OOn Competitive Analysis for Polling Systems
Jin Xu ∗ Natarajan Gautam † Abstract
Polling systems have been widely studied, however most of these studies focus on polling systems withrenewal processes for arrivals and random variables for service times. There is a need driven by practicalapplications to study polling systems with arbitrary arrivals (not restricted to time-varying or in batches)and revealed service time upon a job’s arrival. To address that need, our work considers a polling systemwith generic setting and for the first time provides the worst-case analysis for online scheduling policiesin this system. We provide conditions for the existence of constant competitive ratios, and competitivelower bounds for general scheduling policies in polling systems. Our work also bridges the queueing andscheduling communities by proving the competitive ratios for several well-studied policies in the queueingliterature, such as cyclic policies with exhaustive, gated or l -limited service disciplines for polling systems. Keywords—
Scheduling, Online Algorithm, worst-case Analysis, Competitive Ratio, Parallel Queues withSetup Times
This study has been motivated by operations in smart manufacturing systems. As an illustration, considera 3D printing machine that uses a particular material informally called “ink” to print. Jobs of the sameprototype are printed using the same ink, and when a different prototype (for simplicity, say a differentcolor) is to be printed, a different ink is required and the machine undergoes a setup that takes time toswitch inks. The unprocessed jobs of the same prototype can be regarded as a “queue”. This problemcan thus be modeled as a polling system where the server polls a queue, processes jobs, incurs a setuptime, processes another queue, and so on. In practice, besides the ink (material), other factors such asprocessing temperature, equipment setting and other processing requirements that are required by differentjob prototypes will also incur setup times.Another interesting feature of 3D printing is that it is possible to reveal the processing time of each jobupon the job’s arrival. This is because, before getting printed, the printing requirements such as temperature,nozzle route, printing speed, and so on are specified for the job, and using that we can easily acquire theprinting time before processing. Therefore, it is unnecessary to assume that the processing time of a jobis stochastic at the start of processing, even though many other queueing research papers do so. In thispaper, we assume that the processing time of jobs could be arbitrary, and could be revealed deterministicallyupon arrival. Furthermore, the 3D printer that prints customized parts usually receives jobs with differentprocessing requirements. Job arrivals thus could be time-varying, non-renewal, in batches, dependent, or ∗ Corresponding author, Email: [email protected], Industrial and Systems Engineering, Texas A&M University, TX 77843,USA † Email: [email protected], Industrial and Systems Engineering, Texas A&M University, TX 77843, USA a r X i v : . [ c s . PF ] J un igure 1.1: Polling System with Four Queueseven arrivals without a pattern. It motivates us to consider the generic polling system without imposing anystochastic assumptions on future arrivals.In addition to 3D printing, many other examples of such a general polling system can be found incomputer-communication systems, reconfigurable smart manufacturing systems and smart traffic systems [27,8, 31, 46, 7]. In such systems, job arrivals are arbitrary, processing times of jobs are revealed deterministicallyupon arrival, and a setup time occurs when the server switches from one queue to another. We call sucha polling system “general” mainly because we do not impose any stochastic assumptions on job arrivals orservice times. Having a scheduling policy that works well in such a general setting could prevent the systemfrom performing erratically when rare events occur. Knowing the worst-case performance of a policy also aidsin designing reliable systems. There are very few studies discussing the optimal policies or online schedulingalgorithms for the polling system without stochastic assumptions due to the complexity of analysis [48]. Itis still unknown if those scheduling policies designed for specific polling systems work well in the generalsetting. In our paper, we study completion time minimization for the general polling system from an onlineoptimization perspective and obtain the worst-case performance (i.e., the competitive ratios) of several widelyused scheduling policies with known long-run average waiting time performance, such as cyclic exhaustiveand gated policies. We also, for the first time, provide conditions for the existence of constant competitiveratios for online policies in polling systems. Our work bridges the scheduling and queueing communitiesby showing that some queueing policies that work well under stochastic assumptions also work well in thegeneral scheduling setting. As mentioned earlier, unprocessed jobs from the same prototype (or family) can be modeled as a queue. Wethus consider a single server system with k parallel queues, as in Figure 1.1. The processing time p i of the i th arriving job is revealed instantly upon its arrival time r i . The server can serve the jobs that are waitingin queues in any order non-preemptively. However, a setup time τ is incurred when the server switches fromone queue to another. Information ( r j , p j ) about a future job j (for all j ) remains unknown to the serveruntil job j arrives in the system. The objective is to find the service order for jobs and queues to minimizetotal completion time over all jobs, where the completion time of a job is the time period from time 0 tothe time when the job has been served (exits the system). It is assumed that the total number of jobs is anarbitrary large finite quantity. 2 .2 Preliminaries In this subsection we mainly introduce some concepts and terminologies that we use in this paper. Importantnotations in this paper are provided in Table 1.
Machine Scheduling Problems
Using the notation for one machine scheduling problems [18], we write our problem as | r i , τ | (cid:80) C i , where“ ” refers to the single machine in the system, r i and τ in the middle field refer to the release date andsetup time constraints, and (cid:80) C i in the last field is the completion time objective. In the later part of thispaper we will introduce other constraints, which will appear in the middle field, such as τ ≤ θp min and p max ≤ γp min , where p max = sup i p i is the processing time upper bound and p min = inf i p i is the processingtime lower bound. We say the processing time variation is bounded if p max ≤ γp min for constant γ . If noconstraint is specified in this field, it means 1) all jobs are available at time zero, 2) no precedence constraintsare imposed, 3) preemption is not allowed, and 4) no setup time exists. In this paper, we assume jobs andsetup times are non-resumable and all the policies that we discuss are non-preemptive, unless we specificallystate otherwise. Other objectives are C max , the maximal completion time or makespan [26] and (cid:80) w i C i , the weighted completion time [19]. Online and Offline Problems
A job instance I with n ( I ) number of jobs is defined as a sequence of jobs with certain arrival times andprocessing times, i.e., I = { ( r i , p i ) , ≤ i ≤ n ( I ) } . In this work we mainly focus on an online schedulingproblem, where r i and p i remain unknown to the server until job i arrives. In contrast to the online problem,the offline problem has entire information for the job instance I , i.e., ( r , r , ..., r n ( I ) ) and ( p , p , ..., p n ( I ) ) from time 0. The offline problem is usually of great complexity. The offline version of our online problem,i.e., | r i , τ | (cid:80) C i , is strongly NP-hard since the offline problem with τ = 0 is strongly NP-hard [25, 19, 23].However it is important to note that if preemption is allowed, then Shortest Remaining Processing Time (SRPT) is the optimal policy for | r i , pmtn | (cid:80) C i with τ = 0 (see [40]). SRPT is online and polynomial-timesolvable. It can be used as a benchmark for online scheduling policies. Scheduling Policies
A scheduling policy π specifies when the server should serve which job. In our paper we mainly focus ononline (or non-anticipative) policies [50, 6]. Online policies can be either deterministic or randomized. Forexample, SRPT is a deterministic policy. A randomized policy may toss a coin before making a decision, andthe decision may depend on the outcome of this coin toss [6]. Detailed discussions for randomized algorithmscan be found in [41, 10]. In this paper, we focus on deterministic policies.
