Online Energy-Efficient Scheduling for Timely Information Downloads in Mobile Networks

01/10/2019 ∙ by Yi-Hsuan Tseng, et al. ∙ National Taipei University 0

We consider a mobile network where a mobile device is running an application that requires timely information. The information at the device can be updated by downloading the latest information through neighboring access points. The freshness of the information at the device is characterized by the recently proposed age of information. However, minimizing the age of information by frequent downloading increases power consumption of the device. In this context, an energy-efficient scheduling algorithm for timely information downloads is critical, especially for power-limited mobile devices. Moreover, unpredictable movement of the mobile device causes uncertainty of the channel dynamics, which is even non-stationary within a finite amount of time for running the application. Thus, in this paper we devise a randomized online scheduling algorithm for mobile devices, which can move arbitrarily and run the application for any amount of time. We show that the expected total cost incurred by the proposed algorithm, including an age cost and a downloading cost, is (asymptotically) at most e/(e-1) 1.58 times the minimum total cost achieved by an optimal offline scheduling algorithm.

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I Introduction

In recent years, there is a proliferation of technologies requiring timely information. Examples of the technologies range from smart devices (e.g., smartphones) to smart systems (e.g., smart transportation systems). On the one hand, a smartphone would need timely traffic and transportation information for planning the best route. On the other hand, a smart transportation system would need timely information about vehicles’ positions and speeds for planning collision-free transportation.

In those technologies, end devices (e.g., smartphones and vehicles) can be updated by some remote information sources over wired or wireless networks. The information kept at the end devices should be as timely as possible to accomplish their missions (e.g., planning the best route or planning collision-free transportation). Thus, the age of information was recently proposed in [1] to measure the freshness of the information at the end devices. Interestingly, [1] claimed that a throughput-optimal design or a delay-optimal design might not achieve the minimum age of information. Thus, guaranteeing the freshness of information at end devices becomes an issue.

In this paper, we consider the scenario where a mobile device is running an application. The application needs some timely information, which can be downloaded through neighboring access points (APs). However, frequent downloading for minimizing the age of information causes the mobile device a critical power issue. To strike a balance between the age of information and the involved power, the mobile device needs to identify an energy-efficient scheduling algorithm for downloading the latest information.

However, the mobile user can move at will. The connectivity between the mobile device and the APs changes over time; in particular, the resulting channel dynamics is even non-stationary because the mobile device might run the application for a short time. The great uncertainty of the channel dynamics poses the major challenge to the scheduling design. In this paper, we tackle the challenge by developing an online energy-efficient scheduling algorithm, requiring no knowledge about the channel dynamics or the time for running the application.

I-a Contributions

Our main contribution lies in designing and analyzing online scheduling for power-limited mobile devices with unknown movement and unknown time for running the application. The goal is to minimize a total cost including an age cost and a downloading cost. To reach the goal, we leverage primal-dual techniques [2]

for linear programs. However, the methodology cannot be applied immediately because the age of information usually involves a

square term (see [3] for example), resulting in a non-linear program. We successfully address the issue by transforming the scheduling problem into an equivalent scheduling problem in a virtual queueing system. With the transformation, an optimal offline scheduling algorithm can be obtained using a linear program. Then, applying the primal-dual techniques to the linear program, we propose a randomized online scheduling algorithm while showing that the worst-case ratio between the expected total cost incurred by the proposed online algorithm and that incurred by an optimal offline algorithm is (asymptotically) .

I-B Related works

The age of information has been analyzed for several queueing models, e.g., [1, 4, 5, 6, 7]; meanwhile, scheduling for minimizing the average age of information has also been explored, e.g., [8, 9, 10, 11]. Moreover, some works investigated the age-energy tradeoff (with or without energy harvesting) in various scenarios, e.g., [12, 3, 13, 14]. Despite many works on the scheduling design or on the age-energy tradeoff, their design and analysis were based on stochastic models with some stationary assumptions but cannot be applied in non-stationary settings. To fill this gap, in this paper we explore the age-energy tradeoff in a non-stationary setting.

