I Introduction and Related work
[]T he QualityofService (QoS) offered by any wireless network has traditionally been measured along three dimensions, namely, throughput, packet delay, and energy efficiency. There exists an extensive body of literature addressed to optimizing the crosslayer resource allocations to improve the QoS along these axes [35, 28, 32, 30, 25]. However, it has been argued that the standard QoS metrics are primarily geared towards quantifying the degree of utilization of the system resources, and less towards measuring the actual user experience [16]. With the explosive growth of handheld mobile devices, Internet of Things (IoT), realtime AR and VR systems powered by the emerging 5G technology, the Quality of Experience (QoE) for the users plays a major role in today’s network design [5]. In order to integrate QoE with the design criteria, a new metric, called AgeofInformation (AoI), has been proposed recently for measuring the freshness of information available to the endusers [23, 24].
Designing efficient schedulers to minimize the AoI is currently an active area of research. The papers [21] and [20] study the average AoI minimization problem for static User Equipments (UEs) associated with a single Base Station (BS). In these papers, the authors propose a optimal MaxWeighttype scheduling policy (Theorem 12 of [21]). The paper [33] proposes an optimal scheduling policy for the same setup, where the objective is to minimize the maximum AoI of all UEs. All of these papers consider a singlehop network model with static UEs only. The problem of AoI minimization in a multihop network with static UEs has been studied in [34]. The paper [36] considers the problem of designing an AoIoptimal trajectory for a mobile agent which facilitates information dissemination from a central station to a set of ground terminals. The effect of mobility on the capacity of wireless networks has been investigated in the classic work of [15]. It has been shown that mobility, in general, increases the capacity of ad hoc networks. However, to the best of our knowledge, the effect of UEmobility on the AgeofInformation has not been studied before. One of the main objectives of this paper is to study the AoIoptimal scheduling with mobile UEs.
Most of the existing works on wireless networks assume a stationary channel model for analytical tractability. In rapidly varying environments, such as highspeed trains and vehicletovehicle communication, the standard stationary channel model assumption no longer holds in practice. This is particularly true for the 5G mmWave regime ( GHz), which suffers from severe attenuation loss [14, 38]. On the other hand, designing an accurate and analytically tractable nonstationary wireless channel model remains an overarching challenge to the research community [7, 13]. To overcome this difficulty, in the second part of this paper, we propose a simple adversarial channel model for nonstationary environments and study the scheduling problem in this model. In addition to the emerging 5G technology, the adversarial channel model is also useful for ensuring reliable communication in the presence of tactical jammers, where the interferers, in reality, behave adversarially [31, 29].
Our contributions:
We make the following contributions in this paper.

We study the multiuser scheduling problem in stationary and nonstationary environments. The stationary environment is modelled stochastically, and the nonstationary environment is modelled using an adversarial framework. To the best of our knowledge, this is the first paper that considers the AoIoptimal scheduling problem in an adversarial setting.

Our analytical result enables us to precisely characterize the effect of mobility on the overall AoI as a function of the longterm user mobility statistics. The results may also be effectively used for smallcell network planning [4].

In the nonstationary setting of Section III, we show that a simple online scheduling policy achieves competitive ratio. Using Yao’s minimax principle, we show that no online policy can have a competitive ratio better than .

We propose a heuristic scheduling policy in Section IIIB for the scenario where the future channel states can be accurately estimated for the next slots. We validate the efficacy of the proposed policy through numerical simulations.
The rest of the paper is organized as follows. In Section II, we describe the stochastic model and formulate the problem in the stationary regime. Section IIA and III study the problem in the Stationary and Nonstationary environments respectively. In Section IV, we compare the performance of the proposed scheduling policies via numerical simulations. Section V concludes the paper with some pointers to open problems.
Ii AoI Minimization in Stationary Environments
In this section, we first describe the stochastic system model and then formulate the optimal scheduling problem. In the rest of the paper, the abbreviation UE will refer to any generic user equipment, and the term BS will refer to a Base Station. The area covered by a BS will be referred to as a Cell.