Competitive Ratios
A competitive ratio is the ratio between the solution obtained by an online policy and the benchmark . Inthis paper, the optimal solution to the offline problem is our benchmark. We say the competitive ratio for anonline policy is ρ if sup I C π ( I ) C ∗ ( I ) ≤ ρ , where C π ( I ) is the completion time for job instance I by a deterministicscheduling policy π and C ∗ ( I ) is the optimal completion time of the offline problem. We say a competitiveratio is tight if there exists an instance I such that C π ( I ) C ∗ ( I ) = ρ. r i Release date, the time when job i arrives in the system p i Processing time for job ip max Maximum processing time, p max = max i { p i } p min Minimum processing time, p min = min i { p i } τ Setup time is τ for all queues I = { r i , p i } , for ≤ i ≤ n ( I ) Job instance, set of jobs withinformation about release dateand processing time for all jobsin it C π ( I ) Total completion time for jobsin instance I under policy π C ∗ ( I ) Total completion time for jobsin instance I under the optimalpolicy of the offline problem C πi The completion time for job i under policy π n ( I ) Number of jobs in job instance Iγ Processing time variation, aconstant such that p max ≤ γp min θ A constant such that τ ≤ θp min k Number of queues in the system κ =max { k +1 k γ, k + 1 } Competitive ratio forcyclic-based and exhaustive-likepolicies, a constantTable 1: NotationsProblem Deterministic RandomizedLower Bounds Upper Bounds Lower Bounds Upper Bounds | r i , pmtn | (cid:80) C i | r i , pmtn | (cid:80) w i C i [37] | r i | (cid:80) C i ee − [41] ee − [10] | r i | (cid:80) w i C i ee − [41] 1.686 [17] | r i , p max p min ≤ γ | (cid:80) w i C i √ γ +1 − γ [44] √ γ +1 − γ [44] Unknown Unknown | r i , p max p min ≤ γ | (cid:80) C i Numerical [43] γ − √ γ ( γ − [43] Unknown UnknownTable 2: Competitive Ratios for single-machine Scheduling Problem without Setup Times (i.e., τ = 0 ) Single-machine scheduling problems with setup times or costs have been widely studied from various perspec-tives. A detailed review of the literature can be found in [4, 2, 1, 3, 35]. Other studies considering machinesetup can be found in [45, 20, 33, 32, 38]. However, these papers focus on solving the offline problem whereall the release times and processing times are revealed at time .Numerous research papers have shed light on the online single-machine scheduling problems, withoutconsidering setup times. Table 2 summarizes the current state of art of competitive-ratio analysis overexisting online algorithms for these problems. In addition, a recent paper [30] provides a new method toapproximate the competitive ratio for general online algorithms.The online makespan minimization problem for the polling system is considered in [12] and an O (1) policy is proved to exist. However, the competitive ratio provided in [12] is very large. A 3-competitiveonline policy for the polling system with minimizing the completion time is provided in [52], but only forthe case where k = 2 and jobs are identical. To the best of our knowledge, online policies for general pollingsystems with setup time consideration have not been well studied.4s we mentioned before, there are articles that study polling systems from a stochastic perspective byassuming job arrivals, service times and setup times are stochastic. Long-run average waiting time, queuelength and other metrics of policies are considered for different types of stochastic assumptions [48]. Exactmean waiting time analysis for cyclic routing policies with exhaustive, gated and limited service disciplines forpolling system of M/G/ type queues have been provided in [14, 42, 36, 51, 47, 15]. Service disciplines withinqueues (such as FCFS, SRPT and others) are discussed in [49], but routing disciplines are not discussed. Theoptimal service policy for the symmetric polling system is provided by [28], where “symmetric” means thatqueues and jobs are stochastically identical. However, the optimal solution for the general polling systemremains unknown [8, 48]. Approximating algorithms for the polling system are very few, and none of thosewidely-used policies have been shown to work well in general settings.The contribution of this paper is fourfold: (1) Our work for the first time analyzes polling systemswithout stochastic assumptions, evaluates policy performance by competitive ratios, provides the conditionsfor existence of competitive ratios in polling systems, and proves competitive ratios for some well-studiedpolicies such as cyclic exhaustive and gated policy. (2) Our work bridges the queueing and schedulingcommunities by showing that some widely-used queueing policies also have decent performance in terms ofthe worst-case performance in online scheduling problems. (3) We provide a lower bound for the competitiveratio for general online policies. (4) We also provide new online policies that balance future uncertainty andutilize known information, which may open up a new research direction that would benefit from revealinginformation and reducing variability. This paper is organized as follows: in Section 2 we provide some generalresults for scheduling policies in polling systems; in Section 3 we consider policies which are based on a cyclicrouting discipline; in Section 4 we consider policies which are processing-time based; we make concludingremarks in Section 5. A polling system with zero setup time is the most basic single server system, in which the server only needsto decide when to serve which job, without considering the queue indices. From [40, 16] we know that theoptimal online policy is SRPT in this case. The reason is that by doing this, small jobs are quickly processed,thus the number of waiting jobs is reduced. However, it becomes problematic if setup time is non-zero. If asmall job arrives at the queue which requires a setup, then processing this small job before other jobs maynot be beneficial any more. Deciding which queue to set up and which job to process first thus becomechallenging in such a polling system.In the queueing literature, there are many policies which are defined by service disciplines and routingdiscipline s. The service discipline of each policy determines the time to switch out from a queue and theorder to serve jobs. The routing discipline determines the queue to serve next, when the server switches outfrom a queue. We first introduce some special routing disciplines which are widely studied in the literature. • Static disciplines: The server visits each queue (could be empty or non-empty) following a static routingtable, i.e., a predetermined routing order (see [24, 9]).
Cyclic is one of the most basic static routingdisciplines, under which the server would visit queues one by one, and returns to the first queue oncea cycle has completed. • Random disciplines: The server visits each queue (could be empty or non-empty) in a random manner(see [24, 11, 22]). 5
Purely queue-length based disciplines: An example is
Stochastic Largest Queue (SLQ) policy which isdefined by [28], where the server under SLQ would choose the largest queue to route to when switchingoccurs. • Purely processing-time based disciplines: The next queue to visit could be based on the minimal,maximal, total or average processing time of jobs waiting in the queue. An example is
ShortestProcessing Time (SPT) policy (see [49]), without considering queue indices.Interestingly, these disciplines have the same competitive ratio lower bound, of k , the number of queues. Theorem 1.
