Minimum Age TDMA Scheduling

05/26/2019
by   Tung-Wei Kuo, et al.
National Chengchi University
0

We consider a transmission scheduling problem in which multiple systems receive update information through a shared Time Division Multiple Access (TDMA) channel. To provide timely delivery of update information, the problem asks for a schedule that minimizes the overall age of information. We call this problem the Min-Age problem. This problem is first studied by He et al. [IEEE Trans. Inform. Theory, 2018], who identified several special cases where the problem can be solved optimally in polynomial time. Our contribution is threefold. First, we introduce a new job scheduling problem called the Min-WCS problem, and we prove that, for any constant r ≥ 1, every r-approximation algorithm for the Min-WCS problem can be transformed into an r-approximation algorithm for the Min-Age problem. Second, we give a randomized 2.733-approximation algorithm and a dynamic-programming-based exact algorithm for the Min-WCS problem. Finally, we prove that the Min-Age problem is NP-hard.

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I Introduction

We consider systems whose states change upon reception of update messages. Such systems include, for example, web caches [2], intelligent vehicles [3], and real-time databases [4]. The timely delivery of update messages is often critical to the smooth and secure functioning of the system. Moreover, since any given update is likely dependent on previous updates, the update messages should not be delivered out of order. In most cases, the system does not have exclusive access to a communication channel. Instead, it must share the channel with other systems. Hence, the transmission schedule plays a crucial role in determining the performance of the systems that share the channel.

This scenario can be modeled by multiple sender-receiver pairs and a channel shared by these sender-receiver pairs. The sender sends update messages to the receiver through the shared channel, and the receiver changes its state upon reception of an update message.111The sender may serve as a relay or hub for the system and thus may not be responsible for generating update messages. This paper discusses the design of transmission scheduling algorithms for such channels. Specifically, we assume that the channel has a buffer in which the update messages are stored, and a transmission schedule for the messages in the buffer must be determined.222The buffer may be a logical one that stores the inputs to a scheduler. In this paper, we refer to a system that changes its state upon reception of an update message as a receiver.

To keep the state of a receiver as fresh as possible, it is important to keep the age of the receiver as small as possible. Specifically, the age of a receiver is the age of the receiver’s most recently received message , i.e., the difference between the current time and the time at which is generated. Most prior research analyzes the age of a receiver through stochastic process models [5, 6, 7, 8, 9, 10, 11, 12, 13]

, where the randomness comes from the state of the channel or the arrival process of update messages. In this paper, we take a combinatorial optimization approach to minimize the overall age of all receivers on a reliable channel. In particular, we study the problem defined by He

et al., who considered a scenario in which the transmission scheduling algorithm is invoked repeatedly [14]. Specifically, after the scheduling algorithm computes a schedule, the channel then delivers the messages according to the schedule. New messages may arrive while the channel is delivering the scheduled messages. These new messages are stored in the buffer and scheduled for transmission during the next invocation of the algorithm.

The scheduling algorithm should be designed with the characteristics of the channel in mind. For example, He et al. considered a wireless channel, in which various senders might interfere with one another [14]. They also considered a Time Division Multiple Access (TDMA) channel, in which the channel delivers one message at a time. They identified some conditions in which optimal schedules can be obtained by sorting the sender-receiver pairs according to the number of messages to be sent to the receiver [14]. However, even if the channel is TDMA-based, it remained open whether the problem can be solved optimally in polynomial time. In this paper, we therefore focus on TDMA channels. In the remainder of this paper, we refer to this scheduling problem on a TDMA channel as the Min-Age problem.

In this paper, we cast the Min-Age problem as a job scheduling problem called the Min-WCS problem. The Min-WCS problem has a simple formulation inspired by a geometric interpretation of the Min-Age problem. The simplicity of the formulation also facilitates algorithm design. As we will see in Section VII, one may solve variants of the Min-Age problem by modifying the geometric interpretation and then solving the corresponding job scheduling problem.