Ii System overview

In this section, we start with describing our network model in Section II-A, followed by defining an age model in Section II-B. Then, we formulate a scheduling problem involving the age-energy tradeoff in Section II-C.

Ii-a Network model

Fig. 1: Network model.

Consider a mobile network in Fig. 1, where a mobile device is moving in an area with some access points (APs). The device is running an application that requires timely information. The information at the device can be updated by downloading the latest information through the APs.

Divide time into slots and index them by , where is the total time for running the application. At the beginning of each slot, all the APs can immediately obtain the latest information from an information source, i.e., we neglect the transmission time between the APs and sources (through wired networks as shown in Fig. 1) while focusing on the bottleneck between the APs and the device (through wireless networks). Because of the device’s mobility, the channel between the device and the APs changes over slots. Let be the state indicating the connectivity in slot : if the device can successfully download the latest information through an AP in slot ; if it cannot access any AP in slot . By we define the connectivity pattern of the mobile device, which is arbitrary with potential non-stationary property.

For each slot with , the device can decide whether to download the latest information. Let be the decision of the device in slot , where if the device decides to download, and if it decides not to download. A scheduling algorithm specifies decision for every slot . A scheduling algorithm is called an offline scheduling algorithm if connectivity pattern  (along with running time ) is given as a prior. In contrast, a scheduling algorithm is called an online scheduling algorithm if the connectivity pattern is unavailable; instead, it knows the present connectivity state only.

Ii-B Age model

Let be the age of information at the device at the end of slot , i.e., after decision is made. Suppose that, if the device successfully downloads the latest information, then the age of information becomes zero; otherwise, the age of information increases linearly with slots. Then, the dynamics of the age of information is

(1)

where the second case means that the age increases by one if the device cannot download the latest information, because either the device accesses no AP or the device decides not to download. Moreover, we assume that initially the device has the latest information, i.e., .

Ii-C Problem formulation

To investigate the tradeoff between the age of information and the power consumption for downloading the latest information, we define an age cost and a downloading cost as follows. We consider the cost associated with the age of information in slot  as , i.e., a linear age cost function. Suppose that the device uses a constant power to download the latest information, incurring a constant cost for each downloading. Regarding non-linear age cost functions and dynamic power allocation, please see Section VI.

Given connectivity pattern , we define the total cost under scheduling algorithm by

(2)

where the first term is the downloading cost in slot  and the second one is the resulting age cost in slot . In this paper, we aim to develop an online scheduling algorithm  such that the total cost can be minimized for all possible connectivity patterns .

However, without knowing the connectivity pattern, an online scheduling algorithm is unlikely to achieve the minimum total cost (obtained by an optimal offline scheduling algorithm). We characterize our online algorithm in terms of the competitiveness against an optimal offline algorithm, defined as follows.

Definition 1.

For connectivity pattern , let be the minimum total cost for all possible offline scheduling algorithms. Then, an online scheduling algorithm  is called -competitive if

for all possible connectivity patterns , where is called the competitive ratio of the online scheduling algorithm.

With a -competitive online scheduling algorithm, the resulting total cost can be guaranteed to be at most  times the minimum total cost, regardless of connectivity pattern and running time .

Iii Primal-dual formulation

In this paper, we approach the scheduling problem by leveraging primal-dual techniques [2] for linear programs. However, the total age in Eq. (2) would yield square terms (see [3] for example), resulting in a non-linear total cost function. To resolve the non-linearity, in Section III-A we transform the age model into an equivalent virtual queueing system. Then, by exploiting the virtual queueing system, in Section III-B we successfully formulate a linear program for solving the offline scheduling problem, while proposing a primal-dual formulation. The primal-dual formulation will be used later in Section IV for designing and analyzing our online scheduling algorithm.

