For the network below, the objective is to find the length of the shortest path from node $$P$$ to node $$G.$$ Let $${d_{ij}}$$ be the length of directed are from node $$i$$ to node $$j$$. Let $${s_j}$$ be the length of the shortest path from $$P$$ to node $$j.$$ Which of the following equations can be used to find $${s_G}$$?

In an $$M/M/1$$ queuing system, the number of arrivals in an interval of length $$T$$ is a Poisson random variable (i.e., the probability of there being $$n$$ arrivals in an interval of length $$T$$ is $${{{e^{ - \lambda T}}{{\left( {\lambda T} \right)}^n}} \over {n!}}$$). The probability density function $$f(t)$$ of the inter-arrival time is given by

A moving average system is used for forecasting weekly demand. $${F_1}\left( t \right)$$ and $${F_2}\left( t \right)$$ are sequences of forecasts with parameters $${m_1}$$ and $${m_2}$$, respectively, where $${m_1}$$ and $${m_2}\left( {{m_1} > {m_2}} \right)$$ denote the numbers of weeks over which the moving averages are taken. The actual demand shows a step increase from $${d_1}$$ to $${d_2}$$ at a certain time. Subsequently,

Match the type of gears with their most appropriate description

Type of gear
(P)$$\,\,\,\,\,$$ Helical
(Q)$$\,\,\,\,\,$$ Spiral Bevel
(R)$$\,\,\,\,\,$$ Hypoid
(S)$$\,\,\,\,\,$$ Rack and pinion

Description
1.$$\,\,\,\,\,$$ Axes non parallel a and non intersecting
2$$\,\,\,\,\,$$. Axes parallel and teeth are inclined to the axis
3.$$\,\,\,\,\,$$ Axes parallel and teeth are parallel to the axis
4.$$\,\,\,\,\,$$ Axes are perpendicular and intersecting and teeth are inclined to the axis
5.$$\,\,\,\,\,$$ Axes are perpendicular and used for large speed reduction
6.$$\,\,\,\,\,$$ Axes parallel and one of the gears has infinite radius