Unlike the case for the clearing model of Chapter 3, we have no explicit expressions for the thresh- old of the optimal policy. Therefore, it is particularly important to explore the performance of heuristic policies. The problem we investigate in this chapter has infinite state space. To numerically compute the optimal cost, we would need to solve the Bellman’s equation which necessitates truncation of the state space. Hence, we would only able to get an approximation for the optimal cost. Since estimation of the error due to truncation is not possible, we carried out a simulation study instead of numerical experiments. First, we describe the three policies of interest.
No-Triage Policy (N T)Jobs are served in random order. No job goes through triage.
Triage-Prioritize-Class-1 Policy (T P1) Each job goes through triage in random order. If a job is classified as class-1, it is served right away; otherwise, the job is put aside to be served later, and triage the next untriaged job. When there are no untriaged jobs in the system, class-2 jobs are served. Service is preemptive.
Threshold-type Policy (T h) Serve a class-1 job once it is identified, and serve a class-2 job only when there are no other types of jobs, i.e.,i=k1 = 0. Whenk1 = 0andi >0,skip triage only if k2 ≥L(i), where L(i) = r(˜u−u) r2u i− ru˜ r2u, (4.9)
andr, r2are defined in (4.2);u˜is defined in Theorem 4.3.1.
PolicyN T is the first-come-first-serve policy, and PolicyT P1 is the counterpart of the Triage- Prioritize-Class-1 Policy in the clearing system. The only difference is that the service of less impor- tant jobs, i.e., class-2 jobs, may be preempted by new arrivals. PolicyT his the same as the optimal policy for the clearing system in Chapter 3 except the parameters are adapted to that of Chapter 4.
The parameters of the simulation is described below. The service times are generated from ex- ponential distributions with mean service time τ = 1. The probability of a new arrival job being type-1,p, is fixed at0.3.The conditional probabilities of correct classification for type-1 and type 2 arev1 = 0.9, v2 = 1,respectively. We choseλfrom the set{0.1,0.3,0.5,0.7,0.9}, and chose h1 from{1,3,5,7,9}whileh2 is fixed at1.The triage time is exponentially distributed anduis chosen from{0.1,0.2,0.3}.For each combination of(λ, h1, u),the long-run average cost under PolicyN T is computed by
gN T = ρ
1−ρ ·r, (4.10)
whereρ=λτ.We simulate the system evolution under PolicyT h. We pick2×104 time units as the warmup period by Welch’s method. Then, we run the simulation forTs = 107 time units in addition to the warmup period. We divide Ts into 1000 equal batches, each batch with104 time units, and record the total cost incurred during each batch. We use the average cost of each batch, which can be easily obtained, to compute 95% confidence interval for the long-run average cost of the system under PolicyT h.The mean of the confidence interval is denoted bygT h.
