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 2 every 10 minutes and service time follows exponential distribution with a mean of 15 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 present system is an (M/M/2) queue. Borrowers arrive according to a Poisson process at the rate of 2 every 10 minutes, so the arrival rate is (\lambda = 0.2) per minute. The mean service time is 15 minutes, hence the service rate per counter is (\mu = 1/15 = 0.0667) per minute. With two counters, the traffic intensity is [ \rho = \frac{\lambda}{2\mu} = \frac{0.2}{2\times 0.0667} = 1.5. ] Since (\rho > 1), the present __________ __________ ____ _________ _______ ______ _________ ____ _________ ________ ________.
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Do the stability analysis of the following competing species system of equations with diffusion and advection
where V1 and V2 are advection velocities in x direction of the two populations with densities N1 and N2 respectively. a1 is the growth rate, b1 is the predation rate, d1 is the death rate, C1 is the conversion rate. D1 and D2 are diffusion coefficients. The initial and boundary conditions are:
at
and
where are the equilibrium solutions of the given system of equations.
Interpret the solution obtained and also write the limitations of the model.
Return distributions of the two securities are given below:
| Return | Probabilities | |
| X | Y |
|
| 0.20 | 0.15 | 0.30 |
| 0.15 | 0.08 | 0.25 |
| 0.10 | 0.05 | 0.15 |
| 0.11 | 0.09 | 0.25 |
Which one of the following portfolios cannot lie on the efficient frontier as described by Markowitz?
| Portfolio | Expected return | Standard deviation |
| W | 10% | 25% |
| X | 5% | 7% |
| Y | 17% | 37% |
| Z | 12% | 13% |
A tumour is developing from the organ of a human body with concentration with growth and decay control parameters 7.2 and 2.7 respectively. In how many days the size of the tumor will be twice?
Let be a portfolio of two securities X and Y. Find the values of w1 and w2 in the following situations:
i) and P is risk free.
ii) and variance P is minimum.
iii) Variance P is minimum and and
.
Do the stability analysis of the following model formulated to study the effect of toxicant on prey-predator population and interpret the solution.
Where all the variables and constants are same as defined in the system (32)-(35) except for the following
A company manufacturing soft drinks is thinking of expanding its plant capacity so as to meet future demand. The monthly sale for the past 6 years are available. State, giving reasons, the type of modelling you will use to obtain good estimates for future demand so as to help the company make the right decisions. Also state four essentials and four non-essentials for the problem.