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M/G/1 and M/G/1/K systems: another look
Dmitri A. Moltchanov
http://www.cs.tut.fi/kurssit/TLT-2716/
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
OUTLINE:
• Description of M/G/1 system;
• Methods of analysis;
• Choosing state of the system;
• Method of supplementary variables;
• Imbedded Markov chain approach;
• Description of M/G/1/K queuing system;
• Imbedded Markov chain for M/G/1/K system.
Lecture: M/G/1 and M/G/1/K systems: part II 2
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
1. Description of M/G/1 queuing systemM/G/1 queuing system stands for:
• single server;
• infinite number of waiting positions;
• Poisson arrival process;
– interarrival times are exponentially distributed.
• generally distributed service times:
– practically, any distribution.
What we also assume:
• ’first come, first served’ (FCFS);
• what it may affect: waiting time!
Lecture: M/G/1 and M/G/1/K systems: part II 3
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
1.1. Arrival and service processes
Arrival process:
• Poisson with parameter (mean value) λ
• interarrival times are exponential with mean 1/λ;
• PDF, pdf and LT are:
A(t) = 1− e−λt, a(t) = λe−λt, A(s) =λ
s+ λ. (1)
Service process:
• service times are exponential with mean 1/µ;
• PDF, pdf and LT are:
B(t), b(t), B(s). (2)
Lecture: M/G/1 and M/G/1/K systems: part II 4
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
2. Methods of analysisThere are a number of methods. We consider four most popular methods:
• residual life approach: simple to understand:
– only mean values can be obtained.
• transform approach based on imbedded Markov chain: harder to deal with;
– distributions of desired performance parameters can be obtained;
– the idea: find points at which Markov property holds and use transforms.
• direct approach based on imbedded Markov chain: harder to deal with;
– distributions of desired performance parameters can be obtained;
– the idea: find points at which Markov property holds and use convolution.
• method of supplementary variables: the most complicated one:
– distributions of desired performance parameters can be obtained;
– the idea: look at arbitrary points and make them Markovian.
Lecture: M/G/1 and M/G/1/K systems: part II 5
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
3. Choosing state of the systemLet N(t) be the number of customers at time t:
• do we know how the system evolves in time after t in ∆t?
• arrival may occur with rate λ∆t:
– it is independent from previous arrival: memoryless property (A1 = A2)!
• departure may occur with probability µ∆t:
– D2 is not the same as D1!
• this process is no longer Markovian!
1 1 2( ) : ( ) ( )D t D t D t¹
t
A2(t)
D2(t)
t
A1(t)
Figure 1: State of M/G/1 queuing system.
Lecture: M/G/1 and M/G/1/K systems: part II 6
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
3.1. Transform method based on imbedded Markov chain
State of the system: M/G/1 queuing system
• number of customers in the system at special time instants d+1 , d+2 , . . . :
– choose these instants such that we should not track time since previous service started;
– which instants: just after service completion of customers (D(t+ x|x) = 0)!
t
d1
+d2
+
( ) ( ) ( 0) (0)( | ) 0
1 ( ) 1 (0)
D t x D x D t DD t x x
D x D
+ - + -+ = = =
- -
0x ®
( ) 1 tA t e
l-= -
( )D t
( ) , 0,1,...Q
S t k k= =
Figure 2: State of M/G/1 queuing system.
Lecture: M/G/1 and M/G/1/K systems: part II 7
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
3.2. Alternative state description
State of the system: M/G/1 queuing system
• number of customers in the system and time since previous service started (N(t), D2(t)):
– in this case we know how the system evolves in time;
– we know the distribution of time till the next arrival: it is the same as initial one;
– we know the distribution of time till the next departure: we track it:
D(t+ x|x) = Pr{(T ≤ t+ x)|T > x} = (D(t+ x)−D(x))/(1−D(x)). (3)
t
t
( ) 1 tA t e
l-= -
( ) ( )( | )
1 ( )
D t x D xD t x x
D x
+ -+ =
-( ) , 0,1,...
