Introduction to Queueing Systems
2.8 PROPERTIES OF THE POISSON PROCESS .1 Superposition Property
2.8.5 Poisson Arrivals During a Random Time Interval
Consider the number of arrivals (N) in a random time interval I. Assuming that I is distributed with a probability density function A(t) and I is independent of the Poisson process, then
P N( = =k) ∞
∫
P N( =k I| =t A t dt) ( )0
But P N k I t t
k e
k
( | ) ( ) t
= = = λ! −λ
Hence P N k t
k e A t dt
k
( ) ( ) t
! ( )
= =∞
∫
−0
λ λ
last arrival next arrival
t = 0 t0
t
time
Figure 2.14 Conditional inter-arrival times
Taking the z-transform, we obtain
N z t
k e A t dt z
e tz
k
k
k
t k
t k
k
( ) ( )
! ( )
( )
=
=
=
∞ ∞
−
∞
−
=
∞
∑ ∫
∫ ∑
0 0
0 0
λ λ
λ
λ
!! ( ) ( ) )
)
A t dt e A t dt
A z
= z t
= −
∞
∫
−( − 0λ λ
λ λ
*(
where A*(l − lz) is the Laplace transform of the arrival distribution evaluated at the point (l − lz).
Example 2.5
Let us consider again the problem presented in Example 2.4. When this pas- senger arrives at the station:
a) What is the probability that he will board a train in the next 5 minutes?
b) What is the probability that he will board a train in 5 to 9 minutes?
Solution
a) From Example 2.4, we have l = 1/10 = 0.1 min−1, hence for a time period of 5 minutes we have
λ
λ
λ
t
P train in e t
k
t k e
= × =
= − = − =
5 0 1 0 5
0 5 0 5
0 0
0 5 0
. .
[ ( )
!
( . )
!
.
and
min] ..607
He will board a train if at least one train arrives in 5 minutes; hence P[at least 1 train in 5 min] = 1 − P[0 train in 5 min]
= 0.393
b) He will need to wait from 5 to 9 minutes if no train arrives in the fi rst 5 minutes and board a train if at least one train arrives in the time interval 5 to 9 minutes. From (a) we have
P[0 train in 5 min] = 0.607
PROPERTIES OF THE POISSON PROCESS 67
and P[at least 1 train in next 4 min] = 1 − P[0 train in next 4 min]
= −1 − 0 4 =
0 0 33
0 4 0
e . ( . )
! .
Hence, P[0 train in 5 min & at least 1 train in next 4 min]
= P[0 train in 5 min] × P[at least 1 train in next 4 min]
= 0.607 × 0.33 = 0.2
Example 2.6
Pure Aloha is a packet radio network, originated at the University of Hawaii, that provides communication between a central computer and various remote data terminals (nodes). When a node has a packet to send, it will transmit it immediately. If the transmitted packet collides with other packets, the node concerned will re-transmit it after a random delay t. Calculate the throughput of this pure Aloha system.
Solution
For simplicity, let us make the following assumptions:
(i) The packet transmission time is one (one unit of measure).
(ii) The number of nodes is large, hence the total arrival of packets from all nodes is Poisson with rate l.
(iii) The random delay t is exponentially distributed with density function be−bt, where b is the node’s retransmission attempt rate.
Given these assumptions, if there are n node waiting for the channel to re-transmit their packets, then the total packet arrival presented to the channel can be assumed to be Poisson with rate (l + nb) and the throughput S is then given by
S = (l + nb)P[a successful transmission]
= (l + nb)Psucc
From Figure 2.15, we see that there will be no packet collision if there is only one packet arrival within two units of time. Since the total arrival of packets is assumed to be Poisson, we have
Psucc= e−2(l+nb) and hence
S = (l + nb)e−2(l+nb)
Problems
1. Figure 2.16 shows a schematic diagram of a node in a packet switch- ing network. Packets which are exponentially distributed arrive at the big buffer B according to a Poisson process. Processor P is a switch- ing processor that takes a time t, which is proportional to the packets’
length, to route a packet to either of the two links A and B. We are interested in the average transition time that a packet takes to com- plete its transmission on either link, so how do you model this node as a queueing network?
2. Customers arrive at a queueing system according to a Poisson process with mean l. However, a customer entering the service facility will visit the exponential server k times before he/she departs from the system. In each of the k visits, the customer receives an exponentially distributed amount of time with mean 1/km.
(i) Find the probability density function of the service time.
(ii) How do you describe the system in Kendall notation?
3. A communication line linking devices A and B is operating at 4800 bps.
If device A sends a total of 30 000 characters of 8 bits each down the line in a peak minute, what is the resource utilization of the line during this minute?
4. All the telephones in the Nanyang Technological University are con- nected to the university central switchboard, which has 120 external lines to the local telephone exchange. The voice traffi c generated by its employees in a typical working day is shown as:
2 units of time
Figure 2.15 Vulnerable period of a transmission
Packets
link A
link B
B P
Figure 2.16 A schematic diagram of a switching node
PROPERTIES OF THE POISSON PROCESS 69
Incoming Outgoing Mean holding time Local calls 500 calls/hr 480 calls/hr 2.5 min
Long-distance calls 30 calls/hr 10 calls/hr 1 min
Calculate the following:
(i) the total traffi c offered to the PABX.
(ii) the overall mean holding time of the incoming traffi c.
(iii) the overall mean holding time of the outgoing traffi c.
5. Consider a car-inspection centre where cars arrive at a rate of 1 every 30 seconds and wait for an average of 5 minutes (inclusive of inspection time) to receive their inspections. After the inspection, 20% of the car owners stay back and spend an average of 10 minutes in the centre’s cafeteria. What is the average number of cars within the premise of the inspection centre (inclusive of the cafeteria)?
6. If packets arrive at a switching node according to a Poisson process with rate l, show that the time interval X taken by the node to receive k packets is an Erlang-k random variable with parameters n and l.
7. Jobs arrive at a single processor system according to a Poisson process with an average rate of 10 jobs per second. What is the probability that no jobs arrive in a 1-second period? What is the probability that 5 or fewer jobs arrive in a 1-second period? By letting t be an arbitrary point in time and T the elapsed time until the fi fth job arrives after t, fi nd the expected value and variance of T?
8. If X1, X2, . . . , Xn are independent exponential random variables with parameters l1, l2, . . . , ln respectively, show that the random vari- able Y = min{X1, X2, . . . Xn} has an exponential distribution with parameter l = l1+ l2+ . . . + ln.