Job scheduling has been studied for decades. In fact, the Min-WCS problem is a special case of single-machine scheduling with a non-linear objective function under precedence constraints, which has been studied by Schulz and Verschae [15] and Carrasco et al. [16]. Specifically, for any , the algorithm proposed by Schulz and Verschae approximates the optimum within a factor of when the objective function is concave [15]. When the objective function is convex, Carrasco et al. proposed a -speed 1-approximation algorithm for any  [16].333Specifically, let be the optimal objective value. An -speed -approximation algorithm for a minimization problem finds a solution of objective value at most when using a machine that is times faster than the original machine. The solutions proposed by Schulz and Verschae [15] and Carrasco et al. [16]

are based on linear programming rounding. The objective function of the Min-WCS problem is convex, and we give a randomized 2.733-approximation algorithm for the Min-WCS problem without linear programming. We summarize our major results as follows:

Theorem 1: We introduce the Min-WCS problem and prove that, for any constant , every -approximation algorithm of the Min-WCS problem can be transformed into an -approximation algorithm for the Min-Age problem.

Theorem 2: We solve the Min-WCS problem by combining two feasible schedules. Specifically, we propose a deterministic 4-approximation algorithm and a randomized 2.733-approximation algorithm for the Min-WCS problem.

Theorem 3: We give a dynamic-programming-based exact algorithm for the Min-WCS problem. The result implies that the Min-Age problem can be solved optimally in polynomial time when the number of sender-receiver pairs is a constant. The result holds even if there are arbitrarily many messages.

Theorem 4: We show that the Min-Age problem is NP-hard.

Ii Problem Definition

The studied problem is first considered by He et al., and is referred to as the minimum age scheduling problem with TDMA [14]. Throughout this paper, we simply refer to this problem as the Min-Age problem. To make the paper self-contained, we rephrase the definition of the Min-Age problem.

Inputs: We consider sender-receiver pairs, , where and are the sender and receiver of the th sender-receiver pair, respectively. Time is indexed by non-negative integers, and the current time is . These sender-receiver pairs share one transmission channel, which can transmit one message in one unit of time (hence the name TDMA). Each sender has a set of messages to be sent to receiver . Our task is to schedule the transmissions of messages in .

We use (the birthday of ) to indicate the time at which message is generated. Let be the latest message that has been received by by time .444Recall that a receiver is defined as a system that changes its state upon reception of an update message. The system is first assigned a state during the initialization phase. Thus, if has not received any message sent from , is the initial information installed on during the initialization phase. Thus, . Let be the th oldest message in . Thus, for all , .

Output and constraints: The goal is to find a schedule of message transmissions so that the overall age of information (to be defined later) is minimized. Let be the time at which message is received by under schedule . Hence, by the channel capacity constraint, is the time at which the channel starts to send under schedule . Let be the time needed to send all the messages. A feasible schedule has to satisfy the following constraints.

  1. Due to the channel capacity constraint, is a one-to-one and onto mapping from to .

  2. Since a message may depend on previous messages, the schedule must follow the order of message generation. Specifically, for all , . In other words, for each sender-receiver pair, the transmission schedule must follow the first-come-first-served (FCFS) discipline.

Age: Let be the latest message received by receiver at or before time under schedule . The age of at time is the age of at time , i.e., . Like [14], we assume that, once receives all messages in , the age of becomes zero. Intuitively, under this assumption, a scheduling algorithm that minimizes the overall age would have the side benefit that the last message of each sender-receiver pair is sent as early as possible (under the FCFS discipline). More supporting arguments for this assumption can be found in [14]. Specifically, the age of at time under schedule , , is defined as follows.

otherwise.

Notice that is not used when evaluating the age of . Moreover, is referred to as the initial age of receiver . In Section VII, we will discuss the case where the age of does not become zero even if receives all messages in .

Objective function: In the Min-Age problem, the goal is to minimize the overall age, which adds up the ages of all receivers at all time indices. Specifically, the goal is to find a feasible schedule that minimizes

Fig. 1: An example of the Min-Age problem.
Example 1 (Min-Age Problem).

