=
=
=
W
Wq
Lq
P (q(t) < 2/6) = 1 − 0.75 · e−20/60·(1/1) = 0.4626.
The probability asked for is
2
3
Since X(t) has only a finite number of states, it’s a stable process
independently from the values of λ and μ.
Its state diagram looks like:
X(t) is modelled as a Birth & Death Process with states {0, 1, ..., K}.
Unfortunately, it’s computationally slightly harder to deal with.
We denoted the waiting time in the queue by q(t).
q(t) has the cumulative distribution function 1 − aey(μ−λ).
An M/M/1 queue with limited size K is a lot more realistic than the one
with infinite queue.
Stat 330 (Spring 2015): slide set 22
1
On average, a job has to spend 45 min in the queue. What is the probability
that a job has to spend less than 20 min in the queue?
Wq λq = 0.75 · 3 = 2.25
The M/M/1/K queue
Stat 330 (Spring 2015): slide set 22
1
= 0.25 hours = 15 min
μ
=
Ws
L
3
= = 1 hour
λ
3
W − Ws = 0.75 hours = 45 minutes
E[X(t)] =
=
L
a
0.75
=
=3
1−a
0.25
Printer Queue (continued) A certain printer in the Stat Lab gets jobs with
a rate of 3 per hour. On average, the printer needs 15 min to finish a job.
Let X(t) be the number of jobs in the printer and its queue at time t. We
know already: X(t) is a Birth & Death Process with constant arrival rate
λ = 3 and constant death rate μ = 4.
The properties of interest for this printer system then are:
The M/M/1 Queue: Example (cont’d)
Last update: March 22, 2015
Stat 330 (Spring 2015)
Slide set 22
Stat 330 (Spring 2015): slide set 22
The M/M/1 Queue: Example
kak · p0 = ... =
L
λ̄
5
Stat 330 (Spring 2015): slide set 22
Stat 330 (Spring 2015): slide set 22
where λ̄ is the average arrival rate into the system.
W =
4
a
(K + 1)aK+1
−
1−a
1 − aK+1
Similarly, the expected length of the queue is Lq = Wq · λa.
Theerefore, the expected total waiting time is then W = L/λa
λa = λ − pK λ = (1 − pK )λ.
The adjusted rate λa of individuals entering the system is:
We have to adjust λ by the rate of individuals who are turned away.
6
7
The probability (or fraction of time) that the owner will be alone is
1−a
0.2
p0 = 1−a
K+1 = 1−0.85 = 0.2975.
The number of customers in the shop can be modeled as a B & D
Process with arrival rate λ = 0.2 per minute and μ = 0.25 per minute
and upper size K = 4.
1. What fraction of time will the owner be in the shop on his own?
Convenience Store: In a small convenience store there’s room for only
4 customers. The owner himself deals with all the customers - he likes
chatting a bit. On average it takes a customer 4 minutes to pay for his/her
purchase. Customers arrive at an average of 1 per 5 minutes. If a customer
finds the shop full, he/she will go away immediately.
k=0
K
At this point we have to be careful when dealing with limited systems: λ̄ is
NOT equal to the arrival rate λ.
kpk =
For the expected total waiting time W , we used Little’s theorem:
The rate of individuals being turned away therefore is pK · λ.
Since an incoming individual is turned away, when the system is full, the
probability for being turned away is pK .
It’s therefore a good strategy to try and minimize this number.
From a marketing perspective they are the ”expensive” ones - they are most
likely annoyed and less inclined to return.
Another interesting property of a queuing system with limited size is the
number of individuals that get turned away.
Example
k=0
K
Stat 330 (Spring 2015): slide set 22
The M/M/1/K queue (cont’d)
The M/M/1/K queue (cont’d)
=
L = E[X(t)] = 0 · p0 + 1 · p1 + 2 · p2 + ... + K · pK
The expected number of individuals in the system L then is:
where a = μλ , the traffic intensity and S = 1 + a + a2 + ... + aK =
1−a
=
1 − aK+1
= = a k p0
p0 = S −1
pk
The steady state probabilities pk are:
1−aK+1
1−a .
Stat 330 (Spring 2015): slide set 22
The M/M/1/K queue (cont’d)
Stat 330 (Spring 2015): slide set 22
a
(K + 1)aK+1
= 1.56.
−
1−a
1 − aK+1
W =
L
= 1.56/(0.2 − 0.0243) = 8.88 minutes .
λa
4. What is the expected time a customer has to spend for check-out?
8
p4λ = 0.84·0.2975·0.2 per minute = 0.0243 per minute = 1.46 per hour
3. What is the rate of individuals being turned away?
L=
2. What is the expected number of customers in the store?
Example (cont’d)