Single-Queue Markovian Systems
4.8 CONSIDERATIONS FOR APPLICATIONS OF QUEUEING MODELS
But we know that
N=λT= ×λ µ 1
therefore
N C P N mP
N C C m P
b b
b
= − − −
= + −
+ −
1 1
1 1
µ λ λ
ρ ρ
ρ ρ
[ ( ) ( )]
( ) (4.77)
where
ρ λ=µ
If the population size C of the arriving customers is the same as the number of servers in the system, i.e. C = m, then
N= m
+ ρ ρ
1 (4.78)
(iii) The system time
T = 1
µ (4.79)
4.8 CONSIDERATIONS FOR APPLICATIONS OF
process of some other real-life problems, but so long as the discrepancy is small, it can be treated as the fi rst-cut approximation.
The exponential service time assumption appears to be less ideal than the Poisson assumption but still offers fairly good results as far as voice networks are concerned. However, it may be inadequate in modelling packets or mes- sages in data networks as the length of a packet is usually constrained by the physical implementation. The length could even be a constant, as in the case of ATM (Asynchronous Transfer Mode) networks. Nevertheless, the exponen- tial distribution can be deemed as the worst-case scenario and will give us the fi rst-cut estimation.
The M/M/1 queue and its variants can be used to study the performance measure of a switch with input buffer in a network. In fact, its multi-server counterparts have traditionally been employed in capacity planning of tele- phone networks. Since A K Erlang developed them in 1917, the M/M/m and M/M/m/m models have been extensively used to analyse the ‘lost-calls-cleared’
(or simply blocked-calls) and ‘lost-calls-delayed’ (queued-calls) telephone systems, respectively. A ‘queued-calls’ telephone system is one which puts any arriving call requests on hold when all the telephone trunks are engaged, whereas the ‘blocked-calls’ system rejects those arriving calls.
In a ‘blocked-calls’ voice network, the main performance criterion is to determine the probability of blocking given an offered load or the number of trunks (circuits) needed to provide certain level of blocking. The performance of a ‘queued-calls’ voice network is characterized by the Erlang C formula and its associated expressions.
Example 4.6
A trading company is installing a new 300-line PBX to replace its old exist- ing over-crowded one. The new PBX will have a group of two-way external circuits and the outgoing and incoming calls will be split equally among them.
It has been observed from past experience that each internal telephone usually generated (call or receive) 20 minutes of voice traffi c during a typical busy day. How many external circuits are required to ensure a blocking prob- ability of 0.02?
Solution
In order for the new PBX to handle the peak load during a typical busy hour, we assume that the busy hour traffi c level constitutes about 14% of a busy day’s traffi c.
Hence the total traffi c presented to the PBX:
CONSIDERATIONS FOR APPLICATIONS OF QUEUEING MODELS 135
= 300 × 20 × 14% ÷ 60
= 14erlangs
The calculated traffi c load does not account for the fact that trunks are tied up during call setups and uncompleted calls. Let us assume that these amount to an overhead factor of 10%.
Then the adjusted traffi c = 14 × (1 + 10%) = 15.4erlangs Using the Erlang B formula:
P m
k
m
m
k m
k
= ≤
∑
=( . ) / ! ( . ) / ! 15 4 .
15 4
0 02
0
Again, we solve it by trying various numbers for m and we have
m P
m P
m m
= =
= =
22 0 0254
23 0 0164
. .
Therefore, a total of 23 lines is needed to have a blocking probability of less than or equal to 0.02.
Example 4.7
At the telephone hot line of a travel agency, information enquiries arrive according to a Poisson process and are served by 3 tour coordinators. These tour coordinators take an average of about 6 minutes to answer an enquiry from each potential customer.
From past experience, 9 calls are likely to be received in 1 hour in a typical day. The duration of these enquiries is approximately exponential. How long will a customer be expected to wait before talking to a tour coordinator, assum- ing that customers will hold on to their calls when all coordinator are busy?
On the average, how many customers have to wait for these coordinators?
Solution
The situation can be modelled as an M/M/3 queue with
ρ λ
= µ =
× =
m
9 60 3 /1 6 0 3
( / ) .
P m k
m m k
k
m k m
k k
= +
−
= +
=
− −
=
∑
∑
0
1 1
0 2
1 1 0 9
( )
!
( )
! ( . )
! (
ρ ρ
ρ 0
0 9 3
1 1 0 3 0 4035
1 1 0
3 1
0
. )
! .
. ( )
! .
−
=
= −
=
−
P P m
d m ρ m
ρ 4
4035 0 9 3
1 1 0 3 0 07
1 0 03
0 03 9 60 0 2
3
× ×
−
=
= − =
= = =
( . )
! .
.
. .
/ .
N P
W N
q d
q
ρ ρ
λ mminute
Example 4.8
A multi-national petroleum company leases a certain satellite bandwidth to implement its mobile phone network with the company. Under this implemen- tation, the available satellite bandwidth is divided into Nv voice channels operating at 1200 bps and Nd data channels operating at 2400 bps. It was fore- cast that the mobile stations would collectively generate a Poisson voice stream with mean 200 voice calls per second, and a Poisson data stream with mean 40 messages per second. These voice calls and data messages are approxi- mately exponentially distributed with mean lengths of 54 bits and 240 bits, respectively. Voice calls are transmitted instantaneously when generated and blocked when channels are not available, whereas data messages are held in a large buffer when channels are not available:
(i) Find Nv, such that the blocking probability is less than 0.02.
