Approximation Results in Parallel Machines Stochastic Scheduling

We consider scheduling a batch of jobs with stochastic processing times on parallel machines. We derive various new formulae for the expected flowtime and weighted flowtime under general scheduling rules. Smith's Rule, which orders job starts by decreasing ratio of weight to expected processing time provides a natural heuristic for this problem. We obtain a bound on the worst case difference between the expected weighted flowtime under Smith's Rule and under an optimal policy. For a wide class of processing time dista'ibutions, this bound is of order O(1 ) and does not increase with the number of jobs. 1. In t roduc t ion One of the simplest yet most useful results in scheduling theory is that flowtime (sum of all the waiting and processing times of all the jobs) is minimized by the SPT (shortest processing time first) rule. This result holds for a single processor [28] as well as for parallel processors [22]. For the often more applicable weighted flowtime objective function (where the waiting and processing times are weighted by different cost per unit time for each job), the optimal sequence of jobs on a single processor is in decreasing order of weight to processing time ratios [29], the so-called "Smith's Rule". Minimization of weighted flowtime on several parallel machines is, however, an NP-hard combinatorial optimization problem for any fixed number of machines greater than 1 [10,20]. Scheduling the starting times of the jobs according to Smith's Rule provides a suboptimal heuristic for this problem, whose worst case performance is 1.2071 times the optimal value [18,35]. In the present paper, we examine a stochastic version of these problems, where we assume that the processing requirements of the jobs are not known in advance but are drawn from some known probability distributions. The single-processor results generalize easily to stochastic processing times: SEPT (shortest expected processing time first) and Smith's Rule (decreasing order of weight to expected processing time ratio), respectively, minimize the expected flowtime and the expected weighted flowtime. *This research was supported in part by NSF Grant ECS-8712798. 9 J.C. Bahzer A.G., Scientific Publishing Company 196 G. Weiss, Parallel machines stochastic scheduling Results for parallel machines are much more complex. SEPT remains optimal for flowtime in a wide range of problems, although the proofs are no longer elementary. SEPT is optimal when job processing times are exponentially distributed [2,14,24,36]. It is also optimal when the job processing time distributions are all tails of a single IHR (increasing hazard rate) distribution [31 ]. Both these results are special cases of the remarkable result of Weber, Varaiya and Walrand [32] that SEPT is optimal whenever the processing time distributions of all the jobs are stochastically comparable in pairs. That result seems the most general possible. It does not extend to weaker comparison conditions, and SEPT fails to be optimal in general (see Pinedo and Weiss [25] for counterexamples). Very little can be said on optimality of Smith's Rule for expected weighted flowtime on parallel machines: K/~rnpke [17] discusses some weighted cases. If SEPT and Smith's Rule are no longer optimal on parallel machines, two questions arise: What is the optimal policy and how far are SEPT and Smith's Rule from it? On the first question, I believe the search for the optimal policy in the general case is futile. First, the problem is NP-hard; it seems likely that one can find a parametric family of distributions for which even minimization of expected flowtime is NP-hard. However, the difficulties go beyond the computational effort involved: if SEPT or Smith's Rule are not optimal, then the optimal policy may be extremely complicated to describe and to implement it may involve dynamic scheduling with inserted idle time, and it may depend on the entire processing time distribution of each job rather than on a few parameters. Needless to say, the data to estimate these distributions in such detail will rarely be available in practice. Pinedo and Weiss [25] discuss some very simple problems which have quite complicated optimal solutions; it is easy to imagine problems with optimal solutions of almost any degree of complexity. How far SEPT and Smith's Rule are from optimal can be measured in two ways: in terms of the expected value of the objective function, or in terms of the differences between the policies (this difference can be expressed by, for example, counting the number of jobs which a realization of the optimal policy does not start according to Smith's Rule). In the present paper, we give a complete answer in terms of the objective function. This answer is extremely favorable to Smith's Rule and SEPT. Using the weight, the mean processing time, and one additional simply calculated parameter of the processing time distribution of each of the jobs, we calculate a bound on the difference between the expected objective function values. As long as the values of these three relevant parameters remain bounded, the bound on the expected difference does not grow with the number of jobs n. Thus, under the assumption that the weights and the processing time distributions of all the jobs satisfy some uniform boundedness conditions, we have that even though the expected (weighted) flowtime will as a rule increase as O(n2), the expected difference in objective value between SEPT (or Smith's Rule) and the optimal policy will remain bounded by a constant, independent of n. G. Weiss, Parallel machines stochastic scheduling 197 The uniform boundedness is clearly essential here; without it, the worse case performance ratio for deterministic jobs is 1.2, so that difference is O(nZ). However, an assumption of uniform boundedness on processing time distributions is in general much less restrictive than for the deterministic case. In many problems, all that will be required is uniform bounds on the means and variances of the processing times: this still permits any mixture of actual processing time values to occur and seems a reasonable requirement. The results of this paper are derived in sections 4-11. While deriving the bounds for the objective function, we obtain some insights into the nature of parallel processing and some useful formulae for expected flowtime and weighted flowtime. The results are summarized in section 2, and a plausibility argument is given in section 3. With the question of how well SEPT or Smith's Rule will do in terms of the expected value of the objective function settled, it remains to ask how different are the actual policies from the optimal. The plausibility argument indicates that SEPT may not be optimal toward the end of the schedule. We conjecture that SEPT and Smith's Rule have a turnpike property (for a definition of turnpike optimality, see Shapiro [28]) asymptotically, for large n, most of the optimal decisions will be according to SEPT (or Smith's Rule). Such a turnpike result does indeed hold, and will be the subject of a forthcoming paper. A turnpike optimality result for a special case of a preemptive scheduling problem has been proven in Coffman, Hofri and Weiss [3]. We conclude the paper with a discussion of some possible extensions to preemptive scheduling problems, the relation to Gittins' index, and to various control problems in section 12. 2. Summary of results Jobs 1 . . . . . n require processing times X~ . . . . . X n, nonnegative, which are drawn independently from some given probability distribution functions /71 . . . . . F but their actual values are not known in advance. Throughout this paper, we assume that X 1 . . . . . X are independent random variables, and that they are furthermore independent of when and how the jobs are performed, that is, they are independent of the machines and of the schedule. We assume nothing special about the form of F 1 . . . . . F ; we let #l . . . . . # , and ~ . . . . . cr z be their means and variances, and assume finite third moments. Machines 0 . . . . . M first become available to start the processing of these jobs at time U0o . . . . . UM0, with Uoo + . . . + UMO = 0. Jobs are then processed by the machines in parallel, with no preemptions and no inserted idle times. Let C~ . . . . . C be the completion times of the jobs. Two objective functions are of interest: the flowtime Y'i = ~ Ci which is the sum of all the completion times (waiting plus processing times) of all the jobs and, more generally, the weighted flowtime Y'7=~ Wi Ci where the jobs are weighted by individual costs per unit time W~ . . . . . W . 198 G. Weiss, Parallel machines stochastic scheduling Two policies suggest themselves for these objective functions: SEFF start jobs in the order of shortest expected processing time first, and the so-called "Smith's Rule", denoted by SR start jobs in decreasing order of weight to expected processing time ratio. On a single processor, SEPT and SR minimize the expected flowtime and the expected weighted flowtime, respectively. On M + 1 parallel processors, this is not generally the case. In the present paper, we prove that while SEPT and SR may not be optimal, they are very nearly optimal. In section 3, we give a plausibility argument for these results. The results in section 4 are combinatorial in nature and apply to any processing times X~ . . . . . X n, deterministic or stochastic. (a) Decomposition of flowtimes: Assume jobs are started in the order 1 . . . . . n, with processing times X~ . . . . . X , and are processed by machines 0 . . . . . M with no preemptions or inserted idle time. In analogy with the definitions of U00 . . . . . UM0, we define for any 0 _< m < n: Let Uom <_... < UMm be the ordered times at which the machines complete the service of the first m jobs to start. Let D~,~ = U.,~ Uom, i = 1 . . . . . M (where m = 0, 1 . . . . . n) these are the ordered remaining processing times of jobs on the M busy machines when a machine becomes available for the start of the (m + 1)st job. Then, the weighted flowtime can be decomposed as (see (4.9) in section 4): = M + l M +------Z Wj X j y_, D,j_ . j=l j=l k=j j=l i=1 (b) The job processing time remainders Dij, i = 1 . . . . . M, j = 0, 1 . . . . . n, can be calculated by a Markovian recursion: D~i . . . . . DMi are the ordered values of the M largest values minus the smallest value among X., D~i_ 1 . . . . . DMi-t" Let = Di~ M ( M + 1) i=I ' 2 1 Dii2 i=l for 0 _< j _< n. Note that S. 2J are the sample variances of Uoi . . . . , UM~ (or equivalently of O, D~i . . . . . DMi); in particular, So z, S 2 measure the "raggedness" of the beginning and the end of the multiprocessor schedule. The Dii's satisfy the key formula (see (4.12) in section 4):