Lemma 3
For any policy , we have
(21) 
And as a consequence the same relation also holds at optimality, namely, .
Proof:
See Appendix E.
Iv VirtualQueue Based Policy
We now propose a policy that solves the peak age minimization problem (15). Note that a policy can decide on the activation set , at time , based on the entire history . However, we do not need the entire history to make a choice at time but only a representation of it.
To do so, we construct virtual queue , which reduce by (at most) upon a successful transmission over link and increased otherwise. These queue lengths determine the ‘value’ of scheduling link in time slot . Therefore, a set that maximizes is activated in slot . This virtualqueue based policy, call it , is described below. Here, is any chosen constant.
Age based policy Start with for all . At time ,

Schedule activation set given by
(23) 
Update as
for all , where .
We now prove that the policy is nearly peak age optimal up to an additive factor.
Theorem 2
Proof:
Let and for all and . Also, let be the objective function in our optimization problem (15). The proof is divided into three parts:
Part A: For all time , we have
(25) 
Part B: The virtual queue is mean rate stable, i.e., for all we have
(26) 
Part C: If is mean rate stable then
(27) 
and
(28) 
The proofs of Part A, B, and C are given in Appendix F. Since the virtual queues are mean rate stable, by Part B, (27) and (28) are true. From (25) and (28) we get the result in (24).
Further, if we set
(29) 
for each , then , with policy , solves the optimization problem (15), up to an additive factor. To see this, notice that from (27), we know that satisfies the inequality constraint in (15). Now, consider the objective function evaluated at :
(30) 
where the first equality is because is a continuous decreasing function in , while the second inequality follows directly from Jensen’s inequality as is convex. Substituting (25) in (30) we get
(31) 
Theorem 2 shows that even when the channel statistics are not known the optimal peak age can be achieved, up to an additive factor of , with arbitrary precision. The precision can be chosen by selecting . For example, we may obtain peak age of at most by setting .
V AgeBased Policy
We now consider an age based policy, which schedule links as a function of links’ age . Lemma 2 provided an alternate characterization of average age under policy given by
(32) 
where and . We now propose an agebased policy that schedules set with maximum weight .
Agebased Policy The policy activates links in slot given by:
(33) 
for all .
We, now, prove bounds on the peak and average age of policy . More specifically, we show that the average and peak age of policy is within a factor of from the respective optimal values.
Theorem 3
The agebased policy policy is at most factor4 peak and average age optimal, i.e.,
(34) 
and
(35) 
where and .
Proof:
To obtain the bound we define two functions and , where is a representation of our objective function, which is age at time , while is the drift of a certain Lyapunov function . We obtain a bound on where
denote the vector of all
. Telescoping over slots then yields the result. The detailed proof is given in Appendix G.We note that can be chosen to improve the additive factor of optimality. The best bounds, for both peak and average age, occur when , for which both and are maximized. In the next section, we evaluate the agebased policies for different choices of . We also compare it with the virtualqueue based policy from Section IV.
Vi Numerical Results
Consider a network of links, in which at most links can be activated at any given time. We numerically study the performance of our proposed scheduling policies for this network. We set for all links . We assume links to be either ‘good’, in which case , or ‘bad’ in which case . We use to denote the number of bad links in the network. We simulate the policies , , and the optimal policy for the unknown channel case, proposed in [19], over time slots.
In Figure 4 and 5, we plot perlink peak and average age, namely and , as a function of . Here, we have chosen the parameters for the virtualqueue policy , and for the agebased policy . We observe that the peak and average age of the virtualqueue based policy and the agebased policy nearly coincide.
Also plotted in Figures 4 and 5, is the case when the channel state is not observed, i.e., scheduling decisions are made only using history . We plot the peak age optimal policy of [19], while in Figure 5, we also plot a lowerbound on average age that can be achieved by any such policy [19], since is not average age optimal. We observe that the gap between the optimal policy in the unknown channel case and policies , of the known channel case is large when is small, and diminishes as increases. Smaller implies more network interference, as fewer links can be activated simultaneously. This shows that there is a significant utility, in terms of age reduction, in knowing the channel state especially when the network suffers from large interference.
In Figure 6 and 7 we plot perlink peak and average age as a function of the fraction of nodes with bad channel, namely . We observe that the gap between the optimal policy in the unknown channel state case, and our policies and of the known channel case, widens as the fraction increases. This indicates that if the channel statistics of the network are poor then there is a significant utility, in terms of age reduction, in knowing the channel state information. For example, when all channels are ‘bad’, i.e. , the gap is as large as fold.
Via Choice of Parameters and
We now analyze performance of our proposed policies and over the choice of parameters and , respectively. Here, we set and the number of ‘bad’ channels also to be . For the virtualqueue based policy , we observe that the parameter has nearly no effect on convergence time of the algorithm. To illustrate this, in Figure 8, we plot perlink peak age computed over the first time slots, for two different values of and . We observe that the peak age measured over the first slots converged to the peak age at nearly the same time.
Vii Conclusion
We considered the problem of age minimization for a wireless network under general interference constraints and time varying channels, when the channel state information is perfectly available to the scheduler to make scheduling decisions. We proposed a virtualqueue based policy and an agebased policy to minimize age. We proved that the virtualqueue based policy is nearly peak age optimal, up to a constant additive factor, and that the agebased policy is at most a factor away from age optimality.
Comparison with our previous work, which derived age optimal policies when the channel state information is not available to the scheduler, we demonstrate a fold improvement in age when the channel state information is available to the schedule in a particular network setting. This work, therefore, establishes the utility in obtaining or using the channel state information in scheduling to minimize age.
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