A maintenance service facility has Poisson arrival rates, negative exponential service time and operates on a ‘first come first served’ queue discipline. Breakdowns occur on an average of $$3$$ per day with a range of zero to eight. The maintenance crew can service an average of $$6$$ machines per day with a range of zero to seven. The mean waiting time for an item to be serviced would be

A company produces two types of toys: $$P$$ and $$Q.$$ Production time of $$Q$$ is twice that of $$P$$ and the company has a maximum of $$2000$$ time units per day. The supply of raw material is just sufficient to produce $$1500$$ toys (of any type) per day. Toy type $$Q$$ requires an electric switch which is available @ $$600$$ pieces per day only. The company makes a profit of Rs.$$3$$ and Rs.$$5$$ on type $$P$$ and $$Q$$ respectively. For maximization of profits, the daily production quantities of $$P$$ and $$Q$$ toys should respectively be

Two mating spur gears have $$40$$ and $$120$$ teeth respectively. The pinion rotates at $$1200$$ $$rpm$$ and transmits a torque of $$20N.m.$$ The torque transmitted by gear is

A compacting machine shown in the figure below is used to create a desired thrust force by using a rack and pinion arrangement. The input gear is mounted on the motor shaft. The gears have involutes teeth of $$2$$ $$mm$$ module.

If the drive efficiency is $$80\% ,$$ the torque required on the input shaft to create $$1000N$$ output thrust is