I Introduction
In addition to throughput, delay, and spectral efficiency, the AgeofInformation (AoI) metric has recently emerged as one of the key determinants of the Quality of Service (QoS) offered by the nextgeneration wireless networks. The AoI metric, first introduced in [1], measures the freshness of information available to the users in realtime. Ever since the pioneering work by Kaul et al., there has been an extensive body of work on optimizing and understanding the design implications of AoI in communication systems. See [2] for a comprehensive introduction to the recent advances in this area. In order to keep the analysis tractable, most of the existing papers on AoI assume stationary stochastic system models [3, 4]. Furthermore, the usual performance guarantees given in the literature in connection with AoI are almost always asymptotic in nature. On the contrary, applications where the AoI metric is critical to the system performance, such as the ultrareliable low latency communication (URLLC) and missioncritical communication in cyberphysical systems, typically operate far from the stationary regime [5]. For acceptable performance, these applications also require stringent nonasymptotic upper limits on the ageofinformation. To address this issue, in this paper, we focus on designing robust scheduling algorithms that ensure the maximum information freshness for the endusers, irrespective of the possibly timevarying statistics of the underlying wireless channel. In our recent papers [6, 7, 8], we introduced an adversarial version of Binary Erasure Channel (BEC) model, and showed that a greedy scheduling policy is approximately competitively optimal. These papers assume that the channel states are adversarially chosen and the scheduler does not have access to the current channel state information (CSIT). In the present paper, we extend our previous results to the setting where the channel state information of the current slot is available to the scheduler. The main objective of this paper is to quantify the provable improvement in performance due to the availability of CSIT compared to the setting when the transmitter is oblivious to the current channel state. Due to the complexity of the analysis, we only have been able to theoretically analyze the setting when the number of users ( is either two or three. Our numerical experiments suggest the AoI advantage continues in the presence of CSIT even when the number of users is large. We anticipate that the tools and techniques developed in this paper will be useful to tackle the general problem with an arbitrary number of users. In this paper, we claim the following two main contributions:

For the adversarial channel model, we establish an improved upper bound on the competitive ratio for a greedy online scheduling policy that has access to the current CSIT. We show that the proposed online policy is 2competitive when and competitive when . This improves the previously known tight upperbounds on the competitive ratios (without CSIT), which are known to be (for ) and (for ) respectively (see Theorem 3 of [9] and Theorem 1 of [8], where the competitive ratio is bounded by for any ).

We numerically compare the performance of the online scheduling policy which knows channel states at the current slot with a greedy online scheduling policy which does not have the current channel state information. Our results show that the AoI is substantially reduced with CSIT.
The rest of the paper is organized as follows. In Section II, we describe our adversarial system model and formulate the problem. In Section III, we derive the competitive ratio of a greedy scheduling policy for the case and . In Section IV, we present our simulation results, and finally, in Section VI, we conclude the paper with a brief discussion on possible future research directions.
Ii System Model and Problem Formulation
We consider an online scheduling problem with users located in the coverage area of a single Base Station (henceforth referred to as BS). Time is slotted, and at the beginning of every slot, a fresh update packet arrives at the BS from some external source. Such traffic models are known as the saturated traffic models in the literature [10, 11, 12]. Each of the users are interested in receiving the fresh packet at each slot to keep uptodate with the external source. Once a fresh packet arrives, the BS beamforms and schedules a packet transmission to one of the users according to a scheduling policy . The downlink channels from the BS to the users are assumed to be nonstationary, modeled as an adversarial binary erasure channel, whose states are dictated by an adversary. In particular, the downlink channel state for any user could be either Good or Bad as determined by the adversary. The online scheduling policy , equipped with the channel state information (CSIT), knows the current channel states of all users before the scheduling decision for a slot is made. Making use of the current channel state information, the policy selects a user having a Good channel (if any) and then transmits the latest packet from the BS to the user. The adversary controlling the channel states may know the scheduling policy as well. This adversarial framework was first introduced in our recent papers [6, 7, 8].