If an online policy has routing discipline that is: static, random, purely queue-length based, orpurely processing-time based, then this online policy cannot have a constant competitive ratio that is smallerthan k on | r i , τ | (cid:80) C i .Proof. We assume policy π has a routing discipline that is static, random, or processing time based. Wegive an instance I where there is one job at queue i = 1 , ..., k − at time 0, and n jobs at queue k . Eachjob has processing time . If π is based on a static discipline, we assume the routing order is from queue1 to queue k ; if π is based on random routing discipline, we assume the realization of the random serviceorder is from queue 1 to queue k ; if π is purely processing-time based, it treats all queues equally since theminimal, maximal, total, and average processing times for all queues are equal, and we again let the serverserve from queue 1 to queue k . So the server has to set up k times before reaching queue k , and we have C π ( I ) ≥ τ (( n + k −
1) + ( n + k −
2) + ... + n ) , and C ∗ ( I ) = τ (( n + k −
1) + ( k −
1) + ... + 1) . Letting n → ∞ , we have C π ( I ) C ∗ ( I ) ≥ k .Now we show the lower bound holds for a policy π that is purely queue-length based. Suppose at time0, each queue i = 2 , ..., k has one job with processing time 0, and queue 1 has one job with processing time p. Suppose the server under π serves from queue 1 to queue k, which is consistent with queue-length basedrouting. Then we have C π ( I ) ≥ ( k − p + p + k ( k + 1)2 τ, and C ∗ ( I ) = p + k ( k + 1)2 τ. Letting p → ∞ , we have C π ( I ) C ∗ ( I ) ≥ k. It is important to point out that Theorem 1 provides a competitive ratio bound for policies with somespecial routing disciplines, but the service discipline for these policies could be arbitrary. We will discussmore about policies that are designed based on service and routing disciplines in Section 3.6o reduce setup frequency, a routing discipline may want to avoid setting up empty queues, althoughthere may be arrivals during setup times. Also, an efficient service discipline may prevent the server fromidling when there are unfinished jobs in the system. We thus consider work-conserving policies, under whichthe server would never idle or set up empty queues when the system is non-empty. The next theorem showsthat there exists a constant competitive ratio for all non-preemptive work-conserving policies, under certainconditions.
Theorem 2.
Any non-preemptive work-conserving policy on the polling system with p max ≤ γp min and τ ≤ θp min is at least ( γ + θ ) -competitive with respect to the optimal solution to the offline problem.Proof. Let I be an arbitrary instance, and let ˆ I be the processing time and setup time augmented instance of I such that all job processing times in ˆ I are ˆ p = p max + τ ≤ ( γ + θ ) p min , setup time is 0 and arrival times in ˆ I are the same as I . Note that any non-preemptive work-conserving policy on ˆ I is optimal since processingtimes for jobs in ˆ I are identical, and τ = 0 . Let σ be a non-preemptive work-conserving policy on I , and ˆ σ be a policy that works on ˆ I and serves jobs in the same order as σ does in I , without idling. Then ˆ σ iswork-conserving since σ never idles when there are unfinished jobs in system, and therefore C ˆ σ ( ˆ I ) = C ∗ ( ˆ I ) .We now show C ∗ ( ˆ I ) ≤ ( γ + θ ) C ∗ ( I ) . Let S ∗ i and C ∗ i be the starting and completion times of job i in I underthe optimal solution. Let δ be a schedule that works on ˆ I and finishes each job i at time ( γ + θ ) C ∗ i , we thenhave C δ ( ˆ I ) = ( γ + θ ) C ∗ ( I ) . We now show that δ is a feasible schedule on ˆ I. Notice that schedule δ startsserving job i in ˆ I at time ˆ S δi = ( γ + θ ) C ∗ i − ˆ p = ( γ + θ )( S ∗ i + p i ) − ˆ p ≥ ( γ + θ ) S ∗ i ≥ r i . Since S ∗ i ≥ C ∗ i − , wealso have ( γ + θ ) S ∗ i ≥ ( γ + θ ) C ∗ i − . Therefore ˆ S δi ≥ max { r i , ( γ + θ ) C ∗ i − } , by induction we can show that δ is a feasible schedule on ˆ I . In summary, we have C σ ( I ) ≤ C ˆ σ ( ˆ I ) = C ∗ ( ˆ I ) ≤ C δ ( ˆ I ) = ( γ + θ ) C ∗ ( I ) . Note that Theorem 2 holds only when p max ≤ γp min and τ ≤ θp min . If either inequality does not hold,we may need other scheduling policies to achieve constant competitive ratios, which we will discuss in Section3 and Section 4. In this section we focus on scheduling policies with static routing disciplines. While these policies are easyto implement, they cannot, e.g., prioritize small jobs. To characterize the influence made by job processingtime variation, in this section we assume p max ≤ γp min . When γ is small, all jobs have similar processingtimes. Unlike in [43] where p min is assumed to be non-zero, here we also allow p min = p max = 0 , for whichwe define γ to be 1.In Theorem 1 we have shown that the competitive ratio for a policy with a static or random routingdiscipline is at least k , for the system | r i , τ | (cid:80) C i . Even with additional assumption p max ≤ γp min , we showin the following theorem that the competitive ratio for a policy with a static or random routing discipline isstill lower bounded by k . In this case, cyclic routing is the only static discipline that can achieve this lowerbound. 7 heorem 3. No online policy with a static or random routing discipline can guarantee a competitive ratiosmaller than k for | r i , τ, p max ≤ γp min | (cid:80) C i , and cyclic routing is the only static routing discipline thatcan achieve this lower bound.Proof. Since | r i , τ, p i = 1 | (cid:80) C i is a special case of | r i , τ, p max ≤ γp min | (cid:80) C i , we here only need to showthe lower bound k holds for | r i , τ, p i = 1 | (cid:80) C i . For an arbitrary policy that follows a static routingdiscipline, we suppose the server starts from queue 1, and queue k is the last one visited. Before visitingqueue k for the first time, the server visits queue i for v i times (1 ≤ i ≤ k − . We construct a special jobinstance I by assuming that there is one job arriving at each queue i every time the server visits queue i for i = 1 , ..., k − . Also we suppose there are n k jobs at queue k at time 0 for a large n k , and say they form abatch b k . If we let g ( b k ) = n k ( n k +1)2 , then we have C π ( I ∪ b k ) ≥ C π ( I ) + n k τ ( k − (cid:88) i =1 v i + 1) + g ( b k ) ,C ∗ ( I ∪ b k ) ≤ C π ( I ) + n k τ + g ( b k ) + n ( I )( n k + τ ) , and C π ( I ∪ b k ) C ∗ ( I ∪ b k ) ≥ (cid:80) k − i =1 v i + 1 if we let τ = ( n k ) and n k → ∞ . Since for any static or random routingdiscipline we can construct a job instance like this, to achieve the smallest ratio (cid:80) k − i =1 v i + 1 , we need v i = 1 for i = 1 , ..., k − . Therefore C π ( I ∪ b k ) C ∗ ( I ∪ b k ) ≥ k and only cyclic routing can achieve this lower bound.Theorem 3 shows the advantage of cyclic routing over the other static and random routing disciplines for | r i , τ, p max ≤ γp min | (cid:80) C i . We now focus our discussion on service disciplines at each queue assumingcyclic routing. We first discuss a service discipline called exhaustive , under which the server serves all thejobs in a queue before switching out. We show that the exhaustive service has a competitive advantage overthe other disciplines when all job processing times are identical. Proposition 4.