Channel model
We consider a cellular system where a set of UEs travel around in an area having BSs. Time is slotted, and at every slot, each BS can beamform and schedule a packet transmission to one of the UEs in its coverage area. The wireless link to
from the BS in its current cell is assumed to be a stationary erasure channel with the probability of successful reception of a transmitted packet being
. Hence, when a BS schedules a downlink packet transmission to in its cell, the packet is either successfully received with probability or lost otherwise.Mobility model
We assume that the UE mobility is modelled by a stationary ergodic process. Formally, let the random variable
denote the index of the cell to which is associated with at time ^{1}^{1}1We make the standard assumption that the coverage areas of the cells are mutually disjoint. Hence a UE is associated with only one BS at any time.. Then, according to our assumption, the stochastic process is a stationary ergodic process with the probability that is associated with at any time given by The probability measure denotes the stationary occupancy distribution of the cells by the UEs. The mobility of different UEs is assumed to be independent of each other. Many different mobility models proposed in the literature fall under the above general scheme, including the i.i.d. mobility model, random walk model, and the random waypoint model [12, 1, 19, 3]. See Figure 3 in the Appendix VIA for a schematic.Packet arrival model to BS
We consider a saturated traffic model, where at the beginning of any slot, each BS receives a fresh update packet from a common external source (e.g., a highspeed optical backbone network). Since the UEs are interested in the latest updates only, the BS then deletes any old packet from its buffer and schedules the fresh packet for transmission to some UE following a scheduling policy. The saturated traffic model is standard in applications relying on continuous status updates [8], such as monitoring and surveillance with sensor networks [18], velocity and position updates for autonomous vehicles [22], command and control information exchange in missioncritical systems, disseminating stockindex updates and live game scores.
System states
For slot , let denote the last time before time at which received a packet successfully from any BS. The AgeofInformation of at time is defined as
In other words, the random variable denotes the length of time elapsed since received its last update before time . Hence, the r.v. quantifies the staleness of information available to . See Figure 2 in the Appendix for a typical evolution of . The state of the UEs at time
is completely specified by the AgeofInformation of all UEs, given by the random vector
, and the association of the UEs with the cells, represented by the celloccupancy vector .Policy space and performance metric
A scheduling policy first selects a UE in each cell (if the cell contains any UE), and then schedules the transmission of the latest packet from the BSs to the UEs over the wireless erasure channel described earlier. The scheduling decisions are required to be causal for it to be implementable in realtime. The set of all admissible scheduling policies is denoted by . Our goal in this paper is to design a distributed scheduling policy which minimizes the longterm average AoI of all users. In view of this, we consider the following averagecost problem:
(1) 
Iia Converse and Achievability
The AoI minimization problem given by (1) is an example of an averagecost MDP with countably infinite statespace [6]. Excepting a few cases with special structures (cf. [33]), such problems are notoriously difficult to solve exactly. Moreover, the standard numerical approximation schemes for infinitestate MDPs typically do not provide theoretical performance guarantees. In this paper, we take a different approach to approximately solve the problem (1). In the following Theorem, we obtain a fundamental lower bound to the optimal AoI. Finally, in Theorem 2, we show that a simple online scheduling policy achieves the lower bound within a factor of .
Theorem 1 (Converse)
In the stationary setup, the optimal AoI in (1) is lower bounded as:
(2) 
where the quantity denotes the expected number of cells with at least one UE, where the expectation is taken with respect to the stationary occupancy distribution . In particular, since we also have the following (loose) lower bound which is agnostic of the UE mobility statistics:
Please refer to Appendix VIA for a proof of this theorem.
Discussion
Theorem 1 gives a universal lower bound for the minimum AoI achievable by any admissible scheduling policy . Interestingly, it reveals that the lower bound depends on the mobility of the UEs only through their stationary celloccupancy distribution . Hence, given the stationary distribution , the lower bound (2) is agnostic of the details of the mobility model. The appearance of the quantity in the lower bound should not be surprising as it denotes the typical number of nonempty cells at a slot in the long run. Since a BS can transmit a packet only if at least one UE is present in its coverage area, the quantity , in some sense, represents the multiuser diversity of the system.
Expression for
To get a sense of the lower bound (2), we now work out a closedform expression for for the uniform UE mobility pattern. Using linearity of expectation,
(3)  
Since the cells are disjoint, we readily conclude from (3) that . Recall that denotes the marginal probability that the is in . Since the mobility of the UEs are independent of each other, the expected number of nonempty cells in Eqn. (3) simplifies to:
(4) 
We now evaluate the above expression for the case when the limiting occupancy distribution of each UE is uniform across all BSs, i.e., . The uniform stationary distribution arises, for example, when the UE mobility can be modelled as a random walk on a regular graph [27]. In this case, Eqn. (4) simplifies to
(5) 
For , we have . For , we have the following bounds which are easier to work with
(6) 
For a derivation of the bounds in (6), please refer to Appendix VIB.