Our objective is to design a scheduling policy that competitively optimizes the average freshness of information for all the users. For any time slot , let denotes the last time slot when the user successfully received a packet from the BS. The Age of Information (AoI) for the user at slot is defined as:
In other words, the quantity measures the number of time slots before which the user received the last packet prior to time . The
dimensional vector
represents the collection of AoI for users at time where element of the vector refers to the AoI of the user i.e. . The age increases linearly with time until the user receives a fresh packet. Once a user receives a fresh packet, its AoI instantaneously drops to unity. See Fig. 1 for an illustration of the evolution of AoI.Objective function
Throughout this paper, we consider optimizing the total AoI metric, which is defined as the sum of AoI cost incurred for all users over the entire time horizon under consideration. Hence, the objective function for the AoI minimization can be expressed as:
(1) 
Iia Performance metric
To quantify the performance of any scheduling policy, we use the standard notion of competitive ratio from the online algorithms literature [13, 14]. To be specific, the competitive ratio of any online policy is defined as the ratio between worstcase cost incurred by the policy and the cost incurred by the offline clairvoyant optimal policy OPT, which knows the entire sequence of channel states in advance. Let denote any sequence of channel states. The competitive ratio of an online policy can be expressed as:
(2) 
where, in the above, the supremum is taken over all possible finite length channel state sequences .
IiB Scheduling Policies
In this paper we analyze the performance of the following online policy:
MaxAge with CSIT policy (MaCsit)
At each time slot, the scheduler determines the current channel states of all users using the CSIT. The BS then schedules a fresh packet transmission to the user having the highest age among all users currently having a Good channel. If at any time slot, no channel is in Good state, the MACSIT policy does not schedule a packet transmission to any user.
For bounding the competitive ratio of the MACSIT policy, we need to characterize the Offline optimal policy (OPT). The OPT policy is assumed to know the channel states of all the users for the entire time duration a priori. Hence, the performance of any other scheduling policy is dominated by that of OPT. However, the OPT policy can not be implemented in an online fashion as it assumes the knowledge of the future channel states.
Baseline
In order to determine the benefit of having CSIT, we compare the performance of the MACSIT policy with the MaxAge policy that does not consider the channel state information [6, 8]. Under the MaxAge policy, at each time slot , BS schedules a fresh packet to the user which has the highest age among all the users, irrespective of the current channel states. Hence, if the channel state at the scheduled userend turns out to be Bad, the packet is lost.
Iii Performance Analysis
In this section, we bound the competitive ratio of the MACSIT policy from the above.
A note on determining the Maxage users
To begin with, at any time slot, we first sort the users according to descending order of their ages under the MACSIT policy. The user, who has the highest age among all the users (under the MACSIT policy) is called the Maxage user (ties are broken arbitrarily). Similarly, the user in the sorted list is called the Maxage user. Thus the Maxage user corresponds to in the sorted list at that time slot. Naturally, under a different policy (e.g., OPT) the MaxAge user may not have the highest age among all users.
Next, we recall the concept of a time interval, first introduced in [9],
Definition 1
(Interval) A new interval is said to begin when the Maxage user transmits a packet successfully under the MACSIT policy.
Hence, an interval continues until the channel corresponding to the Maxage user becomes . Let the quantity denote the age of the user^{1}^{1}1here refers to the index of the user which is same for both MACSIT and OPT policy for the entire time duration . Please note that, this does not refer to the index of the user in the sorted list which is prepared at every timeslot to determine the ordering of the Maxage users on the basis of the ages of the users under the MACSIT policy. For example the user 1 at a certain timeslot may become the Maxage user and at another timeslot may become ^{nd} Maxage user and so on, but its index remains same for the time duration for both the policies. at the ^{th} time slot of the ^{th} interval under the MACSIT policy. Also, let denote the age of the user at the time slot of interval under OPT policy. So denotes the age of the user at the first time slot of the interval under the MACSIT policy. The length of the interval is denoted as and the total AoI cost incurred by the MACSIT and the OPT policies on the ^{th} interval are denoted by and respectively. With the above definitions in place, we now proceed to bound the competitive ratio of the MACSIT policy.
