Question

Ternary A server has customers waiting to be served. The service time required by eachcustomer is known in advance: it is ${\mathbf{c}}_{\mathbf{i}}$minutes for customer i. So if, for example, the customers are served in order of increasing i , then the ${\mathbf{i}}_{\mathbf{th}}$customer has to wait ${\mathbf{P}}_{\mathbf{ij}}\mathbf{=}\mathbf{1}{\mathbf{t}}_{\mathbf{j}}$minutes. We wish to minimize the total waiting time.

$\mathbf{T}\mathbf{=}{\mathbf{X}}_{\mathbf{n}}\mathbf{i}\mathbf{=}\mathbf{1}$(time spent waiting by customer ).

Give an efficient algorithm for computing the optimal order in which to process the customers.

Short answer

Therefore, provide customized Huffman method for compressing sequences of letters from such a vocabulary with elements, where even the words appear with known frequency ${\mathrm{f}}_1,{\mathrm{f}}_2,...,{\mathrm{f}}_{\mathrm{n}},$to take use of this new technology.

Step-by-step solution

Introduction

Algorithms providing optimum customer processing ordering:

Its greedy algorithm is utilised to assist the client who serves first in the shortest period of time.

• That is, a large number of consumers are sorted according to their time "t" values, and then served in that order.

It has been discovered that it is no issue what, the total time that deals inside the consumers serve does not alter.

• The total time taken is always equal to the sum of the service times for all of the clients.

Total time

The total time can be calculated as:

$\mathrm{T}={\sum }_{\mathrm{i}=1}^{\mathrm{n}}(\mathrm{number}\mathrm{of}\mathrm{customers}\mathrm{still}\mathrm{waiting}\mathrm{at}\mathrm{time}\mathrm{t})$

• Reduce consequently, this number of customers who should wait is increasing, implying that the optimal course of action is for consumers to be served with greater service time.

Algorithm:

Explanation: Presume that there is indeed a better option than the greedy one.

• When sort data number of customers by time, the Greedy approach is employed. Customers are served once they have been sorted.

• A minimum of one pair of consecutive consumers is required for the best solution. Serving the second consumer takes less time than serving the first.

Take, for example, the requirement that "out of order" must be met by a pair of customers.

• Pair of customers denotes as ${\mathrm{c}}_{\mathrm{i}}$and $c_{i+1}$.

• Service time for the pair of customers denotes as ${\mathrm{t}}_{\mathrm{i}}$and ${\mathrm{t}}_{\mathrm{i}+1}$.

• By assumption, the service time of first customer is greater than the service time of second customer. That is,${\mathrm{t}}_{\mathrm{i}}>{\mathrm{t}}_{\mathrm{i}+1}$.

o Swap the order of the pair of customers will produce the better ordering. So, the second customer ${\mathrm{c}}_{\mathrm{i}+1}$is served before the first customer ${\mathrm{c}}_{\mathrm{i}}$.

• It does not change any waiting time of other customers.

o The waiting time for the customer ${\mathrm{c}}_{\mathrm{i}}$will increase by ${\mathrm{t}}_{\mathrm{i}+1}$and customer ${\mathrm{c}}_{\mathrm{i}+1}$will increase by ${\mathrm{t}}_{\mathrm{i}}$. So, the assumption of ${\mathrm{t}}_{\mathrm{i}}>{\mathrm{t}}_{\mathrm{i}+1}$will reduces the overall waiting time.

• As a result, changing the order of the consumers results in the shortest overall waiting time.

Sorting the number of clients "n" will take a total of n minutes $O(nlo{g}_n)$.

Each and every swap will reduce the total waiting time. So, after all the swaps are performed, the customers will be sorted in ascending order. The optimal solution becomes the greedy solution.

Therefore, the greedy solution must be the optimal solution.