QS t k k= =
Figure 3: State of M/G/1 queuing system given by (N(t), D2(t)).
Lecture: M/G/1 and M/G/1/K systems: part II 8
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
4. Method of supplementary variablesNotes about the method:
• proposed by Kleinrock and Takagi in the middle of 70th;
• very powerful:
– can be used for a number of variations of M/G/1 queuing systems;
– can be used for G/M/1 and its variations too.
• allows to better understand operation of the M/G/1 system;
• one of the most complicated among available for M/G/1 system.
What is the approach:
• track the number of customers and service time that already achieved.
• this process is Markovian: treat it using some approach.
Lecture: M/G/1 and M/G/1/K systems: part II 9
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Do the following:
• consider M/G/1 queuing system at arbitrary time t;
• N(t) be the number of customers:
– evolution of N(t) is no longer Markovian;
– that is why we had to imbed Markov chain at departures.
Let us assume the following:
• X0(t), t ≥ 0 is the time already received by the customer which is in service;
• process {N(t), X0(t), t ≥ 0} is Markovian:
– X0(t) is the supplementary variable that helps in analysis.
Note that by definition for {N(t), X0(t), t ≥ 0} process we assume:
X0(t) = 0, N(t) = 0. (4)
Lecture: M/G/1 and M/G/1/K systems: part II 10
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
t
...X0(t) = 0
t
...
observation point: tiX0(t)
observation point: ti
Figure 4: Service time X0 already received by the customer: X0 6= 0 and X0 = 0.
Lecture: M/G/1 and M/G/1/K systems: part II 11
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Define the steady-state probability that there are k customers in the system:
pk = limt→∞
Pr{N(t) = k}, k = 0, 1, . . . . (5)
Note the following:
• state is now given a couple at time t: {N(t), X0(t), t ≥ 0};
• we have to determine jpdf:
fk(t, x)dx = Pr{N(t) = k, x < X0(t) < x+ dx}. (6)
Considering this fk(t, x) under equilibrium conditions (t→∞):
fk(x) = limt→∞
Pr{N(t) = k, x < X0(t) ≤ x+ dx} = limt→∞
fk(t, x)dx,
f0(x) = 0. (7)
• the latter one is by definition:
– time already received by customer is zero when there are no customers in the system;
– recall we required that the server does not stay idle when there are customers in the system.
Lecture: M/G/1 and M/G/1/K systems: part II 12
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Consider a customer that requires a service time of duration X:
• it has pdf b(x) and PDF B(x);
• bc(x) is the pdf of service time X given that the certain age has already provided (X > x):
bc(x)dx = Pr{x < X < x+ dx |X > x} =b(x)
1−B(x). (8)
– this is a death rate of service time at the age X.
departuredepartureservice time
t
...
X0(t)
Pr{x<X<=x+dx|X>x}
Figure 5: Illustration of the death rate bc(x)dx = Pr{x < X ≤ X + dx|X > x}.
Lecture: M/G/1 and M/G/1/K systems: part II 13
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Consider steady-state operation:
• balance equations for state 0: flow out = flow in:
λp0 =
∫ ∞0
f1(x)bc(x)dx. (9)
– f1(x): jpdf that the state is 1 and service X, X ≥ 0 already provided;
– bc(x): death rate given that age X, X ≥ 0 already provided.
0 1 2 ...
ò¥
01 )()( dxxbxf c
0pl
Figure 6: Applying global balance principle for state 0.
Lecture: M/G/1 and M/G/1/K systems: part II 14
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Consider time instants t and (t+ ∆x):
• at t: {N(t) > 0, X0(t) = x};
• at (t+ ∆x): {N(t+ ∆x) = k,X0(t+ ∆x) = x+ ∆x}.
These event may occur in two ways:
• at t the state was {N(t) = k,X0(t) = x}:– there were no arrival in dx: (1− λ∆x);
– there were no service completion in dx: (1− bc(x)∆x).
• at t the state was {N(t) = k − 1, X0 = x}:– there was an arrival in dx: λ∆x;
– there were no service completion in dx: (1− bc(x)∆x).