Achievability
We now propose an online scheduling policy which approximately minimizes the average AoI (1) for mobile UEs (the abbreviation MMW stands for “Multicell MaxWeight"). Our policy is a multicell generalization of the approximate singleBS scheduling policy proposed in [21]. Moreover, using a tighter analysis, we give an improved factor approximation guarantee for .
The policy
At every slot, each BS schedules a UE under its coverage that has the highest index among all other UEs. The index of is defined as
Theorem 2 (Achievability)
is a approximation scheduling policy for statistically identical UEs with i.i.d. uniform mobility (i.e., and ).
Effect of mobility on AoI
Recall that, a BS can schedule a transmission to only one UE in its cell at every slot. Hence, if all of the UEs remain stationary at a single cell, they all have to contend with each other for scheduling. This naturally increases the average AoI of the UEs. On the other hand, if the UEs are mobile, they can take advantage of multiple downlink transmission opportunities from multiple BSs. This form of multiuser diversity drastically reduces the overall AoI, by improving the network resource utilization. Next, we quantify the effect of mobility on the average AoI.
Define the Mobility Advantage on AoI () to be the ratio of the optimal AoI when all UEs are stationary at a single BS (i.e., .) vs. the optimal AoI when the UEs are mobile. As noted above, for a single BS, we have
From our achievability result in Theorem 2, we know that the lower bound in Eqn. (2) is achievable within a factor of . This implies that
From the equation (6), we have
(7) 
for some constant . Consider the following three scaling regime:

Constant Density: If and scale in such a way that the density of the UEs remains constant, i.e., we see that the average AoI diminishes linearly with the number of BSs, i.e., .

UnderLoaded BS: If , we have

OverLoaded BS: If , we have .
Iii AoI Minimization in NonStationary Environments
In this Section, we consider the problem of AoIoptimal scheduling with static users in a nonstationary environment. Since nonstationary channels are difficult to model and analyze, we propose a new adversarial channel model in this setting. Besides being analytically tractable, all positive results in this model (e.g., Theorem 3) carry over to less adversarial environments.
Channel model
A set of UEs are under the coverage of a single BS (i.e., ). The BS can transmit to any one UE at a slot. The channel state of any at any time slot could be either Good () or Bad (). If the BS schedules a packet to a UE having a Good channel at that slot, the UE decodes the packet successfully. Otherwise, the packet is lost. We assume that, the states of the channels (corresponding to different UEs) are selected by an omniscient adversary from the set of all possible states at every slot. The scheduling policy is online and has no information on the channel states for the current or future slots. We will partially relax this assumption in Section IIIB, by considering a more general class of adversarial channel models with future channel estimations. The cost function over a horizon of slots is given by:
(8) 
The packet arrival model to the BS remains the same as in the stationary environment in Section II.
Performance Metric
As standard in the literature on online algorithms [9, 2], we gauge the performance of an online scheduling policy using competitive ratio (), which compares the cost of with that of an optimal offline policy OPT equipped with hindsight knowledge. More precisely, let be a sequence of length representing the vector of channel states chosen by the adversary for the entire horizon. Then, the competitive ratio of the policy is defined as [2]:
(9) 
where the supremum is taken over all finitelength input sequences , and the cost function is given by (8). In the definition (9), while the online policy has only causal information, the policy OPT is assumed to be equipped with full knowledge on the entire channelstate sequence
Characterization of the optimal offline (Opt) policy
For a given sequence of channel states of length , the optimal offline policy OPT may be obtained by using Dynamic Programming. Let the variable denote the optimal costtogo from time when the AoIs of the the UEs are given by the vector Using standard notations, we have the following backward DP recursion
(10) 
where the minimization in Eqn. (III) is over all UEs having a Good channel at slot . When there is no UE with a Good channel at slot (i.e., ), the second term denoting the future cost is replaced with .
