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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Consider the following data
| x | 2 | 3 | 4 | 5 | 6 |
| y | 8.3 | 16.5 | 30.2 | 65.2 | 125.6 |
A model for insect populations leads to the difference equations
where λ and a are positive constants.
i) Write the equation in the form , and hence identify the growth rate.
ii) Plot the graph of R(Nk) as a function of N2
iii) Express the intrinsic growth rater and the carrying capacity K, for this model, in terms of the parameters, a and λ.
iv) Find the steady-state solution of this model and analyse the solution.
Do the stability analysis of the following model formulated to study the effect of toxicant on one competing species where the environment toxicant concentration is being taken to change w.r.t. time.
along with the initial conditions.
Here,
N1 (t)=Density of prey population
N2(t)=Density of predator population
Co(t) = Concentration of the toxicant in the individual of the prey population
P=Constant environmental toxicant concentration.
a,, a, are the predation rates, r,r, are the growth rates or birth rates, d, is the death rate due to C, m, is the depuration rate, Q, h, k, g are positive rate constants.
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.
The population dynamics of a species is governed by the discrete model
where r and k are positive constants. Determine the steady states and discuss the stability of the model. Find the value of r at which first bifurcation occurs. Describe qualitatively the behaviors of the population for. Since a species becomes extinct if xn ≤1 1 for any n> 1, show using iterations, that irrespective of the size of r> 1 the species could become extinct if the carrying capacity k
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.
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.