Question

In a two-class priority queueing model suppose that a cost of \(C_{i}\) per unit time is incurred for each type \(i\) customer that waits in queue, \(i=1,2 .\) Show that type 1 customers should be given priority over type 2 (as opposed to the reverse) if $$ \frac{E\left[S_{1}\right]}{C_{1}}<\frac{E\left[S_{2}\right]}{C_{2}} $$

Short answer

In a two-class priority queueing model, type 1 customers should be given priority over type 2 customers if \(\frac{E[S_{1}]}{C_{1}} < \frac{E[S_{2}]}{C_{2}}\), where \(E[S_i]\) is the expected waiting time and \(C_i\) is the cost per unit waiting time for each customer type. This is because prioritizing type 1 customers leads to a favorable change in waiting times and an overall reduction in waiting time cost when the inequality holds.

Step-by-step solution

In the given inequality, \(E[S_i]\) represents the expected waiting time for type \(i\) customers in the queue, where \(i=1,2\). Therefore, the ratio \(\frac{E\left[S_{i}\right]}{C_{i}}\) represents the cost per unit time (waiting time) for type \(i\) customers in the queue. By comparing the cost per unit waiting time for both types of customers, the inequality states that type 1 customers should be given priority over type 2 if the cost per unit waiting time for type 1 customers is less than the cost per unit waiting time for type 2 customers. #Step 2: Analyze the benefit of prioritizing type 1 customers#

By prioritizing a type of customer, we can reduce the expected waiting time for that type of customer. The idea behind the priority is to reduce the overall cost for both types of customers' waiting time without sacrificing the service to any type. Let's assume we prioritize type 1 customers over type 2. This will result in a reduction in the expected waiting time for type 1 customers, denoted by \(\Delta S_1\), while possibly increasing the waiting time for type 2 customers, denoted by \(\Delta S_2\). #Step 3: Determine the change in cost per unit waiting time#

When we prioritize type 1 customers, we need to calculate the overall change in customer waiting time cost. We can represent the change in cost as: $$ \Delta C = C_1 \Delta S_1 + C_2 \Delta S_2 $$ Since we are prioritizing type 1 customers, \(\Delta S_1\) will be negative (their waiting time will decrease) and \(\Delta S_2\) will be positive (type 2 customers waiting time may increase). #Step 4: Identify the condition for favorable prioritization#

For the overall cost to be reduced, we need the change in cost \(\Delta C\) to be negative, which means: $$ C_1 \Delta S_1 + C_2 \Delta S_2 < 0 $$ We can rearrange the inequality as: $$ \frac{\Delta S_1}{\Delta S_2} < - \frac{C_2}{C_1} $$ The ratio \(\frac{\Delta S_1}{\Delta S_2}\) represents the change in waiting times when prioritizing type 1 customers. Since \(\frac{E\left[S_{1}\right]}{C_{1}}<\frac{E\left[S_{2}\right]}{C_{2}}\) holds, the change in waiting times will be favorable, and the overall waiting time cost will be reduced. Thus, this proves that type 1 customers should be given priority over type 2 customers when the given inequality holds.