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M/M/C/K/N systems: part II
Lecturer: Dmitri A. Moltchanov
E-mail: dmitri.moltchanov@tut.fi
http://www.cs.tut.fi/kurssit/ELT-53606/
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
OUTLINE:
• M/M/1 queuing system with state-dependent arrivals;
• M/M/m queuing system;
• M/M/m/m queuing system;
• M/M/1/K queuing system;
• M/M/1/∞/K queuing system;
• M/M/∞/-/K queuing system.
• Service time variation in M/-/C/K/N systems.
Lecture: M/M/C/K/N systems: part II 2
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
1. M/M/1 with state dependent arrivalsWhat we assume here:
• rate of the arrival is slowing down when the state goes up;
• where it may occur: some feedback mechanism controlling arrivals.
Parameters are given as follows:
λk =λ
k + 1, k = 0, 1, . . . ,
µk = µ, k = 1, 2, . . . ,
µ0 = 0. (1)
0 1 2 i... ...
2ll
3l
il
1il
+
m m m m m
Figure 1: Birth-death process of M/M/1 queuing system with state dependent arrivals.
Lecture: M/M/C/K/N systems: part II 3
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
1.1. Interesting notes
What is interesting about this queue:
• PASTA property does not hold for such queuing system:
– arrival process is not homogenous Poisson!
• the mean arrival rate to the system is given by follows:
E[λ] =∞∑k=0
λkpk =∞∑k=0
λ
k + 1pk. (2)
– depends on the state of the system;
– we do not know mean arrival rate in advance;
– Little’s result can still be applied with proper E[λ]!
What else we can define:
• state-dependent service times:
– intensity of the service gets smaller when more customers in the system.
Lecture: M/M/C/K/N systems: part II 4
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
1.2. Steady-state distribution
Existence of steady-state distribution (capacity is infinite!):
• E[λ]/µ is always limited irrespective of initial λ
E[λ]
µ<∞. (3)
– even when λ >>> µ!
Solution for steady-state probabilities:
pk = p0
k−1∏i=0
λiµi+1
= p0
k−1∏i=0
λ
(i+ 1)
1
µ= p0
(λ
µ
)k1
k!, k = 1, 2, . . . , (4)
• where we can find p0 from normalizing condition:
p0 = 1−∞∑i=1
pk = 1−∞∑i=1
p0
(λ
µ
)k1
k!= e−(λ/µ). (5)
• λ is the arrival rate that decreases as the state number increases.
Lecture: M/M/C/K/N systems: part II 5
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
1.3. Mean number of customers
Mean number of customers in the system: E[N ]
E[N ] =∞∑k=0
kpk =λ
µ. (6)
Arrival rate to the system:
E[λ] =∞∑k=0
λkpk = µ(1− e−λ/µ). (7)
Mean time spent by customer in the system: E[W ]
• apply Little’s result to get:
E[W ] =E[N ]
E[λ]=
λ
µ2(1− e−λµ ). (8)
Lecture: M/M/C/K/N systems: part II 6
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
2. M/M/m queuing systemNote the following:
• the system has m servers each of which requires 1/µ time to serve customer;
• the birth-death process has the following parameters:
λk = λ, k = 0, 1, . . . ,
µk = kµ, k = 1, 2, . . . ,m,
µk = mµ, k = m+ 1, . . . . (9)
Note: steady-state distribution exists when ρ = λ/(mµ) < 1.
0 1 m-1 m...
1ì 2ì 1)ì-(m mì 1)ì+(m
ë
...
ë ë ë ë
n-1 n ...
ë ë ë
mì mì mì
Figure 2: Birth-death process associated with M/M/m queuing system.
Lecture: M/M/C/K/N systems: part II 7
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
2.1. Steady-state distribution
The equilibrium state distribution is given by product form solution:
pk = p0ρk
k!, k = 1, 2, . . . ,m,
pk = p0ρk
m!mk−m , k = m+ 1, . . . , (10)
• where p0 can be derived from normalizing condition:
p0 =
(m−1∑k=0
ρk
k!+
mρm
m!(m− ρ)
)−1. (11)
Note the following:
• M/M/m queues was (is) extensively used in teletraffic theory;
• example: waiting time to get the free line.