• we assume that multiple events do not occur;
• combining these two we have:
fk(x+ ∆x)dx = (1− λ∆x)(1− bc(x))fk(x)dx+ λ∆x(1− bc(x)∆x)fk−1(x)dx. (10)
Lecture: M/G/1 and M/G/1/K systems: part II 15
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
t
...
X0(t)
xDlone arrival:
xxbc D- )(1no completion:
})(,)({: 0 xxxtXkxtNxt D+=D+=D+D+})(,1)({: 0 xtXktNt =-=
dxxfxxbx kc )())(1( 1-D-Dl
......
dxxfxxbx kc )())(1( 1-D-Dl
,k xD1,k x-
Figure 7: Transition from {N(t) = k − 1, X0 = x} to {N(t+ ∆x) = k,X0(t+ ∆x) = x+ ∆x}.
Lecture: M/G/1 and M/G/1/K systems: part II 16
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
t
...
X0(t)
})(,)({: 0 xxxtXkxtNxt D+=D+=D+D+
xD- l1
})(,)({: 0 xtXktNt ==
dxxfxxbx kc )())(1)(1( D-D- l
xxbc D- )(1no completion:
no arrival:
......,k xD,k x
dxxfxxbx kc )())(1)(1( D-D- l
Figure 8: Transition from {N(t) = k,X0 = x} to {N(t+ ∆x) = k,X0(t+ ∆x) = x+ ∆x}.
Lecture: M/G/1 and M/G/1/K systems: part II 17
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
t→∞: retain ∆x terms and drop those with higher power of ∆x to get:
fk(x+ ∆x) = λ∆xfk−1(x) + (1−∆x(λ+ bc(x)))fk(x), k = 1, 2, . . . (11)
Take limits when ∆x→ 0 we get:
dfk(x)
dx+ (λ+ bc(x))fk(x) = λfk−1(x), k = 1, 2, . . . (12)
To solve for the jpdf fk(x), k = 0, 1, . . . , x > 0, use boundary conditions:
f1(0) = λp0 +
∫ ∞0
f2(x)bc(x)dx k = 1
fk(0) =
∫ ∞0
fk+1(x)bc(x)dx, k = 2, 3, . . . . (13)
Normalization condition is given by:
∞∑k=0
pk = p0 +∞∑k=1
∫ ∞0
fk(x)dx = 1. (14)
Lecture: M/G/1 and M/G/1/K systems: part II 18
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
5. Direct approach based on imbedded Markov chainConsider again M/G/1 queuing system:
• consider imbedded Markov chain approach;
• Markov points are just after service completions.
ô
)(ôr
Figure 9: Imbedded Markov points at M/G/1 queuing system.
State of the system N(t): number of customers in the system just after departure.
Lecture: M/G/1 and M/G/1/K systems: part II 19
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Using the transform approach we proceeded as follows:
• obtain PGF of the number of customers just after departures;
• use Kleinrock’s result to get number of customers at arrival time instants;
• use PASTA property to get number of customers at arbitrary time instants;
In what follows, we will proceed as follows:
• obtain PF (not PGF) of the number of customers just after departures;
• use Kleinrock’s result to get number of customers at arrival time instants;
• use PASTA property to get number of customers at arbitrary time instants;
Let us define the following:
• pi, i = 0, 1, . . . : number of customers in the system at arbitrary time instants;
• qi, i = 0, 1, . . . : number of customers in the system just after departure;
• pi, i = 0, 1, . . . : number of customers in the system that arrival sees.
Lecture: M/G/1 and M/G/1/K systems: part II 20
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Let αk, k = 0, 1, . . . , be the probability of k arrivals in the service time.
arrivals
departuredepartureservice time
t
...ka
Figure 10: k arrivals in a service time.