For the polling system | r i , τ, p min = p max | (cid:80) C i , there always exists an exhaustive disci-pline whose competitive ratio is at least as small as a non-exhaustive discipline.Proof. The case where p i = 0 is trivial. Now we let p i = 1 with appropriate units. Since preemption is notallowed and all the jobs are identical, the server only needs to decide when to switch out and which queue toset up next. If there are jobs in the queue that the server is currently serving, there are two options for theserver: to continue serving the next job in this queue, or to switch to a queue and later come back to thisqueue again. If at time 0 the server is at queue 1 and there is an unfinished job in queue 1, then the server hasto come back after it switches out. Suppose under a non-exhaustive policy π (cid:48) , the server chooses to switch tosome queue(s) and come back to queue 1 at time T . Say the server serves instance I (cid:48) during this period T .Suppose there is an adversary policy which has the same instance at time 0, and this policy chooses to serveone more job in queue 1, and then follows all decisions that policy π (cid:48) has made (including waiting). Notethat every decision policy π (cid:48) made is available to the adversary policy because the adversary policy servesone more job before leaving queue 1. The total completion time under π (cid:48) is C π (cid:48) = 1 + T + C ( I (cid:48) ) , and thecompletion time achieved by the adversary is C ad = 1 + C ( I (cid:48) ) + n ( I (cid:48) ) , where C ( I (cid:48) ) is the completion timeof instance I served by policy π (cid:48) during (0 , T ] . We then have C ad − C π (cid:48) ≤ n ( I (cid:48) ) − T ≤ . Notice that themakespan of these two schemes are the same (including the final setup time of queue 1 for the adversary).8hen by induction, we can show that there always exists an exhaustive discipline that can achieve a smallertotal completion time than a non-exhaustive one.Theorem 3 and Proposition 4 motivate us to consider scheduling policies with cyclic routing and exhaus-tive service. Define a set of policies Π as those policies under which the server 1) serves queues in a cyclicway and skips empty queues when switching, 2) serves each queue exhaustively, 3) stays in queue i for atmost n wi p max amount of time before switching out, where n wi is the number of jobs processed in the server’s w th visit to queue i , and 4) idles at the last queue that it served if all queues are empty. Note that afterserving all the jobs in queue i , the server is allowed to wait in queue i for some extra time to receive morearrivals. If a new arrival occurs at queue i during the time that the server is waiting, then n wi ← n wi + 1 andthe server can process this job at any time before it switches out, as long as the server does not stay in thequeue for time longer than the updated n wi p max . If during the w th visit to queue i , the processing time foreach job in the queue is p max , then the server will switch out immediately once a queue is exhausted. Alsonote that we do not specify the service order of jobs for policies in Π .Notice that the policies in Π does not set up empty queues, so the routing discipline for these policiesneeds to utilize the queue information. We next consider policies which do not require any queue information,so the server under these policies would set up each queue even if the queue is empty before setup is initiated.Let the set of policies Π be defined as for Π except that the server sets up regardless of whether the queueis empty or not. It is important to point out that the policy without waiting (just exhaustively serving)belongs to Π , and the long-run average waiting time and queue length under this policy have been extensivelystudied for M/G/ type queues [14, 42, 36, 51].Another widely studied service discipline is called gated . Under a gated discipline, the server only servesthe jobs that have arrived in the queue before the server finishes setting up the queue, and jobs that arriveafter that time will be left to the next cycle of service. We now let Π be the set of policies with cyclic routingwith skipping the empty queues, and that serves each queue with a gated discipline. Similar to policy set Π , we do not specify the service order for Π either. Once the server has set up queue i , the number of jobsat that time, n wi , will be served during this visit. We also allow the server to wait after clearing a queueunder Π , and the maximal time for staying in queue i in the w th visit is bounded by n wi p max . Similarly,we can have a policy set Π in which policies are cyclic and gated, without skipping empty queues whenswitching. We do not provide the detailed description of Π here since it is similar to policy set Π . Thelong-run average queue length and waiting time under the gated discipline for
M/G/ type queues are alsoprovided in [42, 51], if the server does not skip empty queues when switching.So far we have introduced four policy sets, and we let Π r = ∪ i =1 Π i . In the next theorem we show thatall the policies in Π r have the same competitive ratio for problem | r i , τ, p max ≤ γp min | (cid:80) C i . Theorem 5.
Any policy e ∈ Π r has the competitive ratio κ = max { k +1 k γ, k + 1 } for the polling system | r i , τ, p max ≤ γp min | (cid:80) C i . When γ ≤ k , for arbitrary (cid:15) > , there is an instance I such that C e ( I ) C ∗ ( I ) >κ − (cid:15) = k + 1 − (cid:15) .Proof. The detailed proof in given in the Appendix A.From Theorem 3 we know that the competitive ratio based on static or random routing is at least k , andTheorem 5 shows that any policy from Π r has an approximately tight competitive ratio k + 1 if γ ≤ k . Thisindicates that policies from Π r are nearly optimal among all the policies that are based on static or randomrouting disciplines. It is also important to point out that although gated service disciplines are different from9xhaustive disciplines, one can also regard a gated discipline as an exhaustive-like discipline, as the serverunder a gated discipline would exhaust the jobs that have arrived in the previous cycle. The policies in Π r have a constant competitive ratio for | r i , τ, p max ≤ γp min | (cid:80) C i because of this exhaustive-like servicediscipline, since the server under a policy from Π r would serve as many jobs as possible before switchingout. Some other cyclic policies without using an exhaustive-like service may not have constant competitiveratios for | r i , τ, p max ≤ γp min | (cid:80) C i . We now consider a policy called the l -limited policy. This policy isalso based on the cyclic routing. However, the server under the l -limited policy only serves at most l jobsduring each visit to a queue. A detailed description for this policy and the long-run average waiting timeunder this policy for M/G/ type queues can be found in [42, 15, 47]. Interestingly, as we shall show inProposition 6, no constant competitive ratio is guaranteed by the l -limited policy, regardless whether theserver skips empty queues or not. Proposition 6.
The l -limited policy ( l < ∞ , with or without skipping empty queues) does not have aconstant competitive ratio for | r i , τ, p max ≤ γp min | (cid:80) C i .Proof. We prove this result by giving a special instance I . Suppose there are ( l ∗ n ) number of jobs ( l, n ∈ Z + ) at every queue at time , and each job has processing time p = 1 . At each queue the server sets up andserves l jobs, and this is repeated n times. Let C l ( I ) be the total completion time for the l- limited policy(either with or without skipping empty queues), we have C l ( I ) C ∗ ( I ) = knl ( knl +1)2 + τ l kn ( kn +1)2 knl ( knl +1)2 + τ nl k ( k +1)2 . If we let τ = ( n ) and n → ∞ , then C l ( I ) C ∗ ( I ) → ∞ .Proposition 6 shows that the l - limited policy, which does not belong to Π r , does not have a constantcompetitive ratio for | r i , τ, p max ≤ γp min | (cid:80) C i . Theorem 5 and Proposition 6 show the advantage ofexhaustive-like service disciplines. However, policies in Π r also have their limitations. We next show thatpolicies in Π r do not have constant competitive ratios if p min = 0 < p max (so γ is infinity). Theorem 7.