Iiia Competitive ratio of the MaCsit policy for users
Proposition 1
The competitive ratio of the MACSIT policy for users is upper bounded as .
Proof:
For two users, we can express the difference between the costs incurred by the MACSIT policy and OPT as:
(3) 
where the index in the summation ranges over all slots in the ^{th} interval. We now establish the following Lemma.
Lemma 1
For the Maxage user, the age difference between the MACSIT policy and the OPT policy for every time slot of interval remains constant. For example, if at the interval the user 1 remains the Maxage user then the age difference remains constant throughout the interval .
Proof:
Without any loss of generality, let us assume that the MACSIT policy serves the user 2 at the beginning of the interval, and at the interval, the user 1 becomes the Maxage user.
We establish Lemma 1 on the basis of the following observation.
Both MACSIT and OPT policies can not serve the Maxage user until the channel corresponding to that user becomes Good. Furthermore, whenever the channel becomes , the MACSIT policy will serve the Maxage user immediately and a new interval begins. Thus, within any interval, both the quantities and increase linearly. Hence,
(4) 
Since we assume that user 1 is the Maxage user at the interval, we have . The next interval, i.e., the interval begins when the MACSIT policy serves user 1. Thus we have,
(5) 
We now establish the following useful result.
Lemma 2
For the user other than the Maxage user, the age difference between the MACSIT and the OPT policy is always nonpositive (i.e., for this case).
Proof:
To prove we use the following facts. At the next time slots of the interval whenever the channel corresponding to user 2 becomes , both the MACSIT and the OPT policies serve the user 2. The only scenario when the age of user 2 under the MACSIT policy becomes greater than age of user 2 under OPT i.e. is when the channel corresponding to user 2 becomes and OPT serves the user 2 but MACSIT does not. In other words the OPT policy serves the user 2 while the MACSIT policy serves the user 1. Since we considered user 1 as the Maxage user and if the MACSIT policy serves the user 1, from the definition of interval the next interval i.e. interval starts. This implies at the interval the age of the user 2 under MACSIT will never become more than the age under OPT. Hence,
(6) 
Combining the above two Lemmas, equation (3) may be simplified as
(7) 
For bounding the second term in the above inequality, we need to introduce the notion of ResidueLength as defined below:
Definition 2
(Residuelength) The residuelength is the length of time from the last slot in the previous interval when the Maxage user of the interval got served by the MACSIT policy, counted up to the beginning of the interval.
See Fig. 3 for an illustration of the intervals and the residue lengths. It is not hard to verify that the difference of the ages of the Maxage user under the MACSIT policy and OPT at the beginning of the interval can be upper bounded by the residuelength i.e. . Hence, from Eqn. (7), we have the following upper bound:
(8) 
Finally, to find an upper bound to the competitive ratio, we need to derive a lower bound of the cost of the OPT policy for each intervals. Note that, after the first time slot of any interval, the Maxage user, by definition, encounters consecutive channels. Hence, the cost corresponding to that user under the OPT policy can be lower bounded by .
During the ^{th} interval, the channel corresponding to the user other than the Maxage user (i.e. user 2) does not become after the ^{th} time slot. This fact can be verified from the definition of residuelength. Therefore, the cost for user 2 under OPT for the ^{th} interval can be lower bounded as:
(9) 
Hence, the total AoI cost under the OPT policy (including both users) for the ^{th} interval can be lower bounded as:
(10) 
Summing up the costs over all intervals we have the following bound:
(11) 
Substituting the bound from Eqn. (10) in the inequality above, we have:
(12) 
Now we use the AMGM inequality to get .
