Answer to Question #178733 in Operations Research for Lukiyo

Question

A tractor operator has only one tractor and he operates it on the job order from farmers. The requestors for the jobs arrive with Poisson distribution having interval time of 0.7 day. The average time to do a job is distributed exponentially with mean 0.5 day. Assuming that the tractor can take up the next job immediately on completion of the previous job, determine the following: i. Will a queue be formed? Explain ii. If queue is formed will it statistically stabilize? Explain your answer iii. What is the utilization factor of the tractor? iv. What is the idle time in daily duty of 7 hours? v. What is the mean number of job orders waiting? vi. What is the mean waiting time for job orders in the system? vii. What is the mean waiting time in the queue?

Answer 1

mean $\lambda=0.5$

interval time $t=0.7 day=16.8 hrs$

Probability of 0 jobs $=\dfrac{e^{-\lambda}\times \lambda^0}{0!}=e^{-0.5}=0.601$

(i) Queue will not formed , as the job order arrive is always less than the time taken by the tractor.

(ii) As Queue is not formed , So It doesn't matter If its statistically stablize or not.

(iii) Utilization factor $=\dfrac{piece of equipmenta}{times taken}=\dfrac{0.5}{16.8}=0.0297$

(iv) Idle time in daily duty 7 hours=$\dfrac{7}{24}=0.291day$

(v) Mean number of job order waiting =$\dfrac{0.601}{0.0297}=20.35$

(vi) Mean umber of job order waiting in the system =20.35

(vii) mean waiting time in the queue=$\dfrac{16.8}{7}=2.4hrs$