Policies in Π r do not guarantee constant competitive ratios for | r i , τ | (cid:80) C i .Proof. We prove the theorem by giving a special job instance I . We assume p min = 0 and p max = p so that γ = ∞ . Suppose at time each of queue i = 2 , ..., k has one job with processing time p and queue 1 has nojob. At time τ + (cid:15) there are n jobs arriving at queue 1, with each having processing time . For any policy π ∈ Π r , the server would either setup queue 1 at time 0 then switch to queue 2 at time τ , or setup queue at time 0. In either of the cases the server will be back to queue 1 when queue k is served in the first cycle.Then we have C π ( I ) ≥ k ( k − p + n ( k − p + τ (( n + k −
1) + ( n + k −
2) + ... + n ) , and C ∗ ( I ) = k ( k − p + τ (( n + k −
1) + ( k −
1) + ... + 1) + (cid:15) ( n + k − . lgorithm 1 One Machine Scheduling (OM)1. Simulate SRPT policy on the setup time reduced instance I (cid:101) .2. Schedule the jobs non-preemptively in the order of completion time of jobs by SRPT on I (cid:101) .Letting p = ( n ) and n → ∞ , we have C π ( I ) C ∗ ( I ) → ∞ . Theorem 7 shows the limitation of Π r when the condition p max ≤ γp min is not satisfied. Although theserver under Π r can serve the jobs within each queue following SPT (see[49]) to achieve a smaller expectedwaiting time, the competitive ratio remains κ . When the condition p max ≤ γp min no longer holds for finite γ , one may need policies that utilize the job processing time information to achieve a constant competitiveratio. In the next section we will introduce some processing-time based policies when p max ≤ γp min doesnot hold. In this section we mainly discuss policies in which service and routing disciplines are based on job processingtimes. Service and routing disciplines for these policies are based on job processing times only. To bettercharacterize the competitive ratio for these policies, in this section we assume the setup time τ is bounded bya ratio of the minimal processing time, that is τ ≤ θp min . If τ = p min = 0 , we let θ = 1 . This setting in the3D printing example corresponds to the scenario where jobs of a different color need to be printed, and thetime to set up a new ink is bounded by a constant factor of the minimum possible processing time. Using thestandard notation for scheduling problems (see [18]), we denote this polling problem as | r i , τ ≤ θp min | (cid:80) C i .Notice that when the setup time is small, i.e., θ is small, switching may not be the major contributor to thecompletion time. Thus these processing-time based policies may be efficient when θ is small.We first introduce a benchmark for deriving the competitive ratio of our policies. Usually the competitiveratio ρ is defined by sup I C π ( I ) C ∗ ( I ) ≤ ρ where C ∗ ( I ) is the completion time for I in the offline optimal solution.The offline problem is strongly NP-hard. To non-rigorously show the NP-hardness, we know that if nopreemption is allowed, even the easier problem | r i | (cid:80) C i (without setup time) is strongly NP-hard [25, 19,23]. In this section we use a lower bound of the optimal solution as the benchmark. To get a lower boundfor the optimal solution, we introduce the idea of setup time reduced instance. If instance I is an arbitraryjob instance, then the setup time reduced instance of I , say I (cid:101) , is an instance that has the same jobs (samearrival times and same processing times) as I , but has no setup times. The optimal scheduling policy for I (cid:101) to minimize total completion times is SRPT [40]. This scheduling policy is also online, which is handy forother online policies to emulate. The completion time of instance I (cid:101) under SRPT is denoted by C p ( I (cid:101) ) . Sincesetup time does not exist in I (cid:101) , we have C p ( I (cid:101) ) ≤ C ∗ ( I ) . In this section, we only consider non-preemptivepolicies, but using SRPT as the benchmark. When preemption is not allowed, One Machine (OM) policy isknown to be the optimal online scheduling policy for I (cid:101) [34]. We now apply OM on instance I and prove itscompetitive ratio for polling systems. The description of the OM policy is provided in Algorithm 1. Notethat under OM, a job in I can only be scheduled (started) once it has been completed by SRPT on I (cid:101) . Thecompetitive ratio of OM is provided in Theorem 8. Theorem 8.
OM is a (2 + θ ) -competitive online algorithm for the polling system | r i , τ ≤ θp min | (cid:80) C i . Thecompetitive ratio is tight when using SRPT on the reduced instance as the benchmark. lgorithm 2 Modified One Machine Scheduling (MOM)
Require:
Instance I Denote the queue that the server is serving as queue server while I has not been fully processed do Simulate SRPT on I (cid:101) . Regard the departure time of the i th job in SRPT as the i th arrival time inMOM. if i th arrival is at queue server then ˜ p i = p i else ˜ p i = p i + τ end if Schedule the job with smallest ˜ p i end while return Total completion time C g ( I ) Proof.
Let the completion time of the j th job under OM scheduling be C oj , and the completion time of the j th job completed under SRPT be C pj . Since job j is also the j th job that completes service under SRPT, wehave (cid:80) ji =1 p i ≤ C pj . Then we have C oj ≤ C pj + (cid:80) i : C pi ≤ C pj p i + jτ ≤ (1 + θ ) C pj + (cid:80) ji =1 p i ≤ (2 + θ ) C pj . Since C p ( I (cid:101) ) ≤ C ∗ ( I ) , we get (cid:80) C oi ≤ (2 + θ ) (cid:80) C pi ≤ (2 + θ ) (cid:80) C ∗ i . The competitive ratio is tight when there isonly one job in the instance I which is available at time . Suppose this job has processing time . Then C p ( I ) = 1 , and C o ( I ) = 1 + (1 + θ ) = 2 + θ .The OM algorithm is intuitive, easy to apply and polynomial-time solvable. Despite its simplicity, wemay find it inefficient since setup times are ignored. Although each unnecessary switch only brings a smallamount of delay if θ is small, we may still want to avoid switching too often. Thus we provide another policythat is based on OM, under which the server will avoid unnecessary setups. We call it the Modified OneMachine (MOM) policy, with description in Algorithm 2. Under MOM, we will 1) regard the completiontime of each job under SRPT on I (cid:101) as the new “arrival” time, 2) modify the processing time for job i as p i + τ if it is not located in the queue that the server is serving, and 3) process the job with smallest modifiedprocessing time. After completing a job, the processing time for all the jobs in the system is modified again,and the same mechanism is repeated. Under MOM, the server will prefer the jobs from the queue that it iscurrently serving, thus avoiding frequent switching. We denote the completion time of job i in I by MOMas C gi . Theorem 9.