Furthermore, we have
.
Hence,
(13) 
where we have used the fact that, by definition Hence, .
The above result should be compared and contrasted with Theorem 3 of [8], which proves an upper limit of for the competitive ratio of the MaxAge policy that operates without CSIT.
In the following, we extend the above proof technique for users. The reader will find that although the basic line of analysis remains the same, the details become much more involved in this case.
IiiB Competitive ratio of the MaCsit policy for users
Proposition 2
The competitive ratio of MACSIT policy for users is upper bounded as
Proof: We use the same definition of intervals as in our previous proof. The difference in the AoI costs incurred by the MACSIT and the OPT policies can be expressed as follows:
(14) 
where the index in the summation ranges over all slots in the ^{th} interval. Without any loss of generality, let us assume that user 1 is the Maxage user for the interval under the MACSIT policy. Note that Lemma 1 holds for any number of users (hence, for also). This is because whenever the channel corresponding to the Maxage user becomes , the MACSIT policy serves that user immediately and a new interval begins. Thus, the difference of ages of the Maxage user under the MACSIT and OPT policies remains constant throughout any interval (as both increase linearly throughout an interval). Therefore we can write
(15) 
Using the same definition of residuelengths as before, we can express the above difference as:
(16) 
We denote the time slot when the Maxage user of the ^{th} interval got served by the MACSIT policy as .
To upper bound the quantity we introduce the notion of subintervals, which form a partition of the intervals. The formal definition of a subinterval is given below:
Definition 3
(Subinterval) Within an interval, a new subinterval is said to begin when the MACSIT policy serves the Maxage user among the three users.
Let, denotes the number of subintervals in the ^{th} interval.
We define the SubResidue length of the interval, as follows:
Definition 4
(SubResidue length) The subresidue length of the interval (denoted by ) is the timeelapsed since the last time slot when the ^{nd} Maxage user of the ^{th} subinterval of the ^{th} interval got served by the MACSIT policy, up to the beginning of the ^{th} subinterval.
We illustrate the notion of subintervals and subresidue lengths in Fig. 4. Note that, the above definition is analogous to the definition of intervals.
In every subinterval, the age of user which has the least age under MACSIT policy would be always upper bounded by age of that user under the OPT policy. It directly follows from Lemma 2, discussed in the previous section. Hence, we have the following upper bound:
(17) 
where refers to the length of the subinterval of the interval, and the index runs over all subintervals of the ^{th} interval.
Next, we proceed to lower bound the cost incurred by the OPT policy during the ^{th} interval.
Lower bounding the cost of the Maxage user
Since, the interval continues until the channel corresponding to the Maxage user becomes , the cost incurred by the Maxage user (i.e. user 1 in this case) under the OPT policy is lower bounded by
(18) 
Lower bounding the cost of the ^{nd} Maxage user
The cost incurred by the Maxage user under the OPT policy during the subinterval is . This is true because for the entire duration of the ^{th} subinterval, the channel corresponding to the Maxage user remains .
Lower bounding the cost of the ^{rd} Maxage user
Following the definition of the subresidue lengths, the quantity denotes the last time slot when the MACSIT policy serves the ^{nd} Maxage user of the ^{th} subinterval, counted from the beginning of the ^{th} subinterval. On the ^{th} subinterval, the ^{rd} Maxage user of the ^{th} subinterval becomes the ^{nd} Maxage user. Hence, for the last time slots of the ^{th} subinterval, the cost of the ^{rd} Maxage user of the ^{th} subinterval under the OPT policy is given by .