Comparison with the throughput maximization problem
It is interesting to note that the competitive ratio for the sumthroughput maximization problem in this adversarial model can be arbitrarily bad (i.e., unbounded). It can be understood from the following. Consider a system with two users. If an online scheduler schedules at any slot, the adversary can set the channel corresponding to to Bad and set ’s channel to Good and vice versa. At any slot, the optimal policy schedules the user with the Good channel state. Hence, any online scheduler receives zero throughput, but OPT achieves the full throughput of unity.
Surprisingly enough, Theorem 3 shows that the Max Age (MA) scheduling policy, which schedules a user having the highest age (i.e., Scheduled UE at time ), is competitive for minimizing the AoI.
Theorem 3 (Achievability)
In the adversarial setting with users, the MA policy is competitive for minimizing the average AoI.
Iiia A Lower bound to the competitive ratio
In this section, we use Yao’s minimax principle for obtaining a universal lower bound to the competitive ratio (9) in the adversarial setting. In connection with online problems, Yao’s minimax principle may be stated as follows:
Theorem 4 (Yao’s Minimax principle [2])
Given any online problem, the competitive ratio of the best randomized online algorithm against any oblivious adversary is equal to the competitive ratio of the best deterministic online algorithm under a worstcase input distribution.
Using the above principle, it is clear that a lower bound to the competitive ratio of all deterministic online algorithms under any input channel state distribution yields a lower bound to the competitive ratio in the adversarial setting, i.e.,
(11) 
To apply Yao’s principle in our setting, we construct the following distribution of the channel states: at every slot , a UE is chosen independently and uniformly at random, and assigned a Good channel. The rest of the UEs are assigned Bad channels. The rationale behind the above choice of the channel state distributions will become clear when we compute OPT’s expected cost in Appendix VIE. In general, the cost of the optimal offline policy is obtained by solving the Dynamic Program (III), which is difficult to analyze. However, with our chosen channel distribution , we see that only one UE’s channel is in Good state at any slot. This greatly simplifies the evaluation of OPT’s expected cost. The following Theorem gives the universal lower bound:
Theorem 5 (Converse)
In the adversarial set up, the competitive ratio of any online policy with UEs is lower bounded by Further, for UEs, the lower bound can be improved to
Please refer to Appendix VIE for a proof of this Theorem.
IiiB AoI minimization with Channel Predictions
The converse result in Theorem 5 states that under the adversarial channel model, any online scheduling policy has a worstcase competitive ratio which grows at least linearly with the number of UEs (). This is quite a disappointing result when the number of UEs is large. On the flip side, the fully adversarial channel model may also be too restrictive in practice. To circumvent this situation, we now exploit the physical fact that wireless channels with blockfading may often be estimated quite accurately for a few subsequent future slots [17]. We consider a relaxed adversarial model, where at any slot , the BS can estimate the channels perfectly for a window of the next slots. Here, is an adjustable system parameter that can be adaptively tuned by the policy in accordance with the scale of timevariation of the channels (e.g., fading block length).
Similar to the adversarial model in Section III, we continue to assume that the channel states are binaryvalued and chosen by an omniscient adversary. Thus, the adversarial model discussed in Section III is a special case of this model with the windowsize .
We now propose the following policy which exploits the step lookahead information:
Receding Horizon Control (RHC:) The UE scheduled at each time is chosen by minimizing the total cost for the next timesteps. Hence, the scheduling decision at time is obtained by solving the DP (III) with the boundary condition .
The RHC policy was considered in [26] in the context of loadbalancing in data centers. It was shown that the RHC policy has a competitive ratio of  approaching as the prediction window size is increased. Since the result of [26] is not directly applicable to our problem, we examine the gain for AoI due to channel prediction capabilities via numerical simulations in the next section. Unsurprisingly, RHC reduces to the MA policy when the prediction window
Iv Numerical Simulations
In this Section, we perform numerical simulations to compare the performance of the RHC and MA policies in the adversarial setting. Figure 1 shows the variation of timeaveraged AoI with different number of UEs for . A MonteCarlo simulation with iterations was performed with randomly generated channels, and we plotted the worstcase AoI in Figure 1(a). For each of these iterations, at every time step, the number of Good Channels is selected uniformly at random between and . From the plots, we see that RHC outperforms MA by a large margin even with just a small prediction window of .
Figure 1(b) shows the variation of the AoI with the window size for the RHC policy. The number of UEs is and the simulation is performed for slots. The windowsize is varied from to . Each simulation is repeated for times and we plotted the maximum AoI value at the end of these iterations. We see that increasing the prediction window does not significantly decrease the average AoI.
