Consider a queue featuring either a single server or many servers, which attend(s) toLindepen- dent classes of customers with distinct KPIs of the form discussed in the introduction. Arrivals for thenth class of customers are from a Poisson process at rateλn;n= 1,2, . . . ,L. All service time distributions are known, and they may differ from class to class.
We presume that the queue is operating under a particular service discipline a ∈ A (that is, a rule for selecting the next customer to enter service), where A denotes the set of per- missible work-conserving (server is not idle unless queue is empty) disciplines. The set A
includes FCFS, last-come, first served (LCFS), Random Order of Service (ROS) and both Non-preemptive (NP) and Preemptive Resume (PR) disciplines, among others. We presume that the queue has been operating sufficiently long to have reached stationarity.
LettingWndenote the stationary class-nwaiting time random variable, define the following forn= 1,2, . . . ,L:
• Wn(x)= P(Wn≤ x) is the cumulative distribution function,
• Sn(x)= P(Wn > x)= 1−Wn(x) is the survival function, and
• wn(x) = dWn(x)/dx is the probability density function of the stationary class-nwaiting time distribution.
150 Chapter7. AnOptimizationProblem forQueuesOperating underWaitingTargets
defined below, by aethroughout the paper.
The expected amount of excess waiting time Hn(dn) for a typical class-ncustomer beyond a specified delaydn, henceforth abbreviated the “expected excess”, can be determined from
Hn(dn)= Z ∞ dn (x−dn)wn(x)dx = Z ∞ dn Sn(x)dx. (7.1)
The latter integral above is obtained by integrating the former one by parts. The quantity
λnHn(dn) can be interpreted as the expected amount of excess waiting for class-ncustomers per unit of time.
For such a queue as described above, the general form of the optimization problem can be stated as follows. We seek to minimize a weighted average of the total expected excess waiting per unit time over all permissible customer selection strategies,
mina∈A Z = PnL=1αnλnHn(dn) subject to Wn(dn) ≥ pn; n = 1, 2, . . . , L;
where, αn,n = 1,2, . . . ,L denote the respective weights for the excess waiting for class-n customers. A priori, we observe that the stated problem might be infeasible. If feasible, we observe that there might conceivably be more than one optimal solution.
7.2.1
Formulating the APQ Optimization Problem
In Stanfordet al.(2014), the APQ model first introduced by Kleinrock (1964), was reconsid- ered, and the stationary waiting time distributions for each class of customer were determined. Kleinrock (1964)’s model presumed that waiting customers of theith priority class accrue pri- ority “credit” at a ratebi ≥ 0, where bi ≥ bj if classiis considered to be of higher priority to class j. In what follows, we presume that a lower class index means a higher priority, so that class 1 has top priority and class Lhas the lowest priority. We note that the APQ discipline is work-conserving.
7.2. Formulation of theOptimizationProblem 151
The APQ is a simple yet flexible scheduling model which allows both the priority of the customer (or acuity of the patient in health care applications ) as well as its time spent waiting to be factored into the decision as to which customer next enters service. Settingbi = B>0∀i, a first-come first-served queue that aggregates all customer classes is obtained. SettingbL= 1 whilebi = bi+1∗ M; i = 1,2, . . . ,L−1, a classical non-preemptive priority model is obtained
in the limit as M→ ∞.
To date, analytical solutions for the waiting time distributions exist for two cases: Stanford
et al.(2014) presents the solution for the single-server case where each class may have its own specified service time distribution, while Sharif et al. (2014) addresses the multi-server case when all service times are drawn from the same exponential service time distribution, with service rateµ.
LetAAPQdenote the set of APQ service disciplines. The resulting APQ optimization prob- lem then reduces to the selection of the best accumulation ratesbi; i = 1,2, . . . ,Lin order to minimize the weighted average excess:
mina∈AAPQ Z =
PL
n=1αnλnHn(dn) subject to Wn(dn) ≥ pn ; n = 1, 2, . . . , L;
bn ≥ bn+1; n = 1, 2, . . . , L−1;
bL ≥ 0;
where,αn,n= 1,2, . . . ,Lagain denotes the respective weight for the excess waiting for class-n customers. It is to be noted that the role ofαn’s is not to make higher priority customers gain priority faster (that’s the role of bn’s), rather αn’s provide greater weight to waiting times of the given class of customers, and as such, assess a greater penalty for incurred waits by those customers.
152 Chapter7. AnOptimizationProblem forQueuesOperating underWaitingTargets