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

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 $${s_G}$$?

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,

A journal bearing has a shaft diameter of $$40mm$$ and a length of $$40mm.$$ The shaft is rotating at $$20$$ $$rad/s$$ and viscocity of the lubricant is $$20mPa.s.$$ The clearance is $$0.020mm.$$ The loss of the torque due to viscocity of the lubricant is approximately