Arrivals come from Poisson process αk, k = 0, 1, . . . , we can get αk:
αk =
∫ ∞0
(λx)k
k!e−λxb(x)dx, k = 0, 1, . . . . (15)
Lecture: M/G/1 and M/G/1/K systems: part II 21
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Consider imbedded Markov chain at steady-state t→∞:
• let qij, i, j = 0, 1, . . . , be transition probabilities of going from i to j;
• we can obtain qij, i, j = 0, 1, . . . as
qjk = αk, j = 0,
qjk = αk−j+1, j = 1, 2, . . . . (16)
0 1 2 ...
1a
2a
3
3a
0a 0a 0a 0a
2a
3a
2a2a
Figure 11: Transition probabilities of imbedded Markov chain at steady-state.
Lecture: M/G/1 and M/G/1/K systems: part II 22
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Steady-state probabilities are given by the solution of:
qk =∞∑j=0
qjqjk, k = 0, 1, . . . ,
∞∑k=0
qk = 1. (17)
In vector form it looks much better:
~qQ = ~q : (q0, q1, . . . )
q00 q01 q02 q03 . . .
q10 q11 q12 q13 . . .
0 q21 q22 q23 . . .
0 0 q32 q33 . . ....
......
......
= (q0, q1, . . . )
~q~e = 1 : (q0, q1, . . . )
1
1...
= 1 (18)
Lecture: M/G/1 and M/G/1/K systems: part II 23
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
5.1. Solution: transform approach
Write the previous set of linear equations as follows:
q0 = q0α0 + q1α0, k = 0,
q1 = q0α1 + q2α0 + q1α1, k = 1,
. . .
qk = q0αk + qk+1α0 +k∑j=1
qjαk−j+1, k = 2, 3, . . . . (19)
Multiply kth equation by zk and summing all LHSs and RHSs from k = 0 up to ∞:
Q(z) = q0A(z) + q1A(z) + q2zA(z) + q3z2A(z) + · · · = q0A(z) +
A(z)
z(Q(z)− q0), (20)
• Q(z) is the PGF of the number of customers in the system just after departure.
Rearranging terms gives us:
Q(z) =q0(1− z)A(z)
A(z)− z. (21)
Lecture: M/G/1 and M/G/1/K systems: part II 24
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Note the following:
• if the steady-state exists (ρ = λE[X]) we have:
q0 = 1− ρ, when ρ = λE[X]. (22)
– we obtained this result previously;
– we can obtain it also noting the property Q(1) = 1.
How to proceed further:
• use Klienrock’s principle to claim:
ak = dk, k = 0, 1, . . . . (23)
– where ak, k = 0, 1, . . . is steady-state distribution as seen by arrival.
• use PASTA property to claim:
ak = pk, k = 0, 1, . . . . (24)
Lecture: M/G/1 and M/G/1/K systems: part II 25
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
5.2. Solution: direct approach
Using q0 = 1− ρ we can recursively evaluate qk, k = 1, 2, . . . :
q1 =1
α0
q0(1− α0), k = 1,
qk =1
α0
(qk−1 −
k−1∑j=1
qjαk−j − q0αk−1
), k = 2, 3, . . . . (25)
• it gives the steady-state distribution just after departure.
How to proceed further:
• use Klienrock’s principle to claim:
ak = qk, k = 0, 1, . . . . (26)
• use PASTA property to claim:
ak = pk, k = 0, 1, . . . . (27)
Note: we will obtain pk, k = 0, 1, . . . directly.
Lecture: M/G/1 and M/G/1/K systems: part II 26
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Consider the mean time interval between successive imbedded time points:
D = q0
(1
λ+ E[X]
)+ (1− q0)E[X] = E[X] + q0
1
λ. (28)
Probability p0: system is empty at arbitrary time t:
• fraction of time the system is idle between successive imbedded points;
• therefore, we can write:
p0 =q0(1/λ)
E[X] + q0(1/λ)=
q0q0 + ρ
. (29)
Substitute q0 = (1− ρ) to get:
p0 =1− ρ
(1− ρ) + ρ= 1− ρ = q0. (30)
We can continue to get pk = qk, k = 1, 2, . . . .
