Static and dynamic robustness

We consider a multiserver model that consists of two servers and two queues (Beneficiary-Donor model), as shown in Figure 7.1. Jobs arrive at queue 1 and queue 2 according to Poisson processes (or Markov modulated Poisson processes, as defined in Section 3.2) with average arrival rates $\lambda_1$ and $\lambda_2$, respectively. Jobs have exponentially distributed service demands; however, the running time of a job may also depend on the affinity between the particular server and the particular job/queue. Hence, we assume that server 1 (beneficiary server) processes jobs in queue 1 (type 1 jobs) with rate $\mu_1$ (jobs/sec), while server 2 (donor server) can process type 1 jobs with rate $\mu_{12}$, and can process jobs in queue 2 (type 2 jobs) with rate $\mu_2$. We define $\rho_1=\lambda_1/\mu_1$, $\rho_2=\lambda_2/\mu_2$, and $\hat\rho_1=\lambda_1/(\mu_1+\mu_{12}(1-\rho_2))$. Note that $\rho_2<1$ and $\hat\rho_1<1$ are necessary for the queues to be stable under any allocation policy, since the maximum rate at which type 1 jobs can be processed is $\mu_1$, from server 1, plus $\mu_{12}(1-\rho_2)$, from server 2.

The Beneficiary-Donor model has a wide range of applications in service industries such as call centers and repair facilities. For example, in call centers, the donor server may be a bilingual operator, and the beneficiary server may be a monolingual operator [58,105,176,185,186], or the donor server may be a cross-trained or experienced operator who can handle all types of calls, and the beneficiary server may be a specialized operator who is only trained to handle a specific type of calls [58,176]. (See also Chapter 8) In a repair facility, the donor server may be a technician who can handle jobs of any difficulty, and the beneficiary server may be a technician with limited expertise [61].

Figure 7.1: The Beneficiary-Donor model.

We design and evaluate allocation policies for the Beneficiary-Donor model with respect to three objectives. First, as is standard in the literature, we seek to minimize the overall weighted mean response time, $c_1p_1\mbox{{\bf\sf E}}\left[ R_1 \right] + c_2p_2\mbox{{\bf\sf E}}\left[ R_2 \right]$, where $c_i$ is the weight (importance) of type $i$ jobs, $p_i=\lambda_i / (\lambda_1+\lambda_2)$ is the fraction of type $i$ jobs, and $\mbox{{\bf\sf E}}\left[ R_i \right]$ is the mean response time of type $i$ jobs, for $i=1,2$. Here, response time refers to the total time a job spends in the system. Below, we refer to overall weighted mean response time simply as mean response time.

In addition to mean response time, we consider an additional metric, robustness, introducing two types of robustness: static robustness and dynamic robustness. Static robustness measures robustness against misestimation of load; to evaluate static robustness, we analyze the mean response time of allocation polices for a range of loads to see how a policy tuned for one load behaves under different loads. Dynamic robustness measures the robustness against fluctuations in load; to evaluate dynamic robustness, we analyze the mean response time of allocation policies under Markov modulated Poisson processes, where arrivals follow a Poisson process at each moment, but the arrival rate changes over time.