Iii-a Virtual queuing system

Fig. 2: Virtual queueing system.

Fig. 2 illustrates the virtual queueing system transformed from the age model. The virtual queueing system consists of a server, a queue, and some packet arrivals, operating in the same discrete-time system as the mobile network. At the beginning of each slot, a packet arrives at the virtual queueing system, i.e., the -th packet arrives in slot . The server is associated with an ON/OFF channel for each slot, where the channel is ON in slot if and it is OFF if . When the channel is ON in a slot, the server can decide whether to flush the queue in that slot: the server decides to flush the queue in slot if and it decides to idle if . If the channel is ON and the server decides to flush the queue, then the queue becomes empty, i.e., the queue size becomes zero; otherwise, the new arriving packet stays at the queue i.e., the queue size increases by one. The queueing dynamics is identical to the age dynamics in Eq. (1); therefore, the queue size at the end of slot (after decision is made) can be described by .

Suppose that holding a packet at the end of a slot incurs a holding cost of one unit. As packets are left in the queue in slot , the cost for holding all the packets in slot is . Moreover, suppose that each flushing takes a flushing cost of  units. The goal for the virtual queueing system is to identify a scheduling algorithm for minimizing the total holding cost plus the total flushing cost over slots. Note that the total cost involved in the virtual queueing system is exactly in Eq. (2). In summary, the scheduling problem in the virtual queueing system is equivalent to the original scheduling problem.

With the transformation, we can find that the original online scheduling problem is related to the classical online TCP-ACK problem [2], in which a node decides whether to acknowledge received packets for balancing an acknowledgement cost and a delay cost. In fact, the online scheduling problem in the virtual queueing system generalizes the online TCP-ACK problem to noisy channels. In Section VI, we will further generalize to non-linear age cost functions and dynamic power allocation.

Iii-B Primal-dual formulation

According to Section III-A, we will focus on the scheduling problem in the virtual queueing system. To cast the problem into a linear program, we introduce another variable to indicate if the -th packet (arriving in slot ) is in the virtual queueing system in slot , where if it is and otherwise. Since a total of packets arrives at the virtual queueing system by slot , the queue size in slot  can be expressed as , i.e., counting for all the packets arriving by slot . The total cost in Eq. (2) then becomes

where the first term is the flushing cost in slot and the second one is the holding cost in slot .

We propose an integer program for solving the offline scheduling problem in the virtual queueing system.

Integer program:

(3a)
s.t. (3b)
(3c)

In the integer program, the constraint in Eq. (3b) means that the -th packet arriving by slot (i.e., ) either stays at the virtual queueing system in slot (i.e., ) or has been flushed by slot (i.e., for some ).

By relaxing the integral constraint in Eq. (3c) to real numbers, we can obtain the following linear program.

Linear program (primal program):

(4a)
s.t. (4b)
(4c)

The relaxation has no integrality gap (similar to the argument for the ski-rental problem [2]); thus, a solution for in the linear program can minimize the total cost if connectivity pattern is given in advance, i.e., the solution is an optimal offline scheduling algorithm.

Now, we can see advantages of transforming into the virtual queueing system. The transformation along with the auxiliary variable , for all and , can produce a linear objective function in Eq. (4a) and a linear constraint in Eq. (4b). Next, we refer to the linear program as a primal program and express its dual program as follows.

Dual program:

(5a)
s.t. (5b)
(5c)

The primal-dual formulation will be employed in the next section for developing an online scheduling algorithm.

Iv Online scheduling algorithm design

We will develop an online scheduling algorithm using the primal and dual programs formulated in the previous section. For this purpose, we first propose a primal-dual learning algorithm in Section IV-A for obtaining a feasible solution to the primal and dual programs. The primal-dual learning algorithm does not know connectivity pattern ; instead, for each slot it obtains only. At the end of slot , the primal-dual learning algorithm returns a feasible solution to the primal and dual programs. Subsequently, we propose a randomized online scheduling algorithm in Section IV-C

by exploiting the solution from the primal-dual learning algorithm. The underlying idea is that the intermediate solution produced by the primal-dual learning algorithm in each slot can be viewed as the probability of flushing in that slot.