(ii) Find Nd, such that the mean message delay is less than 0.115 second.
Solution
(i) lv= 200 and m v−1= 54/1200 = 9/200 (lv/mv) = 9
CONSIDERATIONS FOR APPLICATIONS OF QUEUEING MODELS 137
P m k
m v
m
k m
v k
=
∑
=( ) / !
( ) / ! λ µ
λ µ /
/
v
v 0
( ) / ! ( ) / ! 9 .
9
0 02
0 N
v
k N
k
v
v
N k
∑
= ≤ therefore Nv≥ 15 (ii) ld= 40 and m d−1= 240/2400 = 0.1(ld/md) = 4
T P
m
d
d
d d
= +
− 1
µ µ λ
1
10+10 0 115 6
− ≤ ≥
P
N d N
d d
λ . and d
Example 4.9
A group of 10 video display units (VDUs) for transactions processing gain access to 3 computers ports via a data switch (port selector), as shown in Figure 4.14. The data switch merely performs connections between those VDUs and the computer ports.
If the transaction generated by each VDU can be deemed as a Poisson stream with rates of 6 transactions per hour, the length of each transaction is approximately exponentially distributed with a mean of 5 minutes.
Calculate:
(i) The probability that all three computer ports are engaged when a VDU initiates a connection;
(ii) The average number of computer ports engaged;
It is assumed that a transaction initiated on a VDU is lost and will try again only after an exponential time of 10 minutes if it can secure a connection initially.
switch CPU
VDU VDU
Figure 4.14 A VDU-computer set up
Solution
In this example, computer ports are servers and transactions are customers and the problem can be formulated as an M/M/m/m system with fi nite arriving customers.
Given l = 6/60 = 0.1 trans/min m−1= 5 min therefore a = 0.5 (i)
P
k
b
k
k
=
=
∑
=10 3 0 5
10 0 5
0 4651
3
0 3
( . ) ( . )
.
(ii) N=
+ × −
+ − ×
= 0 5
1 0 5 10 0 5
1 0 510 3 0 4651 2 248
. .
.
. ( ) .
.
Problems
1. By referring to Section 4.1, show that the variance of N and W for an M/M/1 queue are
(i) Var [N] = r/(1 − r)2 (ii) Var [W] = [m(1 − r)]−2
2. Show that the average number of customers in the M/M/1/S model is N/2 when l = m.
3. Consider an M/M/S/S queueing system where customers arrive from a fi xed population base of S. This system can be modelled as a Markov chain with the following parameters:
λ λ
µ µ
k k
s k k
= −
=
( )
Draw the state-transition diagram and show that the probability of an empty system is (1 + r)−S. Hence fi nd the expected number of cus- tomers in the system.
4. Data packets arrive at a switching node, which has only a single output channel, according to a Poisson process with rate l. To handle congestion issues, the switching node implements a simple strategy of dropping incoming packets with a probability p when the total number of packets in the switch (including the one under transmission) is greater or more than N. Assuming that the CONSIDERATIONS FOR APPLICATIONS OF QUEUEING MODELS 139
transmission rate of that output channel is m, fi nd the probability that an arriving packet is dropped by the switch.
5. A trading company intends to install a small PBX to handle ever- increasing internal as well as external calls within the company. It is expected that the employees will collectively generate Poisson out- going external calls with a rate of 30 calls per minute. The duration of these outgoing calls is independent and exponentially distributed with a mean of 3 minutes. Assuming that the PBX has separate external lines to handle the incoming external calls, how many exter- nal outgoing lines are required to ensure that the blocking probability is less than 0.01? You may assume that when an employee receives the busy tone he/she will not make an attempt again.
6. Under what conditions is the assumption ‘Poisson arrival process and exponential service times’ a suitable model for the traffi c offered to a communications link? Show that when ‘Poisson arrival process and exponential service times’ traffi c is offered to a multi-channel communications link with a very large number of channels, the equi- librium carried traffi c distribution (state probability distribution) is Poisson. What is the condition for this system to be stable?
7. Consider Example 4.1 again. If a second doctor is employed to serve the patients, fi nd the average number of patients in the clinic, assum- ing that the arrival and service rates remained the same as before?
8. A trading fi rm has a PABX with two long-distance outgoing trunks.
Long-distance calls generated by the employees can be approxi- mated as a Poisson process with rate l. If a call arrives when both trunks are engaged, it will be placed on ‘hold’ until one of the trunks is available. Assume that long-distance calls are exponentially dis- tributed with rate m and the PABX has a large enough capacity to place many calls on hold:
(i) Show that the steady-state probability distribution is 2 1 1 ( − )
+ ρ ρ ρk (ii) Find the average number of calls in the system.
9. By considering the Engset’s loss system, if there is only one server instead of m servers, derive the probability of having k customers in the system and hence the average number of customers, assuming equilibrium exists.