Thus, the cost under the OPT policy during the ^{th} subinterval of the ^{th} interval, excluding the cost of the Maxage user is lower bounded by:
For the last subinterval of the ^{th} interval, the cost incurred by 3^{rd} Maxage user under the OPT policy is lower bounded by (since at the ^{th} subinterval subresidue length does not exist). Hence the cost incurred by the 2^{nd} Maxage and the 3^{rd} Maxage user under the OPT at the subinterval is lower bounded by
(19) 
Finally, summing up the cost over all subintervals in the ^{th} interval, we get the following lower bound to the cost incurred by the 2^{nd} Maxage and the 3^{rd} Maxage user under the OPT policy:
(20) 
where is the subresidue length of the last subinterval of the ^{th} interval and is the length of the last subinterval of the ^{th} interval.
There are three scenarios depending on the values residue length i.e. can take,

Case 1 ,

Case 2 ,

Case 3 .
Case 1
Consider the first scenario where . Now consider the Maxage user of interval. The MACSIT policy serves the Maxage user time slots before the beginning of the interval i.e at slot. The OPT policy can serve the Maxage user twice after time slot. Since , the OPT policy can serve the Maxage user once at the beginning of subinterval and next at the beginning of interval. So, the time slots can be divided into two parts. The first part refers to the subresidue length of the last subinterval i.e and the next one refers to the length of final subinterval i.e. . Hence we have
(21) 
Let denotes the first part of and refers to the second part. So for the above case we have
(22)  
(23) 
Hence the lower bound of the OPT policy at Eq. (20) can be rewritten as
(24) 
Since and , we can rewrite Eqn. (17) as:
(25) 
Summing the costs over all intervals from Eq. (25) and using the lower bound of the OPT policy for the Maxage user of Eq. (18) and the lower bound for the 2nd Maxage and the 3rd Maxage user of Eq. (24) we get the following bound:
(26) 
Using the AMGM inequality, we have , and . Hence, from the above, we get
(27) 
Combining above equations we get
(28) 
Lower bounding the subinterval lengths and the subresidue lengths by zero, from the above, we have
(29) 
Since, using the CauchySchwartz inequality, we have Hence, the RHS of the above equation can be further upper bounded as below:
(30) 
We have . Note that the RHS of Eqn. (30) is monotonically increasing for . Hence, we can upper bound the RHS of equation (30) by substituting . Therefore, we get
(31) 
Please see the Appendix section for the proof of Case 2 and Case 3. Hence, for all values of , we get which implies .
Iv Simulation Results
In this section we provide two particular channel configurations for 2 users and 3 users scenario to show the tightness of the bound provided in the IIIA and IIIB sections.
Iva users case
Consider the following channel state sequence for 2 users where the whole sequence is divided into intervals of fixed length where is even. At the beginning of every interval the channel corresponding to the user 1 is Good and the other channel is Bad. For the next slots both channels remain Bad. Next, at the slot both the channels become Good. After that, at the slot, the channel corresponding to user 2 remains Good but other channel becomes Bad. For the next slots both channels remain Bad and finally at the slot both channels become Good. In Fig. 5 the AoI cost ratio between the MACSIT and the OPT policy for this particular channel state configuration has been plotted. It can be seen as interval length grows the cost ratio approaches 2, while in section IIIA we showed that for 2 user case the competitive ratio for the MACSIT policy is upper bounded by 2.
IvB users case
In this case, we consider the interval length to be multiple of 6. Here we mention at which time slot the channels corresponding to the users become Good. At the first time slot of the interval the channels corresponding to user 1 and 2 are only Good. For and time slots the channels corresponding user 1 and user 3 remain Good respectively. At the time slot, the channels corresponding to user 1 and user 3 become Good and at the next time slot, the channel corresponding to the user 1 only remains Good. After that, at the slot, the channels corresponding to user 2 and user 3 become Good. At next two time slots i.e. and slots the channels corresponding to user 2 and user 1 remain Good respectively. Next at the time slot the channels corresponding to user 1 and user 2 become Good and at the next time slot, the channel corresponding to user 2 only remains Good. After that at slot, the channels corresponding to user 1 and 3 become Good. For the next two time slots i.e. and slots the channels corresponding to user 3 and user 2 remain Good respectively. Next at time slot, the channels corresponding to user 2 and user 3 become Good and at the next time slot, the channel corresponding to user 3 only remains Good. In all other time slots the users which are not mentioned, the channels corresponding to those users remain Bad. For this particular scenario the AoI cost ratio between the MACSIT and the OPT policy has been plotted in Fig. 6. As grows, the cost ratio approaches 2.25, while in section IIIB we showed that for 3 user case the competitive ratio for the MACSIT policy is upper bounded by 2.67.