Lecture: M/G/1 and M/G/1/K systems: part II 27
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
6. M/G/1/K queuing systemM/G/1/K queuing system is characterized by:
• arrivals come from the Poisson process with mean λ;
• generally distributed service times with PDF B(t), pdf b(t) and LT B(s);
• single server;
• limited capacity: limited number of waiting positions.
Note: M/G/1/K queuing system is a very powerful model for packet switching systems.
Departures
...
Arrivals
K-1 waiting positions
Server
Losses
Figure 12: Graphical representation of M/G/1/K queuing system.
Lecture: M/G/1 and M/G/1/K systems: part II 28
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Additional notes:
• K represents the maximum number of customers in the system where:
– one can be served;
– (K − 1) at maximum can wait in the buffer.
• M/G/1/K queue is the loss system:
– arrivals that arrive when K are in, just leave without service;
– these customers are often referred to as lost or blocked.
• let PB be the probability that the customer is lost:
– this is the same as probability that arrival sees K in the system;
– the following fraction of customers actually enters the system:
λA = λ(1− PB). (31)
Note the following:
• λA is the actual rate at which customers enter the system;
• λA must be used in Little’s result.
Lecture: M/G/1 and M/G/1/K systems: part II 29
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
6.1. Steady-state distribution
Consider Markov chain imbedded at moments of customer departures:
• at times instants ti, i = 0, 1, . . . there are ni, i = 0, 1, . . . , K − 1 customers;
• the state space of the system is then:
ni ∈ {0, 1, . . . , K − 1}, i ∈ {0, 1, . . . }, (32)
– departure cannot leave the full system!
To get probabilities of transitions we have to distinguish:
• ith customers leaves an empty system ni = 0;
• ith customers leaves a non-empty system ni 6= 0.
In each case we have to distinguish:
• overflow;
• no overflow.
Lecture: M/G/1 and M/G/1/K systems: part II 30
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
state of the system: ni+1 = K-1state of the system: ni = K-r > 0
service timet
...
K customers K-1 customers
1ira
+³
ith
departure (i+1)th
departure
state of the system: ni+1 = K - r - 1 + ai+1state of the system: ni = K-r > 0
service timet
...
K customers K-1 customers
1ira
+<
ith
departure (i+1)th
departure
ove
rflo
wn
oo
ve
rflo
w
Figure 13: The case ni > 0.
Lecture: M/G/1 and M/G/1/K systems: part II 31
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
ith
departure(i+1)
thservice time
t
...
K customers K-1 customers
11
iKa
+< -
state of the system: ni = 0 state of the system: ni+1 = ai+1
(i+1)th
departure
ith
departure(i+1)
thservice time
t
...
K customers K-1 customers
11
iKa
+³ -
state of the system: ni = 0 state of the system: ni+1 = K-1
(i+1)th
departure
no
ove
rflo
wove
rflo
w
Figure 14: The case ni = 0.
Lecture: M/G/1 and M/G/1/K systems: part II 32
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
We can write that:
• when ith customers leaves an empty system ni = 0:
ni+1 = min(αi+1, K − 1), ni = 0. (33)
• when ith customers leaves a non-empty system ni 6= 0:
ni+1 = min(ni − 1 + αi+1, K − 1), ni = 1, 2, . . . , K − 1. (34)
What we are doing next:
• let pd,k, k = 0, 1, . . . , K − 1 are steady-state probabilities as seen by departure;
• we can take transform approach;
• we are going to find pd,k, k = 0, 1, . . . , K − 1 directly.
