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Chapter 8: Problem 43

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

Expert verified
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

01

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#
02

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.

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Most popular questions from this chapter

A group of \(n\) customers moves around among two servers. Upon completion of service, the served customer then joins the queue (or enters service if the server is free) at the other server. All service times are exponential with rate \(\mu .\) Find the proportion of time that there are \(j\) customers at server \(1, j=0, \ldots, n\).

Customers arrive at a two-server station in accordance with a Poisson process with a rate of two per hour. Arrivals finding server 1 free begin service with that server. Arrivals finding server 1 busy and server 2 free begin service with server 2. Arrivals finding both servers busy are lost. When a customer is served by server 1 , she then either enters service with server 2 if 2 is free or departs the system if 2 is busy. \(\mathrm{A}\) customer completing service at server 2 departs the system. The service times at server 1 and server 2 are exponential random variables with respective rates of four and six per hour. (a) What fraction of customers do not enter the system? (b) What is the average amount of time that an entering customer spends in the system? (c) What fraction of entering customers receives service from server \(1 ?\)

Customers arrive at a two-server system according to a Poisson process having rate \(\lambda=5\). An arrival finding server 1 free will begin service with that server. An arrival finding server 1 busy and server 2 free will enter service with server \(2 .\) An arrival finding both servers busy goes away. Once a customer is served by either server, he departs the system. The service times at server \(i\) are exponential with rates \(\mu_{i}\), where \(\mu_{1}=4, \mu_{2}=2\) (a) What is the average time an entering customer spends in the system? (b) What proportion of time is server 2 busy?

Let \(D\) denote the time between successive departures in a stationary \(M / M / 1\) queue with \(\lambda<\mu .\) Show, by conditioning on whether or not a departure has left the system empty, that \(D\) is exponential with rate \(\lambda\). Hint: By conditioning on whether or not the departure has left the system empty we see that $$ D=\left\\{\begin{array}{ll} \text { Exponential }(\mu), & \text { with probability } \lambda / \mu \\ \text { Exponential }(\lambda) * \text { Exponential }(\mu), & \text { with probability } 1-\lambda / \mu \end{array}\right. $$ where Exponential \((\lambda) *\) Exponential \((\mu)\) represents the sum of two independent exponential random variables having rates \(\mu\) and \(\lambda\). Now use moment-generating functions to show that \(D\) has the required distribution. Note that the preceding does not prove that the departure process is Poisson. To prove this we need show not only that the interdeparture times are all exponential with rate \(\lambda\), but also that they are independent.

Carloads of customers arrive at a single-server station in accordance with a Poisson process with rate 4 per hour. The service times are exponentially distributed with rate 20 per hour. If each carload contains either 1,2, or 3 customers with respective probabilities \(\frac{1}{4}, \frac{1}{2}\), and \(\frac{1}{4}\), compute the average customer delay in queue.
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