MOM is a (2 + θ ) -competitive online algorithm for | r i , τ ≤ θp min | (cid:80) C i . The competitiveratio is tight when using SRPT on the reduced instance as the benchmark.Proof. Note both MOM and OM simulate SRPT on I (cid:101) and schedule job i only after job i has been processedin SRPT. So we can regard OM as FCFS for a job instance with arrival times { C pi , i = 1 , , ... }, while MOMserves the job with the smallest modified processing first in this instance with the same arrival times. Wehave C gj ≤ C pj + (cid:80) ji =1 ˜ p i , where ˜ p i is the modified processing time of job i . Since MOM schedules the availablejobs in the descending order of ˜ p i , we have C gj ≤ C pj + (cid:80) ji =1 ˜ p i ≤ C pj + (cid:80) i : C pi ≤ C pj ( p i + τ ) ≤ (2 + θ ) C pj . Wegive the same example as in Theorem 8 to show the tightness of competitive ratio: Suppose there is onlyone job with p = 1 in instance I , available at time . Then C p ( I ) = 1 , and C g ( I ) = 1 + (1 + θ ) = 2 + θ .Though MOM avoids some switching, OM and MOM have the same competitive ratio when using SRPTon reduced instance as the benchmark. 12ssumption Competitive Ratio p max ≤ γp min , τ ≤ θp min min { θ, γ + θ, max { k +1 k γ, k + 1) }} τ ≤ θp min θp max ≤ γp min max { k +1 k γ, k + 1 } Unbounded Processing Time and Setup Time ≥ Table 3: Competitive Ratios for Different CasesNext we show the lower bound for competitive ratios of the problem | r i , τ = θp min | (cid:80) C i . Notice thisis a special case for the problem | r i , τ ≤ θp min | (cid:80) C i . Theorem 10. If τ = θp min and θ ≥ , then there is no online algorithm whose competitive ratio is smallerthan θ + 1 , using SRPT on the reduced instance as the benchmark.Proof. If there is one job with processing time p min in the system, we have C π ( I ) C p ( I ) ≥ (1+ θ ) p min p min = 1 + θ . If τ = p min = 0 (so θ = 1 ), then the lower bounded ratio is θ + 1 = 2 as provided in [21].A natural question is whether this lower bound is the best lower bound that one can have. The answerremains open. There could be either an online policy whose competitive ratio is exactly equal to this lowerbound, or a larger lower bound which is closer to the ratio (2 + θ ) .So far we have shown the existence of constant competitive ratios under different assumptions for pollingsystems as summarized in Table 3. We find that when either p max ≤ γp min or τ ≤ θp min holds, we can haveconstant competitive ratios. In fact, in many practical scenarios either the processing time is bounded or thesetup time is bounded or both. Also, when p max ≤ γp min and τ ≤ θp min both hold, any work-conservingpolicies, as well as policies from Π r , OM, and MOM all have constant competitive ratios. When processingtime variation is relatively small compared with the setup time, i.e., γ is small and θ is large, then policiesfrom Π r can have a smaller competitive ratio than OM or MOM. On the other hand, when θ is small while γ or k is large, using OM or MOM may be more efficient as setup times do not contribute much to thetotal completion time. When ≤ γ < , a work-conserving policy outperforms OM and MOM, and it alsooutperforms policies from Π r if θ < k γ and γ ≥ k . In this paper we consider scheduling policies in the polling system without stochastic assumptions. Conditionsfor the existence of constant competitive ratios are discussed and competitive ratios for several well-studiedpolling system scheduling policies are provided. Specifically, we show that for | r i , τ, p max ≤ γp min | (cid:80) C i system, an online policy needs to have a cyclic routing discipline and exhaustive-like service discipline toachieve a constant competitive ratio. We provide a policy set Π r such that every policy from Π r has aconstant competitive ratio κ for problem | r i , τ, p max ≤ γp min | (cid:80) C i . We further provide processing-timebased policies which have constant competitive ratios in system | r i , τ ≤ θp min | (cid:80) C i . We show that if therouting discipline for an online policy is static, random, purely queue-length based, or purely processing-timebased, then the competitive ratio of this policy cannot be smaller than k . However, it remains unknownwhether there exists an online policy with a constant competitive ratio for the problem | r i , τ | (cid:80) C i without any bounding conditions for processing times and setup times.13 eferences [1] Allahverdi, A.
The third comprehensive survey on scheduling problems with setup times/costs.
European Journal of Operational Research 246 , 2 (2015), 345–378.[2]
Allahverdi, A., Ng, C., Cheng, T. E., and Kovalyov, M. Y.
A survey of scheduling problemswith setup times or costs.
European Journal of Operational Research 187 , 3 (2008), 985–1032.[3]
Allahverdi, A., and Soroush, H.
The significance of reducing setup times/setup costs.
EuropeanJournal of Operational Research 187 , 3 (2008), 978–984.[4]
Altman, E., Khamisy, A., and Yechiali, U.
On elevator polling with globally gated regime.
Queueing Systems 11 , 1 (1992), 85–90.[5]
Anderson, E. J., and Potts, C. N.
Online scheduling of a single machine to minimize total weightedcompletion time.
Mathematics of Operations Research 29 , 3 (2004), 686–697.[6]
Bansal, N., Kamphorst, B., and Zwart, B.
Achievable performance of blind policies in heavytraffic.
Mathematics of Operations Research 43 , 3 (2018), 949–964.[7]
Boon, M. A., Adan, I. J., Winands, E. M., and Down, D.
Delays at signalized intersectionswith exhaustive traffic control.
Probability in the Engineering and Informational Sciences 26 , 3 (2012),337–373.[8]
Boon, M. A., Van der Mei, R., and Winands, E. M.
Applications of polling systems.
Surveys inOperations Research and Management Science 16 , 2 (2011), 67–82.[9]
Boxma, O. J., Levy, H., and Weststrate, J. A.
Efficient visit orders for polling systems.
Perfor-mance Evaluation 18 , 2 (1993), 103–123.[10]
Chekuri, C., Motwani, R., Natarajan, B., and Stein, C.
Approximation techniques for averagecompletion time scheduling.
SIAM Journal on Computing 31 , 1 (2001), 146–166.[11]
Chung, H., Un, C. K., and Jung, W. Y.
Performance analysis of markovian polling systems withsingle buffers.
Performance Evaluation 19 , 4 (1994), 303–315.[12]
Divakaran, S., and Saks, M.
An online algorithm for a problem in scheduling with set-ups andrelease times.
Algorithmica 60 , 2 (2011), 301–315.[13]
Epstein, L., and van Stee, R.
Lower bounds for on-line single-machine scheduling.
TheoreticalComputer Science 299 , 1 (2003), 439–450.[14]
Ferguson, M. J., and Aminetzah, Y. J.
Exact results for nonsymmetric token ring systems.
IEEETransactions on Communications 33 , 3 (1985), 223–231.[15]
Gautam, N.
Analysis of queues: methods and applications . CRC Press, 2012.[16]
Gittins, J., Glazebrook, K., and Weber, R.
Multi-armed bandit allocation indices . John Wiley& Sons, 2011.[17]
Goemans, M. X., Queyranne, M., Schulz, A. S., Skutella, M., and Wang, Y.
Single machinescheduling with release dates.
SIAM Journal on Discrete Mathematics 15 , 2 (2002), 165–192.1418]
Graham, R. L., Lawler, E. L., Lenstra, J. K., and Kan, A. R.
Optimization and approximationin deterministic sequencing and scheduling: a survey.
Annals of Discrete Mathematics 5 (1979), 287–326.[19]
Hall, L. A., Schulz, A. S., Shmoys, D. B., and Wein, J.
Scheduling to minimize averagecompletion time: Off-line and on-line approximation algorithms.
Mathematics of Operations Research22 , 3 (1997), 513–544.[20]
Hinder, O., and Mason, A. J.
A novel integer programing formulation for scheduling with familysetup times on a single machine to minimize maximum lateness.
European Journal of OperationalResearch 262 , 2 (2017), 411–423.[21]
Hoogeveen, J. A., and Vestjens, A. P.