Iv-a Primal-dual learning algorithm

We propose the primal-dual learning algorithm in Alg. 1 for obtaining a feasible solution to the primal and dual programs in the online fashion. All the variables are initialized to be zeros in Line 1. In each new slot , Alg. 1 updates the values of all the variables according to the present channel state . If the channel is ON, then Alg. 1 updates the variables in Lines 11; if it is OFF, then Alg. 1 does in Lines 11. The updates are performed by iteration from (i.e., the first packet) until (i.e., the -th packet).

First, consider slot with . The value of in Line 1 implies if the -th packet has been flushed by slot . If , then the -th packet has been flushed; thus, no variable needs to be updated. On the contrary, if , then all the associated variables get updated in Lines 11.

More precisely, if the condition in Line 1 holds, then variable is updated to be the value of in Line 1 to satisfy the primal constraint in Eq. (4b). Moreover, the intuition of updating in Line 1 is that the value of can imply the probability of flushing the queue in slot . The more packets stay at the queue, the higher the flushing probability is, i.e., the higher the value of is. Thus, for those packets potentially staying at the queue in slot  (i.e., satisfying the condition in Line 1), Alg. 1 increases the value of according to Line 1. The constant in Line 1 is specified as in Line 1 for satisfying the dual constraint in Eq. (5b) (see Lemma 5 for detail). Regarding , it is updated to be one for maximizing the dual objective function in Eq. (5a).

Second, consider slot with . The value of remains unchanged (i.e., ) because the server has to idle in that slot. The value of , for , is the same as that in the previous slot (see Line 1) since the server can do nothing for those packets arriving prior to slot . However, is set to be one in Line 1 since the -th packet arriving in slot has to stay at the queue. In addition, similar to the case of , Alg. 1 sets to be one.

Remark 2.

Though a scheduling algorithm needs the values of only, Alg. 1 learns the values of all variables , , and . The values of other variables and will facilitate to analyze the primal objective value in Eq. (4a) computed by Alg. 1. See Theorem 7 later for detail.

1 , , for all and ;
;
  // is a constant.
/* For each new slot , the variables are updated as follows: */
2 if  then
       /* Consider those -th packets arrving by slot . */
3       for  to  do
4             if   then
5                   ;
6                   ;
7                   ;
8                  
9             end if
10            
11       end for
12      
13 else  //
14       for  to  do
15             if  then
16                   ;
17                  
18             else
19                   ;
20                  
21             end if
22            ;
23            
24       end for
25      
26 end if
Algorithm 1 Primal-dual learning algorithm.

Iv-B Analysis of Alg. 1

In this section, we demonstrate the feasibility of the solution produced by Alg. 1, while analyzing the resulting primal objective value. Since the value of variable is updated over slots, we use to represent the value of variable at the end (i.e., after update) of slot . Similarly, let and represent the corresponding values at the end of the -th iteration of slot .

In addition, for each -th iteration of slot , we use , for all , to represent the value of at the end of that iteration, while using to represent the value of at the beginning of that iteration. The two notation sets are just employed for simplifying the following proofs; in fact, in the -th iteration of slot , only one variable can be updated, but for all keeps unchanged.

We first establish the primal feasibility in the next lemma.

Lemma 3.

Alg. 1 produces a feasible solution to the primal program (4).

Proof.

See Appendix A. ∎

To examine the dual constraint in Eq. (5b), we need the following technical Lemma 4. In that lemma, for each slot  we consider an ordered set whose order follows the processing steps in Alg. 1. Namely, the set can be expressed by , where such that , , . We want to emphasize that the set does not capture all the iterations since slot , e.g, it excludes .

Lemma 4.