V Comparison between the MaCsit and the Max Age policy
In this section we provide numerical results to show the advantage of having CSIT. Through simulations we compare the performance of MACSIT policy and the MaxAge policy [6, 8] which does not have CSIT. In this case we consider the channel states corresponding to each user to be independent and identically distributed.
In Fig. 7, Fig. 8 and Fig. 9, the time averaged AoI cost () for MACSIT policy and Max Age policy when , and have been plotted respectively.
In Fig. 10, the ratio between the average AoI cost of Max Age policy and that of MACSIT policy for these three cases has been shown. From the plots we can conclude that when the policy is equipped with CSIT the performance improves drastically.
Vi Conclusion
The paper investigates the fundamental limits of AgeofInformation for static users over adversarial environments when the scheduling policy is assumed to know the CSIT at the current slot. Theoretically we provide upper bound on the competitive ratio when the number of users is either 2 or 3. Through simulations, we showed that the greedy scheduling policy performs substantially better over adversarial setting when the policy is equipped with the channel state information at the current slot. Finding an upper bound on the competitive ratio for arbitrary number of users is an interesting open problem.
Vii Acknowledgement
This work is partially supported by the grant IND417880 from Qualcomm, USA and a research grant from the Govt. of India for the potential CenterofExcellence Intelligent Networks under the IoE initiative. The computational results reported in this work were performed on the AQUA Cluster at the High Performance Computing Environment of IIT Madras.
References
 [1] S. Kaul, R. Yates, and M. Gruteser, “Realtime status: How often should one update?” in INFOCOM, 2012 Proceedings IEEE. IEEE, 2012, pp. 2731–2735.
 [2] Y. Sun, I. Kadota, R. Talak, and E. Modiano, “Age of information: A new metric for information freshness,” Synthesis Lectures on Communication Networks, vol. 12, no. 2, pp. 1–224, 2019.
 [3] I. Kadota, A. Sinha, and E. Modiano, “Optimizing age of information in wireless networks with throughput constraints,” in IEEE INFOCOM 2018IEEE Conference on Computer Communications. IEEE, 2018, pp. 1844–1852.
 [4] I. Kadota, A. Sinha, E. UysalBiyikoglu, R. Singh, and E. Modiano, “Scheduling policies for minimizing age of information in broadcast wireless networks,” IEEE/ACM Transactions on Networking, pp. 1–14, 2018.
 [5] J. Park, S. Samarakoon, H. Shiri, M. K. AbdelAziz, T. Nishio, A. Elgabli, and M. Bennis, “Extreme urllc: Vision, challenges, and key enablers,” arXiv preprint arXiv:2001.09683, 2020.
 [6] S. Banerjee, R. Bhattacharjee, and A. Sinha, “Fundamental limits of ageofinformation in stationary and nonstationary environments,” in 2020 IEEE International Symposium on Information Theory (ISIT), 2020, pp. 1741–1746.
 [7] R. Bhattacharjee and A. Sinha, “Competitive algorithms for minimizing the maximum ageofinformation,” SIGMETRICS Perform. Eval. Rev., vol. 48, no. 2, p. 6–8, Nov. 2020. [Online]. Available: https://doi.org/10.1145/3439602.3439606
 [8] A. Sinha and R. Bhattacharjee, “Optimizing the ageofinformation for mobile users in adversarial and stochastic environments,” arXiv preprint arXiv:2011.05563, 2020.