V Conclusion and Future Work
This paper investigates the fundamental limits of AgeofInformation in stationary and nonstationary environments from an online scheduling pointofview. In the stochastic setting, a optimal scheduling policy has been proposed for mobile UEs. For the nonstationary regime, a new adversarial channel model has been introduced. Upper and lower bounds for the competitive ratio have been derived for the adversarial model. As an immediate extension of this work, the effect of mobility in the nonstationary environment may be considered. The gap between the upper and lower bounds of the competitive ratio may be tightened. Also, it will be interesting to obtain the competitive ratio for step lookahead policies as a function of the predictionwindow .
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Vi Appendix
Via Proof of Theorem 1
Proof:
In the proof below, we first follow a samplepathbased argument to obtain an almost sure lower bound to AoI. Finally, we use Fatou’s lemma [37] to convert the almost sure bound to a bound in expected AoI, as defined in Eqn. (1).
Consider a sample path under the action of any arbitrary scheduling policy up to time . See Figure 2. For , let the r.v. denote the number of packets received up to time , the r.v. denote the time interval between receiving the ^{th} packet and the ^{th} packet, and the r.v. denote the time interval between receiving the last (^{th}) packet and the timehorizon . Hence, we have
(12) 
Since the AoI of any UE increases in step of one at each slot until a new packet is received (and then it drops to one again), the average AoI up to time may be lower bounded as:
(13)  
where in (a) we have used Eqn. (12), and in (b) we have defined and used Jensen’s inequality afterwards. Rearranging the Eqn. (12), we can express the random variable as:
With this substitution, the term within the bracket in Equation (13) evaluates to
(14) 
where the last inequality is obtained by minimizing the resulting expression by viewing it as a quadratic in the variable .
Hence, from Eqns. (13) and (14), we obtain the following lower bound to the average AoI under the action of any admissible scheduling policy:
(15) 
Next, we analyze the resource constraints of the system to further lower bound the RHS of the inequality (15). Let the r.v. denote the total number of transmission attempts made to by all BSs up to time . Also, let the r.v. denote the fraction of time that contained at least one UE in its coverage area. Since, a BS can attempt a downlink transmission only when there is at least one UE in its coverage area, the total number of transmission attempts to all UEs by the BSs is upper bounded by the following global balance condition:
(16) 
where . Plugging in Eqn. (16), we can further lower bound the inequality (15) as:
An application of the CauchySchwartz inequality on the RHS yields:
(17) 
Note that, successfully received packets out of a total of packet transmissionattempts made by the BSs via the erasure channel with success probability . Without any loss of generality, we may fix our attention on those scheduling policies only for which
. Otherwise, at least one of the UEs receive a finite number of packets, resulting in infinite average AoI. Hence, using the Strong law of large numbers
[37], we obtain:(18) 
Moreover, using the ergodicity property of the UE mobility, we conclude that almost surely:
where we recall that denotes the stationary cell occupancy distribution defined earlier. Thus, we have almost surely
(19)  
where the function denotes the expected number of nonempty cells where the expectation is evaluated w.r.t. the stationary occupancy distribution . Hence, putting equations (18) and (19) together with the lower bound in (17), we have almost surely:
(20) 
Finally,
where the inequality (a) follows from Fatou’s lemma. This concludes the proof of Theorem 1. Note that the proof continues to hold even when the mobility of the UEs are not independent of each other.
ViB Derivation of the bounds in Eqn. (6)
For , we have the following bounds:
(21) 
where The inequality (b) is standard. To prove the inequality (a), consider the concave function
for some . Since a concave function of a real variable defined on an interval attains its minima at one of the end points of the closed interval, and since , we have if , i.e., , i.e., . Thus, the inequality (a) holds for with . The inequality (21) directly leads to the bounds in Eqn. (6).