Optimal on-line algorithms for single-machine scheduling.In
International Conference on Integer Programming and Combinatorial Optimization (1996), Springer,pp. 404–414.[22]
Jan-pieter, L. D., Boxma, O. J., and van der Mei, R. D.
On two-queue markovian pollingsystems with exhaustive service.
Queueing Systems 78 , 4 (2014), 287–311.[23]
Kan, A. R.
Machine scheduling problems: classification, complexity and computations . Springer Science& Business Media, 2012.[24]
Konheim, A. G., Levy, H., and Srinivasan, M. M.
Descendant set: an efficient approach for theanalysis of polling systems.
IEEE Transactions on Communications 42 , 234 (1994), 1245–1253.[25]
Lawler, E. L., Lenstra, J. K., Kan, A. H. R., and Shmoys, D. B.
Sequencing and scheduling:Algorithms and complexity.
Handbooks in Operations Research and Management Science 4 (1993),445–522.[26]
Lenstra, J. K., Shmoys, D. B., and Tardos, E.
Approximation algorithms for scheduling unrelatedparallel machines.
Mathematical Programming 46 , 1-3 (1990), 259–271.[27]
Levy, H., and Sidi, M.
Polling systems: applications, modeling, and optimization.
IEEE Transactionson Communications 38 , 10 (1990), 1750–1760.[28]
Liu, Z., Nain, P., and Towsley, D.
On optimal polling policies.
Queueing Systems 11 , 1-2 (1992),59–83.[29]
Lu, X., Sitters, R., and Stougie, L.
A class of on-line scheduling algorithms to minimize totalcompletion time.
Operations Research Letters 31 , 3 (2003), 232–236.[30]
Lübbecke, E., Maurer, O., Megow, N., and Wiese, A.
A new approach to online scheduling:Approximating the optimal competitive ratio.
ACM Transactions on Algorithms (TALG) 13 , 1 (2016),15.[31]
Miculescu, D., and Karaman, S.
Polling-systems-based autonomous vehicle coordination in trafficintersections with no traffic signals.
IEEE Transactions on Automatic Control (2019).[32]
Mosheiov, G., Oron, D., and Ritov, Y.
Minimizing flow-time on a single machine with integerbatch sizes.
Operations Research Letters 33 , 5 (2005), 497–501.1533]
Ng, C., Cheng, T. E., Yuan, J., and Liu, Z.
On the single machine serial batching schedulingproblem to minimize total completion time with precedence constraints, release dates and identicalprocessing times.
Operations Research Letters 31 , 4 (2003), 323–326.[34]
Phillips, C., Stein, C., and Wein, J.
Minimizing average completion time in the presence of releasedates.
Mathematical Programming 82 , 1-2 (1998), 199–223.[35]
Ruiz, R., and Vázquez-Rodríguez, J. A.
The hybrid flow shop scheduling problem.
EuropeanJournal of Operational Research 205 , 1 (2010), 1–18.[36]
Sarkar, D., and Zangwill, W.
Expected waiting time for nonsymmetric cyclic queueing systems –exact results and applications.
Management Science 35 , 12 (1989), 1463–1474.[37]
Schulz, A. S., and Skutella, M.
The power of α -points in preemptive single machine scheduling. Journal of Scheduling 5 , 2 (2002), 121–133.[38]
Shen, L., Dauzère-Pérès, S., and Neufeld, J. S.
Solving the flexible job shop scheduling problemwith sequence-dependent setup times.
European Journal of Operational Research 265 , 2 (2018), 503–516.[39]
Sitters, R.
Competitive analysis of preemptive single-machine scheduling.
Operations Research Letters38 , 6 (2010), 585–588.[40]
Smith, D. R.
A new proof of the optimality of the shortest remaining processing time discipline.
Operations Research 26 , 1 (1978), 197–199.[41]
Stougie, L., and Vestjens, A. P.
Randomized algorithms for on-line scheduling problems: how lowcan’t you go?
Operations Research Letters 30 , 2 (2002), 89–96.[42]
Takagi, H.
Queuing analysis of polling models.
ACM Computing Surveys (CSUR) 20 , 1 (1988), 5–28.[43]
Tao, J., Chao, Z., and Xi, Y.
A semi-online algorithm and its competitive analysis for a singlemachine scheduling problem with bounded processing times.
Journal of Industrial and ManagementOptimization 6 , 2 (2010), 269–282.[44]
Tao, J., Chao, Z., Xi, Y., and Tao, Y.
An optimal semi-online algorithm for a single machinescheduling problem with bounded processing time.
Information Processing Letters 110 , 8-9 (2010),325–330.[45]
Vallada, E., and Ruiz, R.
A genetic algorithm for the unrelated parallel machine scheduling problemwith sequence dependent setup times.
European Journal of Operational Research 211 , 3 (2011), 612–622.[46] van der Mei, R. D., and Roubos, A.
Polling models with multi-phase gated service.
Annals ofOperations Research 198 , 1 (2012), 25–56.[47]
Van Vuuren, M., and Winands, E. M.
Iterative approximation of k-limited polling systems.
Queue-ing Systems 55 , 3 (2007), 161–178.[48]
Vishnevskii, V., and Semenova, O.
Mathematical methods to study the polling systems.
Automationand Remote Control 67 , 2 (2006), 173–220.[49]
Wierman, A., Winands, E. M., and Boxma, O. J.
Scheduling in polling systems.
PerformanceEvaluation 64 , 9 (2007), 1009–1028. 1650]
Wierman, A., and Zwart, B.
Is tail-optimal scheduling possible?
Operations Research 60 , 5 (2012),1249–1257.[51]
Winands, E. M., Adan, I. J.-B. F., and van Houtum, G.-J.
Mean value analysis for pollingsystems.
Queueing Systems 54 , 1 (2006), 35–44.[52]
Zhang, L., and Wirth, A.
Online machine scheduling with family setups.
Asia-Pacific Journal ofOperational Research 33 , 04 (2016), 1650027.
A Proof for Theorem 5 in the Main Paper
In this section we mainly provide the proof for Theorem 5 of our paper. We first introduce a fact that willbe useful later in our proof.
Fact 11.
For positive numbers { a i , b i } ni =1 , we have (cid:80) ni =1 a i (cid:80) ni =1 b i ≤ max ni =1 { a i b i } . Proof.
