Hwk-Week2 Matthew Farris September 10, 2017 Chapter 2: Basics of Queueing Theory Problem 2 To solve this problem, we can use the basics of M/M/1 system, however we must tweak the service time from exponential to a continously uniformly distributed service time. This means we can use the PollaczekKhinchine formula, and then follow through with Little’s Law and the other relations to get the other variables First we find Wq using the formula: Wq =
λ(σ 2 +1/µ2 ) 2(1−λ/µ)
We know the standard deviation for a uniform distribution is: p σ = (b − a)2 /12) which is equal to: p σ = (1.9 − .1)2 /12) ⇒ 0.52 min *I think this has a unit of minutes? Plugging this variable, the arrival rate and the estimate service rate (still 1) we get: Wq =
.8(.5192 +1/12 ) 2(1−.8/1)
⇒ 2.54 min
From the reading we know that we can calculate Lq simply: Lq = λWq ⇒ 0.8X2.54 ⇒ 2.03 people in queue Next, we can find W from the below Equation W = Wq + E(S) ⇒ 2.54 + 1 ⇒ 3.54 min Applying our relations again, we can find L just by utilizing λ L = λW ⇒ 0.8 ∗ 3.54 ⇒ 2.83 people in the systen And finally we can find our steady-state utilization: ρ = λ/µ ⇒ .8/1 = .8 Comparing our values from problem one we can see the following:
W Wq L Lq p
Prob_1
Prob_2
5.0 4.0 4.0 3.2 0.8
3.54 2.54 2.83 2.03 0.80
1
Problem 3 For this problem we will tackle it in much the same way as the previous, albeit a bit more painstakingly. The expected value is again 1 so, first we will find the standard deviation of the triangular distribution. p σ = (a2 + m2 + b2 − am − ab − bm)/18) ⇒ 0.37 Next we find wq λ(σ 2 +1/µ2 ) 2(1−λ/µ)
Wq =
⇒
.8(.5192 +1/12 ) 2(1−.8/1)
⇒ 2.27 min
and use all the formulas above to find the rest of the values: Lq = λWq ⇒ 0.8X2.27 ⇒ 1.82 people in queue W = Wq + E(S) ⇒ 2.27 + 1 ⇒ 3.27 min L = λW ⇒ 0.8 ∗ 3.54 ⇒ 2.62 people in the systen ρ = λ/µ ⇒ .8/1 = .8 Prob_1
Prob_2
Prob_3
5.0 4.0 4.0 3.2 0.8
3.54 2.54 2.83 2.03 0.80
3.27 2.27 2.62 1.82 0.80
W Wq L Lq p
Problem 5 Now we move on to a entity with more than one server, in this instance 3. Following the basic formulas, we first need to have a singular value to compute all the others, and fortunately, we have another plug-and-play formula. First we can calculate p(0) with the following formula: p(0) =