course: management science queuing system c) Kim Him, mekanik di Silvar Muftler

Question

course: management science
queuing system

c) Kim Him, mekanik di Silvar Muftler Shop, boleh memasang mufflers baru pada kira-kira 1 am setiap 20 minit. Pelanggan yang mahukan perkhidmatan mereka tiba di kedaí pada purata 2 setlap jam, berikutan taburat Poisson. Mereka dlservis secara keluar yang pertama masuk, pertama dan datang dari penduduk yang sangat besar pembeli mungkin. Kira ciri-ciri operasi sistem perkhidmatan berikut Kim Him, the mechanic &t; Silver Mufler Shop, is able to install new mufflers at about 1 every 20 minutes. Custonners seeking their services arive at the shop on the everage of 2 per hour, totowing a Poisson distributiod. They are served on a first-in, first out basis and come from a very large population of possible buyers. Caloulate the following operating characteristics of the service system: i. Purata bilangan berada di dalam sistem Average number in the system (3 markahl marks) . Kebarangkalian bahawa tidak ada pelajar dalam sistem. The probebility that there are no students in the system (3 markah/ marks) ii. Purata bilangan pelajar beratur Average number of studerts waiting in linte (3 markahi marks) Purata Masa berada di dalam barisan. Average time waiting in line iv. (3 markahl marks) v. Penggunaan sistem. System utilization (3 markahl marks)

Explanation / Answer

Solution

We will solve this question using theory of M.M/1 Queue System.

Back-up Theory

An M/M/1 queue system is characterized by arrivals following Poisson pattern with average rate ?, [this is also the same as exponential arrival with average inter-arrival time = 1/ ?] service time following Exponential Distribution with average service time of (1/µ) [this is also the same as Poisson service with average service rate = 1/µ] and single service channel.

Let n = number of customers in the system and m = number of customers in the queue.

[Trivially, n = m + number of customers under service.]

Let (?/µ) = ?

The steady-state probability of n customers in the system is given by Pn = ?n(1 - ?) ………(1)

The steady-state probability of no customers in the system is given by P0 = (1 - ?) ………(2)

Average queue length = E(m) = (?2)/{µ(µ - ?)} …………………………………………..(3)

Average number of customers in the system = E(n) = (?)/(µ - ?)…………………………..(4)

Average waiting time = E(w) = (?)/{µ(µ - ?)} ……………………………………………..(5)

Average time spent in the system = E(v) = {1/(µ - ?)}……………………………………..(6)

Percentage idle time of service channel = P0 = (1 - ?) …………………………………….(7)

Probability of waiting = 1 - P0 = ? …………………..…………………………………….(8)

Preparatory Work

Given ? = 2/hour, µ = 3/hour, ? = 2/3 = 0.6667 …………………………………………..(8)

i) Average number of customers in the system = E(n) = (?)/(µ - ?) = 2 ANSWER [vide (4)]

ii) Probability none in the system = P0 = (1 - ?) = 1/3 = 0.3333 ANSWER [vide (7)]

iii) Average number of customers waiting in the queue = E(m) = (?2)/{µ(µ - ?)}

= 4/3 = 1.3333 ANSWER [vide (3)]

iv) Average time a customer waits in the queue = E(w) = (?)/{µ(µ - ?)}

= 2/3 hour = 40 minutes ANSWER [vide (5)]

v) System utilization = 1 - P0 = ? = 2/3 = 66.67% ANSWER [vide (7)]

DONE

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