We give an example in [14] with our notation.555The example is shown in Fig. 5 in [14]. We consider two sender-receiver pairs, where and . Specifically,

Consider the schedule shown in Fig. 1 with

Observe that is a one-to-one and onto mapping from to , where and . Moreover, follows the first-come-first-served policy. Hence, is a feasible schedule. . . Hence, .

Iii A Corresponding Job Scheduling Problem and Problem Transformation

In this paper, we cast the Min-Age problem as a job scheduling problem called the Min-WCS problem. We first give the definition of the Min-WCS problem in Section III-A. We then show that the Min-Age problem can be transformed into the Min-WCS problem in Section III-B.

Iii-a The Min-WCS Problem

We consider a job scheduling problem with precedence constraints. That is, the order of job completion has to follow a given precedence relation . Specifically, for any two jobs and , if , then , where is the completion time of job under schedule . We consider chain-like precedence constraints. Specifically, the set of all jobs is divided into job chains, , where is a chain of jobs, . For any feasible job schedule and any , . Throughout this paper, denotes the th job of job chain . is called a leaf job if ; otherwise, it is called an internal job.

We are now ready to define the job scheduling problem considered in this paper. The input consists of job chains, where each job is associated with a non-negative weight . The processing time of every job is one unit of time, and the system only has one machine, which starts processing jobs at time 0. All jobs are non-preemptive. Hence, the completion time of the last completed job is . Since the processing time of each job is one unit of time, a feasible schedule is a one-to-one and onto mapping from the set of all jobs to . The goal is to find a feasible schedule that minimizes , where is the total weighted completion time of all jobs under , and is the total completion time squared of all leaf jobs under . Specifically,

and

In this paper, we refer to this job scheduling problem as the Min-WCS problem.

Iii-B Transformation from the Min-Age Problem to the Min-WCS Problem

In this subsection, we give a method to solve the Min-Age problem by transforming it into the Min-WCS problem. The high-level idea is to construct a corresponding job for each message . Specifically, given a problem instance of the Min-Age problem, we construct a corresponding instance of the Min-WCS problem, where

(1)

and

(2)

The job weight is determined by and . Specifically,

(3)

and

(4)

Note that, since , all weights are non-negative, and thus this is a valid problem instance of the Min-WCS problem. Since we have and in the transformation, in what follows, we omit the subscript of and .

Example 2 (The transformation).

Consider the Min-Age problem instance in Example 1. We transform into the following instance of the Min-WCS problem. has two jobs chains. The first job chain has three jobs, and the second job chain has two jobs. The weights of the first two jobs in are

and

The weight of the last job in is

Similarly, we have and . Recall that, in Fig. 1, . Consider a schedule such that for all , . We then have and . Notice that .

Fig. 2: A geometric interpretation of .

The rationale behind the transformation: We give a geometric interpretation of .666He et al. also gave a geometric interpretation of  [14]. The geometric interpretation proposed in this paper is different from that in [14], and our interpretation naturally suggests a transformation into the job scheduling problem defined in this paper. We use Fig. 2 to explain the idea. Notice that in Fig. 1, is the total area of rectangles shown in Fig. 1. In Fig. 2, we divide the overall age of into white rectangles and gray rectangles. Since we only consider the total area, we right-shift all rectangles by 0.5 unit. For , there are white rectangles, and the width of the th white rectangle is . The height of the th white rectangle is (if ) or (if ). The height can be interpreted as the age reduction after receiving message . Note that, after receiving the last message, the age becomes zero. Hence, the total height of the white rectangles should be , i.e., the initial age of . Therefore, the height of the bottom white rectangle is . After considering age reduction, we still need to increase the age by one after each unit of time. This is captured by the gray rectangles. The width of every gray rectangle is one, and the heights of gray rectangles are . Hence, the total area of the gray rectangles is . Let be any feasible schedule of a Min-Age problem instance . We have

Let be ’s corresponding job scheduling problem instance. Specifically, and satisfy Eq. (1) to Eq. (4). Let be any feasible schedule of . We have

Thus, if holds for all , , we then have .