ViC Proof of Theorem 2
Proof:
Let the scheduling decisions at slot be denoted by the binary control vector , where if and only if the following two conditions hold simultaneously: (1) , i.e., is within the coverage area of at slot , for some , and (2) schedules a transmission to at time ^{2}^{2}2Recall that the random variable denotes the index of the BS is associated with at time .. Since a BS can schedule only one transmission per slot to a UE in its coverage area, the control vector must satisfy the following constraint:
For performance analysis, we consider the following Lyapunov function, which is linear in the ages of the UEs:
(22) 
The conditional transition probabilities for the age of may be written as follows:
where the first equation corresponds to the event when was scheduled and the packet transmission was successful, and the second equation corresponds to its complement event. Hence, for each UE , we can compute :
(23) 
From the equation above, we can evaluate the onestep conditional drift as:
(24)  
Finally, consider the drift minimizing policy MultiCell MW (MMW), under which, each Base Station schedules a user having the highest weight in its cell. For the purpose of the proof, we now define a stationary randomized scheduling policy RAND, under which every BS randomly schedules a UE in its cell with probability ^{3}^{3}3We use the usual convention that summation over an empty set is zero.. Comparing MMW with RAND, we have:
Thus, we have the following upperbound of the drift (24) under the MMW policy:
Taking expectation of the above driftinequality w.r.t. the random celloccupancy vector , we have
(25) 
where Our next task is to evaluate this expectation. Note that, we can alternatively express the random variable as
where
We can evaluate this expectation exactly for the i.i.d. uniform mobility model. Recall that . Hence,
(26) 
In the special case when all UEs are identical, i.e., , the summation (VIC) has a closedform expression. Clearly, for all , we have:
To evaluate the expectation of , we integrate the binomial expansion of in the range to obtain the identity:
ViD Proof of Theorem 3
Proof:
Let us assume that the MA policy had successful transmissions during the entire timehorizon of length . We divide the time horizon into successive intervals, defined naturally as follows. Let be the time index at which the MA policy had its ^{th} successful transmission, , and . Let denote the length of the ^{th} interval between the ^{th} and ^{th} successful transmissions of the MA policy. For notational consistency, we define See Figure 4. We start our analysis with two simple observations  first, whenever a successful transmission is made by the MA policy, the optimal policy OPT also transmits at that slot successfully. Second, the MA policy is a persistent round robin policy, which keeps on scheduling a user (having the highest age) until the transmission is successful. In the immediately following time slot, the MA policy switches to the other user and continues the roundrobin scheduling cycle. See Figure 5 for a typical run.
Hence, under the MA policy, the states of the users (in sorted order) at the beginning of the ^{th} interval is
Since the MA policy continues scheduling the UE having the highest age, at the end of the ^{th} slot of the ^{th} interval, the ages of the UEs (in sorted order) are given by:
Hence, the cost incurred by the MA policy during the ^{th} interval is computed as:
(30)  
(31) 
where in the last step, we have used the AMGM inequality to conclude
Hence, the total AoI cost incurred by the MA scheduling policy over the entire time horizon is upper bounded as:
On the other hand, the cost incurred by OPT during the ^{th} interval is lower bounded as:
(32)  
where we have separately lower bounded the cost incurred by the UE being scheduled by MA (which was consistently seeing Bad channels) and the other UEs. Finally, the cost of the entire horizon may be obtained by summing up the cost incurred in the constituent intervals. Hence, noting that , from Eqns. (30) and (32), the competitive ratio of the MA policy may be upper bounded as follows:
ViE Proof of Theorem 5
Proof:
To apply Yao’s principle, we need to compute the expectations appearing in the numerator and the denominator of Eqn. (11).
ViE1 Upper bound to Opt’s expected cost
Let the random variable denote the total AoIcost incurred by the ^{th} UE up to time . In other words,
Hence, the limiting timeaveraged total expected cost incurred by OPT may be expressed as
(33) 
In the following, we will show that all of the above limits exist with the assumed choice of the underlying probability space. We now use the Renewal Reward Theorem [10] in order to evaluate the RHS of Eqn. (33). Since, under the assumed channel state distribution , only one channel is in Good state, the optimal policy OPT is easy to characterize  at any slot, OPT schedules the user having Good channel. Under this probability space, it can be verified that, for each user , the sequence of random variables constitute a renewal process, with the commencement of scheduling of the ^{th} user constituting renewal instants. A generic renewal interval of length for the ^{th} user consists of two parts  (1) a consecutive sequence of Good channels of length , and (2) a consecutive sequence of Bad channels of length . The AoI cost incurred by the user in any generic renewal cycle may be written as the sum of the costs incurred in two parts:
Let be the probability that that the channel is Good for the ^{th} user at any slot. Hence, from our construction, the random variables and
follows a Geometric distribution having the following p.m.f.
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