A library wants to improve its service facilities in terms of the waiting time of its borrowers. The library has two counters at present and borrowers arrive according to Poisson distribution with arrival rate 1 every 6 minutes and service time follows exponential distribution with a mean of 10 minutes. The library has relaxed its membership rules and a substantial increase in the number of borrowers is expected. Find the number of additional counters to be provided if the arrival rate is expected to be twice the present value and the average waiting time of the borrower must be limited to half the present value.
The current system at the library can be modeled as a M/M/2 queue, where the first "M" stands for the Poisson arrival process (exponentially distributed inter-arrival times), the second "M" represents the exponential service time, and "2" indicates the two counters (servers).
Given data:
- Arrival rate, λ=16 per minute\lambda = \frac{1}{6} \text{ per minute},
- Service rate, μ=110 per minute\mu = \frac{1}{10} \text{ ____ ___ ___ __________ ________ ___ ____ ____.
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ompanies located on the banks of a river, dumping their chemicals waste into river, causing high levels of pollution. Local authorities passed new legislation with very high fines if the pollution in the river exceeds certain specified concentration limits. State, giving reasons, the type of modeling you will use to find a policy for discharging the waste to ensure that the concentration level never exceeds the specified limits. Also state four essentials and two non-essentials for the problem.
In a species of animals a constant fraction of the population = 6.2 are born each breeding season and a constant fraction ẞ=4.5 die. Formulate a difference equation for the population and find out the number of individuals after fifteen seasons given that the initial number is 987. Find the closed form solution of the formulated difference equation. If the growth rate of the population is represented by r then interpret the solution obtained when i) r > 0 and ii) r < 0.
Do the stability analysis of the following competing species system of equations with diffusion and advection
where Vand V, are advection velocities in x direction of the two populations with densities N, and N, respectively. a, is the growth rate, b, is the predation rate, d, is the death rate, C, is the conversion rate. D, and D, are diffusion coefficients. The initial and boundary conditions are:
where are the equilibrium solutions of the given system of equations. Interpret the solution obtained and also write the limitations of the model.
For a multi-channel queuing system with λ = 12/hours, u=5/hours, c=3, po, = 0.056, calculate
i) The average time a customer is in the system
ii) The average number of customers in the system
iii) Whether any time would be saved for customers if the three-channel system with the service rate of 5 per hour is replaced by a single-channel system with an average service rate of 15 per hour?
The owner of a readymade garments store sells two types of shirts: Zee-shirts and Button-down shirts. He makes a profit of Rs. 5 and Rs. 10 per shirt on Zee-shirts and Button-down shirt, respectively. He has two tailors, A and B at his disposal to stitch the shirts. Tailors A and B can devote at the most 7 hours and 15 hours per day, respectively. Both these shirts are to be stitched by both the tailors. Tailors A and B spend 2 hours and 5 hours, respectively in stitching one Zee-shirts, and 4 hours and 3 hours, respectively in stitching a Button-down shirt. How many shirts of both types should be stitched in order to maximize daily profit?
a) Formulate and solve this problem as an LP problem.
b) If the optimal solution is not integer-valued, use Gomory technique to derive the optimal integer solution.
Five securities have the following expected returns
A=20%, B=15%, C=25%, D=22%, E=18%. Calculate the expected returns for a portfolio consisting of all five securities under the following conditions
i) The portfolio weights are of equal percentage in each
ii) The portfolio weights are 32% in A and remaining are equally divided among other four securities.
Ships arrive at a port at the rate of one in every 4 hours with exponential distribution of inter- arrival times. The time a ship occupies a berth for unloading has exponential distribution with an average of 10 hours. If the average delay of ships waiting for berths is to be kept below 14 hours, how many berths should be provided at the port?