 [9] S. Banerjee, R. Bhattacharjee, and A. Sinha, “Fundamental limits of ageofinformation in stationary and nonstationary environments,” in 2020 IEEE International Symposium on Information Theory (ISIT). IEEE, 2020, pp. 1741–1746.
 [10] S. Borst and P. Whiting, “Dynamic rate control algorithms for hdr throughput optimization,” in Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No. 01CH37213), vol. 2. IEEE, 2001, pp. 976–985.

[11]
J. M. Holtzman, “Asymptotic analysis of proportional fair algorithm,” in
12th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications. PIMRC 2001. Proceedings (Cat. No. 01TH8598), vol. 2. IEEE, 2001, pp. F–F.  [12] H. J. Kushner and P. A. Whiting, “Convergence of proportionalfair sharing algorithms under general conditions,” IEEE transactions on wireless communications, vol. 3, no. 4, pp. 1250–1259, 2004.
 [13] A. Fiat and G. J. Woeginger, Eds., Online Algorithms, The State of the Art (the book grow out of a Dagstuhl Seminar, June 1996), ser. Lecture Notes in Computer Science, vol. 1442. Springer, 1998. [Online]. Available: https://doi.org/10.1007/BFb0029561

[14]
S. Albers, Competitive online algorithms
. Citeseer, 1996.
Viii Appendix
Case 2:
In this case, before the interval the last time slot at which the MACSIT policy can serve the Maxage user of interval would lie somewhere at the interval. At that time slot that user has the least age under the MACSIT policy. We denote that time slot as and the Maxage user of interval as . Next we need to determine the time slots where the OPT serves but the MACSIT does not.

After time slot suppose, the OPT serves at some time slot at interval but the MACSIT does not. This is only possible when the 2nd Maxage user and get Good channels.Since at the interval, has the least age under the MACSIT policy, the MACSIT serves the 2nd Maxage user. Hence at that time slot becomes the Maxage user and at the interval it will become the Maxage user. But it is not possible, as is the Maxage user of interval and same user can not become Maxage user at two consecutive intervals.

Another possible scenario is after the Maxage user at the interval gets Bad channels constantly and at the beginning of interval the channels corresponding to both and the 2nd Maxage user become Good and the OPT serves instead of serving the 2nd Maxage user. In this scenario the MACSIT serves the Maxage user and the Maxage user becomes the Maxage user of the interval i.e. . But in this case the OPT policy can serve at max once after the user gets served by the MACSIT policy which implies
(32) At time slot the OPT policy can serve either 2nd Maxage user or the . But if the OPT policy serves the , then for the rest of the time slots of interval and entire interval the cost difference for that user under the OPT and the MACSIT policy remains zero.
Suppose the OPT policy serves the 2nd Maxage user. Since after time slot, both and 2nd Maxage user get Bad channels constantly and , the cost difference between the MACSIT and the OPT policy for the users other than the Maxage user for rest of the interval(33) Hence the cost difference between the MACSIT policy and the OPT policy at interval is
(34) For this particular case at the interval there will not be any subinterval because can not get Good channel at the interval, otherwise the MACSIT policy will serve immediately and this will contradict the assumption . Thus, for interval, we have
(35) Since the OPT policy serves the at the beginning of interval we have
(36) Hence at interval the cost difference between the MACSIT policy and the OPT policy is
(37) Since the OPT policy did not serve the 2nd Maxage user at the beginning of interval the cost of under OPT policy is lower bounded by .
Now consider, , and . Also let , (since ) and .
In the expression of the first summation indicates the lower bound on the cost incurred by under the OPT policy since it got served by the OPT policy for the last time before interval. The rest two summations refer to the lower bound of the cost incurred by under the OPT policy since it got served by the MACSIT policy before interval.
Comments
There are no comments yet.