A coffee kiosk at a transit station opens from 6:00 AM to 8:00 PM. Potential customers...

 

Transcribed Image Text:

A coffee kiosk at a transit station opens from 6:00 AM to 8:00 PM. Potential customers arrive at the coffee kiosk at a Poisson rate of A (number of customers per hour). An entering customer first places an order and pays using a dedicated QR ordering system (no waiting time to place an order). Each customer is allowed to order one drink only. Once an order is placed, the customer will wait for the drink to be served. Two baristas are working at the kiosk, and the first free barista will prepare and serve the drink when an order is received. Suppose each barista works at Poisson rate of (number of customers per hour) to prepare and serve one drink. (a) Set up a birth and death process for the number of customers at the coffee kiosk (in queue for drink preparation and drink is under preparation) and draw its transition rate diagram. (5 marks) (b) Show that the long-run probability of having no customer at the kiosk is given by 2-2 2+2 Hence, state the necessary condition for the limiting probabilities to exists. (5 marks) (c) Show that if the necessary condition for the existence of limiting probabilities in part (b) holds, the expected number of customers at the kiosk is given by 2 (4-2) (5 marks) A coffee kiosk at a transit station opens from 6:00 AM to 8:00 PM. Potential customers arrive at the coffee kiosk at a Poisson rate of A (number of customers per hour). An entering customer first places an order and pays using a dedicated QR ordering system (no waiting time to place an order). Each customer is allowed to order one drink only. Once an order is placed, the customer will wait for the drink to be served. Two baristas are working at the kiosk, and the first free barista will prepare and serve the drink when an order is received. Suppose each barista works at Poisson rate of (number of customers per hour) to prepare and serve one drink. (a) Set up a birth and death process for the number of customers at the coffee kiosk (in queue for drink preparation and drink is under preparation) and draw its transition rate diagram. (5 marks) (b) Show that the long-run probability of having no customer at the kiosk is given by 2-2 2+2 Hence, state the necessary condition for the limiting probabilities to exists. (5 marks) (c) Show that if the necessary condition for the existence of limiting probabilities in part (b) holds, the expected number of customers at the kiosk is given by 2 (4-2) (5 marks)