The above result then suggests the following method to construct a schedule for . First, obtain a schedule of the corresponding Min-WCS problem instance . We then view as the transmission order of in . Specifically, we set . The following lemma establishes the relation between and . Throughout this paper, we use and to denote problem instances of the Min-Age problem and the Min-WCS problem, respectively.

Lemma 1.

Let and be any two schedules of and , respectively. If and satisfy Eq. (1) to Eq. (4), and , for all , , then

  1. is feasible if and only if is feasible.

  2. .

Proof.

By the above discussion, we already have . Since for all , , is a one-to-one and onto mapping from to if and only if is a one-to-one and onto mapping from the set of all jobs to . On the other hand, it is easy to see that follows the first-come-first-served policy for each sender-receiver pair if and only if follows the chain-like precedence constraint. Thus, is feasible if and only if is feasible. ∎

The next lemma establishes the relation between the optimums of a Min-Age problem instance and the corresponding Min-WCS problem instance.

Lemma 2.

Let and be the optimal schedules of and , respectively. If and satisfy Eq. (1) to Eq. (4), then .

Proof.

Let be a schedule such that for all , . Similarly, let be a schedule such that for all , . By Lemma 1, we have

Finally, since

and

we have . ∎

Theorem 1.

For any constant , if there exists a polynomial-time -approximation algorithm for the Min-WCS problem, then there exists a polynomial-time -approximation algorithm for the Min-Age problem.

Proof.

The -approximation algorithm for the Min-Age problem proceeds as follows. First, given a problem instance of the Min-Age problem, the algorithm constructs a corresponding instance of the Min-WCS problem by the aforementioned transformation. Obviously, the transformation can be done in polynomial time. We then apply the -approximation algorithm for the Min-WCS problem on to get a schedule . We construct a schedule for by setting for all , . By Lemmas 1 and 2, is feasible and . ∎

Iv Approximation Algorithms for the Min-WCS Problem

By Theorem 1, to solve the Min-Age problem, it suffices to solve the Min-WCS problem. Notice that the objective function of the Min-WCS problem is the sum of two functions, and . When the objective function becomes (respectively, ), we refer to the problem as the Min-WC problem (respectively, the Min-CS problem). Both the Min-WC problem and the Min-CS problem can be solved optimally in polynomial time. Given an instance of the Min-WCS problem, the high-level idea of our algorithm is to first solve the corresponding instances of the Min-WC problem and the Min-CS problem. Throughout this paper, we use (respectively, ) to denote the optimal schedule of the Min-WC problem (respectively, the Min-CS problem). We then interleave with to approximate the Min-WCS problem. We first discuss the solutions of the Min-WC problem and the Min-CS problem in Section IV-A. We then present our algorithm for the Min-WCS problem in Section IV-B.

Iv-a Algorithms for the Min-WC Problem and the Min-CS Problem

Iv-A1 The Min-WC Problem

The Min-WC problem is a special case of the minimum total weighted completion time scheduling problem subject to precedence constraints, which has been studied over many years [17, 18, 19]. When the precedence constraints are chain-like, the problem can be solved in polynomial time [18, 19]. Recall that, in our problem, the processing time of every job is one. The algorithm for the Min-WC problem proceeds as follows. For each job , define the job’s priority as . To minimize the total weighted completion time, the machine should first process the job with the highest priority. We still need to follow the precedence constraints. Hence, to determine the next processing job, we only consider the first unprocessed job in each job chain, and we choose the one that has the highest priority. Algorithm 1 summarizes the pseudocode.

1 for  to  do
2        the set of the first unscheduled job in each job chain
3       
4       
5       
Algorithm 1 An Algorithm for the Min-WC Problem
Lemma 3 (Lawler [18]).

Algorithm 1 solves the Min-WC problem optimally in polynomial time.

Example 3 (Algorithm 1).

Consider the problem instance in Example 2. We have and . Since , Algorithm 1 first schedules and sets . The job completion order under is .

Iv-A2 The Min-CS Problem

By a simple interchange argument, it is easy to see that the shortest job chain should be completed first in the Min-CS problem. Algorithm 2 summarizes the pseudocode. We have the following lemma.

Lemma 4.