Without loss of generality, assume a n b n = max ni =1 { a i b i } , then for any i we have a n b n ≥ a i b i , thus a n b i ≥ a i b n holds for all i = 1 , ..., n . Since a n b n − (cid:80) ni =1 a i (cid:80) ni =1 b i = a n (cid:80) ni =1 b i − b n (cid:80) ni =1 a i b n ( (cid:80) ni =1 b i ) = (cid:80) ni =1 ( a n b i − a i b n ) b n ( (cid:80) ni =1 b i ) ≥ , we have (cid:80) ni =1 a i (cid:80) ni =1 b i ≤ a n b n = max ni =1 { a i b i } .In Theorem 5 we want to show sup I C e ( I ) C ∗ ( I ) ≤ ρ for some constant ρ. However, by Fact 11 we only need toshow that this inequality holds for the instance processed in each busy period of the server under an onlinepolicy e ∈ Π . Here we first introduce the concept of busy periods under the online policy. If there is atleast one job in the system, we say the system is busy, otherwise it is empty. When the system is empty,the server is not serving under the online policy. The status of the system under the online policy can bedescribed as a busy period following an empty period, and then following by a busy period, and so on. Thereare two types of busy periods that we are interested in. A Type I busy period (denoted as I-B) is the busyperiod in which the server resumes work without setting up. This is because the server was idling at the lastqueue it served (say queue i ) after the previous busy period, and the first arrival in the new busy period alsooccurs at queue i . A Type II busy period (denote as II-B) is the busy period in which the server resumeswork with a setup, because the new arrival occurs at a queue different from the queue where the server wasidling. Note that the total job instance may be processed by the online policy in multiple busy periods, andin the proof we only consider the subset of this job instance that is processed in a certain busy period, whichwe call busy period instance. Also note that the very first busy period must be a II-B, since the server hasto set up a queue before processing at the very beginning.Without loss of generality, we say a cycle (round) starts when the server under the online policy visitsqueue 1 and ends when it visits queue 1 the next time. If at some time point a queue, say queue i , is emptyand skipped by the server, we still say that queue i has been visited in this cycle, with setup time 0. Wedefine the job instance served by the server in its w th visit to queue as b wi (for i = 1 , ..., k ), and we call each b wi a batch . Batch b wi is a subset of job instance I . In each cycle, there are k batches served by the onlinepolicy, and some, but not all may be empty (if all of them are empty then the system is empty and theserver would idle at queue 1). We let I w = ∪ ki =1 b wi . For each batch b wi with the number of jobs n ( b wi ) = n wi , S wi is the earliest time when a job from b wi starts being processed under the online policy, and R wi is theearliest release date (arrival time) over all jobs from batch b wi . Notice that R wi ≤ S wi . Each batch b wi maybe processed by the optimal offline policy in a different way from the online policy. Suppose E w ∗ i is the17arliest time when a job in batch b wi starts service under the optimal offline policy. Note E w ∗ i may differfrom S wi . Before time S wi , we know all the batches ( ∪ w − j =1 I j ) ∪ ( ∪ i − l =1 b wl ) have been served by the onlinepolicy. However in the optimal offline policy, only some jobs from these batches have been served, and someother jobs from other cycles of the online policy may have already been served. We suppose q wi number ofjobs in ( ∪ w − j =1 I j ) ∪ ( ∪ i − l =1 b wl ) have been served by the optimal offline policy before time E w ∗ i . We define E wi as the earliest time when the truncated optimal offline policy starts to serve batch b wi , where the idea oftruncated optimal offline policy will be defined later. We let C e ( I ) be the cumulative completion times of alljobs in job instance I under online policy e ∈ Π , and C ∗ ( I ) be the cumulative completion times of all jobsin I under the offline optimal policy. For convenience, we let g ( I ) = n ( I )( n ( I )+1)2 for job instance I , whichis the sum of arithmetic sequence from 1 to n ( I ) . Also, g ( I ) can be regarded as the completion time of I when 1) all the jobs are available at time , 2) each of them has processing time , and 3) no setup timeis considered. All the notations are summarized in Table 4 of this document. Before going to the proof ofTheorem 5, we first provide an example to show how the total completion time is characterized. Example 12.
Suppose there is a job instance I with n ( I ) = n + n jobs which arrive at time R . Each ofthe job has processing time . Under some policy π , suppose the server starts to process the first n jobs,and idles for time W , and then processes the rest of n jobs without idling. The total completion time for I is given by C π ( I ) = ( R + 1) + ( R + 2) + ... + ( R + n ) + ( R + n + W + 1) + ( R + n + W + 2) + ... + ( R + n + W + n )= ( n + n ) R + n W + g ( I ) . Notice the completion time of I is made up of three components. The first component ( n + n ) R isbecause all the jobs in I arrive at time R . The second term n W is because the remaining n jobs wait foranother W amount of time. The third term g ( I ) is the pure completion time if we process jobs one by onewithout idling.Notice that the optimal policy may not always be work-conserving (i.e., never idles when there are jobsin the system). The optimal policy may wait at some queue in order to receive more jobs which will arrivein the future. The truncated optimal solution (of a busy period instance I ) is defined by the completion timefor the optimal offline problem minus the additional completion time caused by idling, which is shown inFigure A.1. There is a waiting (idling) period W between b and b in Figure A.1. The truncated optimalsolution is given by C ∗ ( b ∪ b ∪ b ∪ b ) − W ( n + n + n ) . We use C t ( I ) to denote the total completiontime for a busy period instance I under the truncated optimal schedule. Note that this truncated optimalsolution is only for an instance I served by the online policy during a specific busy period, so the earlieststart time of service of this truncated optimal policy cannot be earlier than the earliest arrival time of I .Also note that the truncated optimal solution is always a lower bound for the real optimal solution. Lemma 13.
Suppose I is a job instance served in a busy period by the online policy, b is a batch in somecycle for the offline problem and p min = 1 , then C t ( I ∪ b ) ≥ C t ( I ) + g ( b ) + En ( b ) + n ( b )( n ( I ) − q ) , where E is the time when the server starts serving batch b in the truncated optimal solution, and q is the number ofjobs in I that are served before time E .Proof. We suppose the optimal solution is given, and now we consider the total completion time of I ∪ b under truncated optimal solution. If all the jobs from b are served after I in the optimal solution, then we18igure A.1: Truncated Optimal Solutionhave C t ( I ∪ b ) ≥ C t ( I ) + g ( b ) + En ( b ) . If not, we let δ ( b ) = C t ( I ∪ b ) − C t ( I ) be the additional completiontime incurred by inserting b into I . Notice that δ ( b ) is minimized when all jobs in b has p min = 1 . If we canshow that δ ( b ) ≥ g ( b ) + En ( b ) + n ( b )( n ( I ) − q ) with every job in b having p min = 1 , we can then prove thelemma. So we assume here that every job in b has p min = 1 . Since the earliest time to process batch b in thetruncated optimal solution is E, if we combine all jobs in b altogether and serve them in one batch from time E to time E + n ( b ) , then δ ( b ) is again minimized since all the jobs in b have the smallest processing time.So in the following we show that by inserting batch b at time E, the additional completion time incurred isat least g ( b ) + En ( b ) + n ( b )( n ( I ) − q ) . By inserting batch b into I from time E to E + n ( b ) , some jobs from I served after E in the original truncated optimal solution (with total number ( n ( I ) − q ) ) are moved afterbatch b , resulting an increase of delay n ( b )( n ( I ) − q ) for these jobs. Besides, inserting a batch b at time E increases the total completion time by g ( b ) + En ( b ) .Lemma 13 is illustrated in Figure A.2. The first schedule in Figure A.2 is the truncated optimal for thebatch b ∪ b ∪ b . The second schedule is the truncated optimal schedule for b ∪ b ∪ b ∪ b , where b isseparated into two parts. If all jobs in b have processing time p min = 1 , it is always beneficial to schedule alljobs of b in the same batch, which is shown as the third schedule in Figure A.2. Notice that in Figure A.2, q = n ( b + b ) = n + n . Lemma 14.