For a slot , assume that the -th element in the ordered set is for some and . If is updated to be one at the end of -th iteration of slot , then at the end of that iteration we have

(6)

for all and .

Proof.

See Appendix B. ∎

We next confirm the dual feasibility using the above lemma.

Lemma 5.

Alg. 1 produces a feasible solution to the dual program (5) if the constant is specified as .

Proof.

According to Lines 1, 1 and 1, we obtain , for all and , satisfying the dual constraint in Eq. (5c).

Considering a fixed slot , we will show that the value of in Eq. (5b) is less than or equal to . Note that , for and with , is updated to be one (in Line 1) if the condition in Line 1 holds. Thus, it suffices to show that at most elements in the ordered set can be updated to be one.

Suppose that the -th element in is for some and . According to Eq. (6) in Lemma 4, if the -th element in is updated to be one, then we have

for all and , where the last equality is based on as stated in the lemma. Thus, all the elements in after are no longer updated since their conditions in Line 1 fail. ∎

Remark 6.

According to the proof of Lemma 5, the computational complexity of Alg. 1 can be reduced by modifying the iteration in Line 1: consider iteration until where is the -th ON slot before slot , i.e., we remove all the iterations before . That is because the iterations before  cannot satisfy the condition in Line 1.

After establishing the feasibility of the solution returned by Alg. 1, the next theorem states the primal objective value in Eq. (4a) computed by Alg. 1.

Theorem 7.

The primal objective value computed by Alg. 1 is bounded above by

for all possible connectivity patterns .

Proof.

Let and be the increment of the primal objective value and that of the dual objective value, respectively, in the -th iteration of slot . Then, we derive and as follows.

  1. If and , then according to Lines 1 and 1, can be expressed as

    Moreover, according to Line 1.

  2. If and , then and because all the variables keep unchanged.

  3. If and , then and .

  4. If and , then , and .

The above four cases can conclude that

for all and .

By and we denote the primal and dual objective values computed by Alg. 1. Then, and ; therefore, the result follows since

where the last inequality is due to the weak duality [2]. ∎

Iv-C Randomized online scheduling algorithm

In this section, we propose a randomized online scheduling algorithm in Alg. 2. The algorithm updates variable in Line 2 in the same way as Alg. 1 does. Moreover, Alg. 2 uses additional variables: the value of in slot is the cumulative value of until (see Line 2); the value of in slot is the cumulative value of until slot (see Line 2). Let and be the corresponding values at the end of slot .

Alg. 2 picks a uniform random number in Line 2. According to Lines 2 - 2, if and there exists a such that , then the server decides to flush the queue in slot , i.e., the device decides to download the latest information in that slot. The intuition of Alg. 2 is that, with the uniform random choice of , the probability of flushing in slot can be derived as in Appendix C.

The next theorem presents the competitive ratio of Alg. 2.

Theorem 8.

The expected competitive ratio of Alg. 2 is

approaching as tends to infinity.

Proof.

The underlying idea is to show that the expected total cost incurred by Alg. 2 is less than or equal to the primal objective value computed by Alg. 1. See Appendix C for detail. ∎

Remark 9.

We remark that if the server flushed the queue in each ON slot with probability of directly, then the resulting expected total cost would be higher than the primal objective value computed by Alg. 1. Look at the second case of the proof of Theorem 8 (see Appendix C). In that case, the -th packet under the scheduling algorithm can stay at the queue in slot with a non-zero probability, yielding a non-zero expected holding cost in slot . Thus, the holding cost for the packet in slot is higher than the term in the primal objective value computed by Alg. 1.

1 for all ;
;
  // is a constant.
2 Choose a random number

with the continuous uniform distribution;

/* For each new slot , do as follows: */
3 if  then
4       for  to  do
5             if  then
6                   ;
7                  
8             end if
9            
10       end for
11      ;
12       ;
       /* The device decides whether to download the latest information as follows: */
13       if  then
14             The server decides to flush the queue;
15             ;
16            
17       else
18             The server decide to idle;
19            
20       end if
21      
22 end if
Algorithm 2 Randomized online scheduling algorithm.