Lecture: M/M/C/K/N systems: part II 8
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
2.2. Performance measures
The most important measure:
• probability that arriving customer will wait which is given by:
C(m, ρ) =∞∑k=m
pk =
mρm
m!(m−ρ)∑m−1k=0
ρk
k!+ mρm
m!(m−ρ)
. (12)
– this formula is extensively tabulated in literature.
Mean number of customers in the buffer: E[NQ]
• we obtain it directly from steady-state distribution:
E[NQ] =∞∑k=m
kpk =pm
1− ρ
∞∑k=0
k(1− ρ)ρn = C(m, ρ)ρ
1− ρ. (13)
Mean waiting time in the buffer: E[WQ]
• using Little’s result we get:
E[WQ] =E[NQ]
λ= C(m, ρ)
ρ
(1− ρ)λ. (14)
Lecture: M/M/C/K/N systems: part II 9
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
3. M/M/m/m queuing systemNote the following:
• there are no waiting positions in this queue;
• capacity of the system is limited and equal to the number of servers.
Parameters of birth-death process are as follows:
λk = λ, k = 0, 1, . . . ,m− 1
λk = 0, k ≥ m− 1
µk = kµ, k = 1, 2, . . . ,m,
µk = 0, k > m. (15)
0 1 m-1 m...
1ì 2ì 1)ì-(m mì
ë ë ë ë
Figure 3: Birth-death process associated with M/M/m/m queuing system.
Lecture: M/M/C/K/N systems: part II 10
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
3.1. Steady-state distribution
Note the following:
• customers enters the queue if at least one position is free!
• otherwise, the customer is immediately rejected: loss system!
Condition of existence of steady-state distribution:
• state-space is limited;
• system is always stable irrespective of λ/mµ!
Defining ρ = λ/µ, the steady-state distribution is given by:
pk = p0ρk
k!, k = 1, 2, . . . ,m,
pk = 0, k = m+ 1, . . . , (16)
• where from the normalizing conditions we get p0:
p0 =
(m∑k=0
ρk
k!
)−1. (17)
Lecture: M/M/C/K/N systems: part II 11
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
3.2. Performance measures
Application of the M/M/m/m queue:
• call service process between telephone exchanges.
Exchange
...
cu
sto
me
rs
Exchange...
m links
Figure 4: Using M/M/m/m as a model in telephone network.
Most important parameter: probability of blocking:
• probability that an arrival finds all server busy forcing customer to leave without service.
Note: blocking is often used to describe the situation when the system is full:
• we say this system is with blocking;
• we say customer is blocked, etc.
Lecture: M/M/C/K/N systems: part II 12
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
To derive probability of blocking:
• arrival process is Poisson: PASTA property holds;
• probability that arrival sees all servers busy = fraction of time all servers busy:
B(m, ρ) =ρm
m!∑m
k=0ρk
k!
, (18)
– which is known as Erlang-C formula.
Using the following recursion it is easy to estimate probabilities B(m, ρ):
B(0, ρ) = 1, B(m, ρ) =ρB(m−1,ρ)
m
1 + ρB(m−1,ρ)m
. (19)
Other performance parameters:
• can be obtained using Little’s result (first, you have to get E[N ]);
• note that the arrival rate at which customers enter the queue is:
λA = λ(1−B(m, ρ)). (20)
Lecture: M/M/C/K/N systems: part II 13
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
4. M/M/1/K queuing systemNote the following:
• frequently used in performance evaluation of packet networks;
• birth-death process associated with M/M/1/K has the following parameters:
λk = λ, k = 0, 1, . . . , K − 1
λk = 0, k ≥ K
µk = µ, k = 1, 2, . . . , K,
µk = 0, k > k. (21)
0 1 K-1 K...
ì ì ì ì
Kë
1-Kë
2ë
1ë
Figure 5: Birth-death process associated with M/M/1/K queuing system.
Lecture: M/M/C/K/N systems: part II 14
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
4.1. Steady-state distribution
Existence of steady-state distribution:
• state space of the system is limited: {0, 1, .., K};
• steady-state distribution exist for all λ and µ irrespective of λ/µ!
