Question

Consider a system where the interarrival times have an arbitrary distribution \(F\), and there is a single server whose service distribution is \(G\). Let \(D_{n}\) denote the amount of time the \(n\) th customer spends waiting in queue. Interpret \(S_{n}, T_{n}\) so that $$ D_{n+1}=\left\\{\begin{array}{ll} D_{n}+S_{n}-T_{n}, & \text { if } D_{n}+S_{n}-T_{n} \geqslant 0 \\ 0, & \text { if } D_{n}+S_{n}-T_{n}<0 \end{array}\right. $$

Short answer

In the given equation for the waiting time of customers in a single server system with interarrival time distribution \(F\) and service time distribution \(G\), the variables \(S_n\) and \(T_n\) can be interpreted as follows: - \(S_n\) represents the service time for the \(n\)th customer. - \(T_n\) represents the interarrival time between the arrival of the \(n\)th customer and the arrival of the \((n+1)\)th customer.

Step-by-step solution

The variable \(D_n\) represents the amount of time the \(n\)th customer spends waiting in the queue. Our goal is to interpret the variables \(S_n\) and \(T_n\) in the given equation. #Step 2: Identify the role of Sn and Tn in the equation#

The equation is: $$ D_{n+1}=\left\{\begin{array}{ll} D_{n}+S_{n}-T_{n}, & \text { if } D_{n}+S_{n}-T_{n} \geqslant 0 \\\ 0, & \text { if } D_{n}+S_{n}-T_{n}<0 \end{array}\right. $$ This is a piecewise function, which means that the equation for \(D_{n+1}\) depends on whether or not the expression \(D_n + S_n - T_n\) is non-negative. #Step 3: Analyzing the two cases#

We have two cases: 1. If \(D_n + S_n - T_n \geq 0:\) In this case, the waiting time for the \((n+1)\)th customer is equal to the waiting time of the \(n\)th customer, plus some additional time \(S_n\), minus some other time \(T_n\). Here, the \((n+1)\)th customer's waiting time depends on the \(n\)th customer's waiting time, which means the server is still working on the \(n\)th customer when the \((n+1)\)th customer arrives. Hence, \(S_n\) would represent the service time of the \(n\)th customer, and \(T_n\) would represent the interarrival time between the \(n\)th and \((n+1)\)th customers. 2. If \(D_n + S_n - T_n < 0:\) In this scenario, the \((n+1)\)th customer's waiting time is zero, meaning they have no delay. This tells us that the server finished working on the \(n\)th customer before the \((n+1)\)th customer arrives. In this case, \(S_n\) represents the service time of the \(n\)th customer, and \(T_n\) is the interarrival time between the \(n\)th and \((n+1)\)th customers, similar to the previous case. #Step 4: Interpreting Sn and Tn#

Based on the analysis in step 3, we can interpret \(S_n\) and \(T_n\) as follows: - \(S_n\) is the service time for the \(n\)th customer. - \(T_n\) is the interarrival time between the arrival of the \(n\)th customer and the arrival of the \((n+1)\)th customer.