Table 4.1 presents our simulation results. The confidence intervals of the long-run average cost for each scenario are displayed in columns labeledgT h.To the right of each of thegT h column, we present the percentage improvement,η, by PolicyT hover PolicyN T.IfgN T falls into the confidence interval ofgT h,the improvement is insignificant and we simply setη = 0.00;otherwise, it is defined as
η= gN T −gT h
gN T
×100. (4.11)
From Table 4.1, we observe that the improvement is insignificant when (i) the cost rates of the two types are close; and/or (ii) when the arrival rate is small, and/or (iii) when the expected triage time is large. These observations are consistent with our intuition that when the two types of jobs are
h1 λ gN T T h, u
= 0.1 T h, u= 0.2 T h, u= 0.3
gT h Imprv% gT h Imprv% gT h Imprv%
1 0.1 0.111 0.111±0.000 0.00 0.111±0.000 0.00 0.111±0.000 0.00 1 0.3 0.429 0.428±0.001 0.00 0.428±0.001 0.00 0.428±0.001 0.00 1 0.5 1.000 0.999±0.003 0.00 0.999±0.003 0.00 0.999±0.003 0.00 1 0.7 2.333 2.334±0.011 0.00 2.334±0.011 0.00 2.334±0.011 0.00 1 0.9 9.000 8.945±0.125 0.00 8.945±0.125 0.00 8.945±0.125 0.00 3 0.1 0.178 0.177±0.000 0.00 0.178±0.001 0.00 0.178±0.001 0.00 3 0.3 0.686 0.670±0.002 2.36 0.685±0.002 0.00 0.684±0.002 0.00 3 0.5 1.600 1.517±0.004 5.22 1.598±0.004 0.00 1.602±0.005 0.00 3 0.7 3.733 3.459±0.018 7.36 3.723±0.016 0.00 3.736±0.018 0.00 3 0.9 14.400 13.607±0.161 5.51 14.303±0.173 0.00 14.407±0.182 0.00 5 0.1 0.244 0.240±0.001 1.77 0.244±0.001 0.00 0.245±0.001 0.00 5 0.3 0.943 0.886±0.002 6.08 0.933±0.002 1.02 0.942±0.002 0.00 5 0.5 2.200 1.945±0.006 11.60 2.121±0.007 3.61 2.201±0.006 0.00 5 0.7 5.133 4.285±0.017 16.53 4.817±0.020 6.16 5.122±0.023 0.00 5 0.9 19.800 16.266±0.218 17.85 18.247±0.210 7.84 19.626±0.249 0.00 7 0.1 0.311 0.305±0.001 2.06 0.310±0.001 0.00 0.311±0.001 0.00 7 0.3 1.200 1.101±0.003 8.23 1.163±0.003 3.07 1.199±0.003 0.00 7 0.5 2.800 2.365±0.006 15.53 2.590±0.008 7.49 2.766±0.009 1.20 7 0.7 6.533 5.048±0.020 22.74 5.763±0.023 11.79 6.336±0.031 3.02 7 0.9 25.200 19.220±0.249 23.73 22.077±0.264 12.39 24.189±0.317 4.01 9 0.1 0.378 0.367±0.001 2.80 0.374±0.001 0.91 0.378±0.001 0.00 9 0.3 1.457 1.316±0.003 9.72 1.390±0.003 4.60 1.449±0.004 0.55 9 0.5 3.400 2.785±0.007 18.09 3.065±0.009 9.84 3.313±0.009 2.57 9 0.7 7.933 5.789±0.020 27.03 6.725±0.030 15.23 7.483±0.038 5.68 9 0.9 30.600 21.706±0.251 29.06 25.769±0.288 15.79 28.482±0.343 6.92
Table 4.1: Comparison in the average cost by using the heuristic policyT has opposed to No-Triage policy.
similar in the sense of the cost rates, then triage brings few benefits and No-Triage Policy performs similar to the optimal policy. When the arrival rate is small, service resources are sufficient and the congestion level in the system is low. Hence, triage is not needed. When triage takes too much time, the additional delay imposed on the jobs can not be justified by the benefits brought by triage. On the contrary, when the differences among the jobs are significant, and/or the traffic intensity is high, and/or triage is fast, PolicyT himproves over No-Triage Policy as much as29%.Not surprisingly, the most significant improvement happens whenh1 = 9, λ= 0.9andu = 0.1,i.e., when the two types of jobs are most different, the system’s congestion level is high and triage can be done rapidly.
CHAPTER 5: EXTENSIONS
In this chapter, we study three extensions. In Section 5.1, we study a clearing model as in Chapter 3 but this time we consider having multiple identical servers instead of a single server. In Section 5.2, we consider a model with arrivals as in Chapter 4, but this time triage is instantaneous and incurs a fixed cost. In Section 5.3, we discuss the case that triage is not optional and is required to be done for each new arrival before services. In each section, we describe the model assumptions and are able to characterize the optimal policy partially or completely.