Algorithm 2 solves the Min-CS problem optimally in polynomial time.

1
2
3 while  do
4       
5       
6        for  to  do
7              
8              
9              
10       
Algorithm 2 An Algorithm for the Min-CS Problem
Example 4 (Algorithm 2).

Consider the problem instance in Example 2. Since , the job completion order under is .

Observe that in Example 3 and Example 4, . It is easy to see that and are thus optimal schedules of the Min-WCS problem. Therefore, the optimal message transmission order in Example 1 is , and the optimal overall age is .

Proposition 1.

Let be any instance of the Min-WCS problem, and let and be the optimal schedules of the corresponding instances of the Min-WC problem and the Min-CS problem, respectively. If , then and are optimal schedules of .

Iv-B Interleaving and Randomly: A Randomized Approximation Algorithm for the Min-WCS Problem

While and solve the Min-WC problem and the Min-CS problem, respectively, neither nor can approximate the Min-WCS problem well. Specifically, we have the following results, whose proof can be found in the appendix.

Proposition 2.

The approximation ratio of Algorithm 1 for the Min-WCS problem is .

Proposition 3.

The approximation ratio of Algorithm 2 for the Min-WCS problem is .

Despite the above negative results, we will show that interleaving and gives an -approximation algorithm for the Min-WCS problem. A critical observation of the Min-WC problem (respectively, the Min-CS problem) is that, if we multiply the optimal scheduled completion time (respectively, ) of every job by a factor (i.e., we delay the optimal schedule by a multiplicative delay factor of ), then the total weighted completion time (respectively, the total completion time squared of all leaf jobs) is increased by a multiplicative factor of (respectively, ). This immediately suggests the following deterministic 4-approximation algorithm: For each job , set . Hence, is a delayed version of with a delay factor less than two777Although different jobs have different delay factors, every job has a delay factor less than two., and the time period is idle for any integer . We call such an idle time period an idle time slot. Moreover, define the finish time of an idle time slot as . Consider another schedule obtained by setting for each job . Hence, is a delayed version of with a delay factor of two. We can view as a schedule obtained by inserting jobs one by one following the order specified in to the idle time slots in . For each job , set . We will show that satisfies the precedence constraints. Finally, we remove the idle time slots in to obtain the final schedule . We then have

and

Thus,

Since is a lower bound of the optimum of the Min-WCS problem, is a 4-approximation solution.

1 the schedule obtained by Algorithm 1
2 the schedule obtained by Algorithm 2
3
4
5 for  to  do
6       

is set to 1 with probability

and is set to 0 with probability
7        if  then
8               forall Job such that  do
9                     
10              
11       
12for  to  do
13        the th completed job under
14        the finish time of the th idle time slot in
15       
16forall  do
17       
18       
19for  to  do
20        the th completed job under
21       
22       
Algorithm 3 An Algorithm for the Min-WCS Problem with Parameter

In hindsight, we first insert idle time slots to and then insert jobs to the idle time slots following the order specified in . To improve the algorithm, we insert idle time slots to randomly. Specifically, let be a number in . Initially, . For every two jobs and that are processed contiguously in (i.e., ), we insert an idle time slot between and with probability . Notice that, in , we never insert two or more contiguous idle time slots, which is a critical property that will be used in the analysis. Algorithm 3 summarizes the pseudocode. Observe that this randomized algorithm degenerates to Algorithm 2 when , and this randomized algorithm degenerates to the aforementioned deterministic 4-approximation algorithm when .

Fig. 3: An example of Algorithm 3.
Example 5 (Algorithm 3).

Consider a Min-WCS problem instance with two job chains where and . Hence, the job completion order under is . Assume that the job completion order under is . Assume . and are shown in Fig. 3.

Since and do not overlap, we never execute two jobs at the same time in . Thus, to prove that is feasible, it remains to prove the following lemma.

Lemma 5.

follows the precedence constraints.

Proof.

For the sake of contradiction, assume that there are two jobs and from the same job chain such that but . We first consider the case where . Hence, we must have (otherwise, and would violate the precedence constraints). Since follows the precedence constraints, . Finally, since