To do it, we have to find transition probabilities of imbedded Markov chain:
pd,jk = Pr{ni+1 = k|ni = j}, j, k ∈ {0, 1, . . . , K − 1}. (35)
Lecture: M/G/1 and M/G/1/K systems: part II 33
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Do the following:
• αk, k = 0, 1, . . . : probability of exactly k arrivals during a service time;
• the number of arrivals during service time is given by Poisson process;
• we can find these probabilities as follows by integration:
αk =
∫ ∞0
(λt)k
k!e−λtb(t)dt. (36)
• using αk you can find transition probabilities:
– case nj = 0:
pd,0k =
{αk, k = 0, 1, . . . , K − 2,∑∞
m=K−1 αm, k = K − 1,, (37)
– case nj > 1, 2, . . . , K − 1:
pd,jk =
{αk−j+1, k = j − 1, j, . . . , K − 2,∑∞
m=K−j αm, k = K − 1,. (38)
Lecture: M/G/1 and M/G/1/K systems: part II 34
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Using pd,jk you can solve the following:
pd,k =K−1∑j=0
pd,jpd,jk, k = 0, 1, . . . , K − 1,
K−1∑k=0
pd,k = 1. (39)
• and you’ll find pd,k, k = 0, 1, . . . , K − 1!
Note the following:
• we have K unknowns and therefore K independent equations are needed;
• take K − 1 equations out of (39) and normalizing condition:
pd,k = pd,0αk +k+1∑j=1
pd,jαk−j+1, k = 0, 1, . . . , K − 2,
K−1∑k=0
pd,k = 1. (40)
Lecture: M/G/1 and M/G/1/K systems: part II 35
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Considering system at steady-state, let:
• pa,k, k = 0, 1, . . . , K: probability that arrival finds k = 0, 1, . . . , K customers in the system:
– it does not matter whether this arrival actually enters the system.
• pk, k = 0, 1, . . . , K: probability that system has k = 0, 1, . . . , K customers at arbitrary time;
• pac,k, k = 0, 1, . . . , K − 1: probability that arrival sees k = 0, 1, . . . , K − 1 customers:
– we assume that this arrival joins the system.
Note the following:
• Klienrock’s result states:
pd,k = pac,k, k = 0, 1, . . . , K − 1. (41)
• PASTA property states:
pk = pa,k, k = 0, 1, . . . , K. (42)
Question: how to find pk?
Lecture: M/G/1 and M/G/1/K systems: part II 36
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Let PB be probability that the arrival is blocked:
• this is the probability that arrival observes K in the system;
• therefore, PB = pK and we can write:
pk = pa,k = (1− PB)pac,k = (1− PB)pd,k, k = 0, 1, . . . , K − 1. (43)
Note also the following:
K∑k=0
pa,k = 1,K−1∑k=0
pac,k = 1. (44)
Since the previous holds we can get:
K−1∑k=0
pa,k = 1− PB =K−1∑k=0
(1− PB)pac,k, (45)
Note: to find pk, k = 0, 1, . . . , K have to find PB!
Lecture: M/G/1 and M/G/1/K systems: part II 37
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
Let us recall the following:
• E[X] be the mean service time of a customer;
• ρ = λE[X] be the offered traffic to the system.
The mean arrival rate of customers actually entering the system is:
λc = λ(1− PB), (46)
The throughput of the system is:
ρc = ρ(1− PB). (47)
These imply that the probability of finding empty system at arbitrary time is:
p0 = 1− ρc. (48)
Using the latter one and previous results we can write:
1− ρ(1− PB) = (1− PB)pd,0, (49)
• where pd,0 has been found earlier.
Lecture: M/G/1 and M/G/1/K systems: part II 38
Teletraffic theory I: queuing theory D.Molthchanov, TUT, 2011
From the last equation we can find probability of blocking:
PB = 1− 1
pd,0 + ρ. (50)
Using pd,k, k = 0, 1, . . . , K−1, and previous results (43, 50) we get pk, k = 0, 1, . . . , K−1:
pk =1
pd,0 + ρpd,k, k = 0, 1, . . . , K − 1, (51)
Now, for example, mean number in the system:
E[N ] =K∑k=0
kpk =1
pd,0 + ρ
K−1∑k=0
kpd,k +K
(1− 1
pd,0 + ρ
). (52)
The effective arrival rate to the queue is given by:
λc = λ(1− PB) =λ
pd,0 + ρ. (53)
• that should be used to obtain other performance parameters.
Lecture: M/G/1 and M/G/1/K systems: part II 39