Steady-state distribution is given by:
pk = p0ρk, k = 1, 2, . . . , K, (22)
pk = 0, k = K + 1, . . . , (23)
• where p0 can be found from the normalizing condition:
p0 =(1− ρ)
(1− ρK+1)(24)
Note: from equilibrium state distribution all mean parameters can be found.
Lecture: M/M/C/K/N systems: part II 15
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
5. M/M/1/∞/K queuing systemNote the following:
• the first case when we consider queuing system with finite population;
• parameters of birth-death process are as follows:
λk = λ(K − k), k = 0, 1, . . . , K − 1
λk = 0, k ≥ K
µk = µ, k = 1, 2, . . . , K,
µk = 0, k > k. (25)
0 1 K-1 K...
ì ì ì ì
Kë
1-Kë
2ë
1ë
Figure 6: Birth-death process associated with M/M/1/∞/K queuing system.
Lecture: M/M/C/K/N systems: part II 16
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
5.1. Steady-state distribution
Existence of steady-state distribution:
• state space of the system is limited: {0, 1, .., K};
• arrival rate decreases when state increases;
• steady-state distribution exist for all λ and µ irrespective of λ/µ!
Defining ρ = λ/µ the steady-state distribution is:
pk = p0ρk K!
(K − k)!, k = 1, 2, . . . , K,
pk = 0, k > K, (26)
• where p0 can be found from normalizing condition:
p0 =
(K∑k=0
ρkK!
(K − k)!
)−1(27)
Note: performance parameters can be found immediately.
Lecture: M/M/C/K/N systems: part II 17
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
6. M/M/∞/-/KNote the following: number of servers need not be more K:
• limited population of customers.
Parameters of birth-death process are given:
λk = λ(K − k), k = 0, 1, . . . , K − 1
λk = 0, k ≥ K
µk = kµ, k = 1, 2, . . . , K,
µk = 0, k > k. (28)
0 1 K-1 K...
1ì 2ì 1)ì-(K Kì
Kë
1-Kë
2ë
1ë
Figure 7: Birth-death process associated with M/M/∞/-/K queuing system.
Lecture: M/M/C/K/N systems: part II 18
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
6.1. Steady-state distribution
Existence of steady-state distribution:
• state space of the system is limited: {0, 1, .., K};
• steady-state distribution exist for all λ and µ irrespective of λ/µ!
Defining ρ = λ/µ, steady-state distribution is given by:
pk = p0ρk K!
k!(K − k)!, k = 1, 2, . . . , K,
pk = 0, k > K, (29)
• where p0 can be found from normalizing condition:
p0 =
(K∑k=0
ρkK!
k!(K − k)!
)−1(30)
The mean arrival rate to the system is:
E[λ] =K∑k=0
λkpk = Kλ1
1 + ρ(31)
Lecture: M/M/C/K/N systems: part II 19
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
6.2. Performance measures
Mean number of customers in the system: E[N ]
E[N ] =Kρ
1 + ρ. (32)
Mean number of customers in the buffer: E[NQ]
E[NQ] = 0. (33)
• since the number of servers is infinite.
Mean time spent by customer in the system: E[W ]
E[W ] =E[N ]
E[λ]=
1
µ. (34)
• since any arriving customer immediately enters the server.
Mean time spent by customer in the buffer – E[Wb]
E[Wb] = 0. (35)
• since the number of servers is infinite.
Lecture: M/M/C/K/N systems: part II 20
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7. Service time variation in M/-/C/K/N systemWhy service time variations:
• exponential distribution gives poor approximation;
• which other distribution we can use:
– Erlang distribution;
– hyperexponential distribution;
– Cox distribution;
– phase type distribution.
How we can analyze these systems:
• analyzing M/G/- queuing system;
– complete lack of memoryless property in service process.
• analyzing M/PH/- queuing systems and its special cases:
– we may still benefit from memoryless property of components.
Lecture: M/M/C/K/N systems: part II 21
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
7.1. Method of stages for M/Er/1
When this method is useful:
• service time is not exponentially distributed;
• service time is a combination of exponentials.