V Numerical studies

Theorem 8 has analyzed Alg. 2 in the worst-case scenario; in contrast, this section will investigate Alg. 2 in average-case scenarios by computer simulations.

We run Alg. 2 for 10,000 slots with for a fixed value of for all . Note that, under the stationary setting, an optimal offline scheduling algorithm for minimizing the long-run average cost is of the threshold-type (see [10]). Figs. 5 - 5 display the time-average cost for the proposed Alg. 2 and the optimal offline scheduling algorithm. We can observe that the ratio between the average cost of Alg. 2 and that of the optimal offline scheduling algorithm is at most 1.20 (when and ); moreover, the ratio can even achieve 1.0048 (when and ). The average of the ratios for those results in Figs. 5 - 5 is 1.07. In summary, Alg. 2 performs much better than what we analyzed in the worst-case scenario (with the expected competitive ratio of 1.58).

Moreover, we compare Alg. 2 with an online scheduling algorithm: for each slot  with , the server flushes the queue when and idles otherwise. The idea is to make a decision in a greedy way such that the resulting cost in the present slot can be minimized. Figs. 5 - 5 also display the time-average cost for the greedy scheduling algorithm. We can observe that the proposed Alg. 2 significantly outperforms the greedy scheduling algorithm, except for the case when and . The exception is because the greedy algorithm happens to take the threshold close to the optimal one.

Fig. 4: Average costs for different values of when .
Fig. 3: Average costs for different values of when .
Fig. 4: Average costs for different values of when .
Fig. 5: Average costs for different values of when .
Fig. 3: Average costs for different values of when .

Vi Extensions

In this section, we extend the proposed Alg. 2 to non-linear age cost functions and dynamic power control. Without loss of generality, we can focus on modifying the primal-dual learning algorithm in Alg. 1.

Vi-a Non-linear age cost functions

Let be a function of the age of information, indicating the cost incurred by age . Without loss of generality, we can assume that ; if not, we can scale up the value of and the age cost function to make the function values becomes integers. In this context, packets arrive at the virtual queueing system at the beginning of slot . Then, we can modify the iterations in both Lines 1 and 1 of Alg. 1 to the iterations from to the total number of packets until slot . With the modification, all the previous feasibility results and the competitive ratio hold.

Vi-B Dynamic power control

Next, consider the scenario where the device can adjust its transmission power according to the present channel quality. We focus on the linear age cost function. Let be the minimum cost of downloading the latest information in slot  if . Suppose that at most power levels can be used and consider with . Then, by replacing the in Line 1 of Alg. 1 with to make the dual program feasible, we can obtain the same feasibility results while achieving the competitive ratio of

approaching again as tends to infinity.

Vii Conclusion

This paper treated a mobile network where a mobile device is running an application. To realize the mission of the application, the device needs to download the latest information through neighboring access points. The paper proposed a randomized energy-efficient scheduling algorithm for the mobile device to decide whether to download for each slot. In particular, the proposed algorithm enjoys the online feature without the knowledge of the channel dynamics or the time for running the application. The proposed scheduling algorithm can asymptotically achieve the expected competitive ratio of

. Interesting extensions of this work include scheduling for multiple information downloads, imperfect channel estimation, and partial information about the connectivity pattern.

Appendix A Proof of Lemma 3

The primal constraint in Eq. (4c) is obviously true. Next, we verify the primal constraint in Eq. (4b) in the following four cases.

  1. If and , then is updated to be in Line 1 of Alg. 1. Therefore, the primal constraint in Eq. (4b) holds since

    where the last inequality is based on for all , because Alg. 1 increases the value of in Line 1 and never decreases it.