Consider the example of queuing system:
• Kendall’s notation: M/Er/1;
• Poisson arrivals, single server, infinite waiting room, Erlang service times.
Server: Erlang
Buffer
l 1m
Figure 8: Illustration of the queue of M/Er/1 type.
Lecture: M/M/C/K/N systems: part II 22
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
How we can represent the service in this system:
• customer first starts getting service at stage 1:
– it is served for exponentially distributed time with mean 1/2µ.
• after completion it enters the stage 2:
– it is served for exponentially distributed time with mean 1/2µ.
• the service time is the sum of two exponentials:
– the result: Erlang distribution with mean 1/µ.
ServerBuffer
l 12m
12m
Figure 9: Illustration of server with two stages.
Note: new customer enters the service after full service completion.
Lecture: M/M/C/K/N systems: part II 23
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
7.2. State of the system
State of the system:
• pair (n, j):
– n: total number of customers in the system;
– j: stage at which the current customer is served.
Do the following:
• SS = {0, 1, . . . }: number of customers in the system;
• SB = {1, 2}: phase of the service process;
• state space of the system is given by Cartesian product:
SS × SB = {0, 1, . . . } × {1, 2}, (36)
Lecture: M/M/C/K/N systems: part II 24
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
7.3. Example: generalized Erlang service times
Assume: µ1 6= µ2: generalized Erlang distribution!
(0,0) (1,1)
(1,2)
(2,1)
(2,2) (3,2)
(3,1)l l l l
lll
1m
1m 1
m2
m2
m2
m ...
Figure 10: Transition diagram of M/Er/1 system with state description given by (n, j).
Write balance equations for this diagram as:
λp00 = p12µ2,
p11(λ+ µ1) = p00λ+ p22µ2,
p12(λ+ µ2) = p11µ1,
p21(λ+ µ1) = p11λ+ p32µ2,
. . . (37)
Lecture: M/M/C/K/N systems: part II 25
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
Rewrite it as follows:
p12 =λ
µ2
p00,
p11 =λ(λ+ µ2)
µ1µ2
p00,
. . . (38)
Analyze as follows:
• express all pii, i = 1, 2, . . . as a function of p00;
• use normalizing condition to find p00.
This approach can be extended to:
• r stages of service time;
• finite number of waiting positions in the buffer;
• more general service time: hyperexponential, Cox, phase-type.
Lecture: M/M/C/K/N systems: part II 26
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
7.4. Alternative state description
What is bad about (n, j) description:
• (n, j) description is two-dimensional;
• when n or j are large it is getting complicated!
Another state description:
• number of uncompleted phases of work in the system:
– each waiting customer adds r phases of work;
– served customer adds j ≤ r phases of work:
• there is one-to-one correspondence between (n, j) and uncompleted phases of work:
(n, j) = (n− 1)r + j, (39)
– j is the number of phases left for customer currently being served;
– r is the total number of phases in service time;
– n is the number of customers in the system.
Lecture: M/M/C/K/N systems: part II 27
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
7.5. Example: Erlang service times
Assume: µ1 = µ2 = µ: Erlang distribution.
0 1 r r+1...
ì ì ì ì
ë
...
ë
Figure 11: Transition diagram of M/Er/1 system with a single state descriptor.
Note: any arrival adds r phases of work to the system.
Denote: pn be the steady-state probability that n phases of work are in the system:
• use global balance principle to get:
p0λ = p1µ,
pn(λ+ µ) = pn+1µ, n = 1, 2, . . . , r − 1,
pn(λ+ µ) = pn−rλ+ pn+1µ, n = r, r + 1, r + 2, . . . . (40)
Lecture: M/M/C/K/N systems: part II 28
Network analysis and dimensioning I D.Moltchanov, TUT, 2013
Do not forget normalizing condition:
∞∑n=0
pn = 1, (41)
How to get the solution:
• one can solve equations to get uncompleted phases of work;
• we do not directly get any interesting quantity;
• after obtaining steady-state distribution we have to find distribution of states.
Lecture: M/M/C/K/N systems: part II 29