  2. If and , then the value of is zero. Therefore, the primal constraint holds since

  3. If and , then is set to be one in Line 1 of Alg. 1, agreeing with the primal constraint.

  4. If and , then (according to Line 1 of Alg. 1) is the same as with where is the previous ON slot of slot , i.e., is the previous ON slot of slot after its arriving slot . Because the primal constraint in slot  holds (by the above three cases), the primal constraint in slot  holds as well.

To conclude, the solution produced by Alg. 1 satisfies Eq. (4b).

Appendix B Proof of Lemma 4

We prove the claim in Eq. (6) by induction on . First, suppose that the first element in the ordered set is updated to be one. Then, the claim in Eq. (6) is true when because

where the last inequality is because the value of increases by as least in Line 1 of Alg. 1.

Second, assume that , for some and , is the -th element in . Suppose that is updated to be one at the end of -th iteration of slot and the claim in Eq. (6) holds.

Then, we consider the -th element in the following. Assume that is the -th element in and it is updated to be one at the end of -th iteration of slot . In that iteration, variable is updated according to Line 1 of Alg. 1, and hence we can obtain

(7)

where (a) is due to the value of , for all , at beginning of the -th iteration of slot is greater than or equal to that at the end of -th iteration of slot ; (b) is due to . Then, we can establish the claim in Eq. (6) by

where (a) is due to and ; (b) is based on the inequality in Eq. (7); (c) results from the induction hypothesis for .

Appendix C Proof of Theorem 8

The proof needs the following technical lemma.

Lemma 10.

The server running Alg. 2 flushes the queue in slot  with probability of .

Proof.

First, if , then the probability of flushing in slot  is . Second, consider slot with . According to Lines 2 - 2 of Alg. 2, the server decides to flush in slot  if there exists a such that . Let

be the continuous uniform random variable whose value is between 0 and 1. Then, under Alg. 

2 the probability of flushing in slot can be derived in the following two cases.

  1. If , then is the only one value of such that can belong to . Then, the probability of flushing in slot is

  2. If , then similar to the first case the probability of flushing in slot is

Then, we complete the proof. ∎

We proceed to compare the expected total cost incurred by Alg. 2 with the primal objective value in Eq. (4a) computed by Alg. 1. First, according to Lemma 10, the expected number of flushing in slot  (under Alg. 2) is . Therefore, the expected flushing cost incurred by Alg. 2 in slot  is , which is less than or equal to the term in the primal objective value computed by Alg. 1.

Second, we show that the expected cost incurred by Alg. 2 for holding the -th packet in slot , for all and , is less than or equal to the term in the primal objective value computed by Alg. 1:

  1. If and , then (similar to Lemma 10) the server running Alg. 2 flushes the queue during the period from slot until with probability of . Therefore, under Alg. 2 the expected number of the -th packet left in the virtual queueing system in slot  is

    where the inequality is because in Alg. 1 the value of , for all , at the beginning of the -th iteration of slot  is less than or equal to that at the end of slot ; additionally, it is less than one in this case. Thus, the expected cost incurred by Alg. 2 for holding the -th packet in slot is less than or equal to the term in the primal objective value computed by Alg. 1.

  2. If and , then under Alg. 2 the -th packet must be flushed by slot since in this case there must exist a such that (similar to Lemma 10). Thus, the cost incurred by Alg. 2 for holding the -th packet in slot  is zero, which is the same as produced by Alg. 1.

  3. If and , then under Alg. 2 the -th packet stays at the queue in slot , yielding one unit of the holding cost. The holding cost is the same as produced by Alg. 1.

  4. If and , then with the slot  in the fourth case of Appendix A; moreover, the expected cost incurred by Alg. 2 for holding the -th packet in slot is the same as that in slot . Since the expected holding cost for the packet in slot  is less than or equal to (according to the above three cases), the holding cost for the packet in slot is less than or equal to as well.

We then can conclude that the expected total cost incurred by Alg. 2 is less than or equal to the primal objective value computed by Alg. 1. The result stated in the theorem immediately follows from Theorem 7.

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