How to cite this paper:Garg, D. (2014) Approximate Analysis of an M/M/1 Markovian Queue Using Unit Step Function. Open Access Library Journal, 1: e973. http://dx.doi.org/10.4236/oalib.1100973
Approximate Analysis of an M/M/1
Markovian Queue Using Unit Step Function
Dhanesh GargDepartment of Mathematics, Maharishi Markandeshwar University, Ambala, India Email: [email protected]
Received 7 August 2014; revised 27 September 2014; accepted 28 August 2014
Copyright © 2014 by author and OALib.
This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/
Abstract
This study analyzes a single server queueing model with a time-dependent arrival rate and service rate is constant. In this model, the incoming arrivals are Poisson stream; service time is exponen-tially distributed and the first-come first-served queueing discipline. We obtain an explicit ex-pression for the state probability distribution with time-dependent arrival rate using unit step function.
Keywords
Transient Analysis, Unit Step Function, Queueing System
Subject Areas: Applied Statistical Mathematics, Numerical Mathematics
1. Introduction
The vast majority of the queueing literature considers only models where the arrivals and service processes are time-homogeneous. Yet it is clear that in many situations it is quite natural to have the arrival rate varying with time. In the past few years a number of interesting literatures that discuss the transient behaviour of the M/M/1 queue, an aspect which is very important for practical applications have been focused by cf. Abate and Whitt
OALibJ | DOI:10.4236/oalib.1100973 2 October 2014 | Volume 1 | e973 step function. In this paper, the incoming arrivals are Poisson stream; service time is negative exponentially dis-tributed and the first-come first-served queueing discipline. For a special form of the traffic intensity, we obtain an explicit analytic expression for the probability distribution function with time-dependent arrival rate using unit step function. The rest of the paper is organized as follows: in Section 2, the mathematical description of the considered queueing model is derived; in Section 3, the transient solution to the model is derived.
2.
Model
Description
We consider a single server queue at which customers arrive to a Poisson stream of rate. The service time is ex-ponentially distributed time-independent rate. Let the system take initially non-empty with FCFS discipline i.e., “j” is in the system at t = 0. In view of the arbitrary rate functions, there is no loss of generality in assuming that we start the process at t = 0, by introduce a new function i.e. unit step function define as
( )
1 if1
0 if
t T
u t
t T
<
− = >
(1) This function if useful in expressing boundary conditions which is depends upon time. The basis of the ap-proximate differential equation derived from the time dependent arrival rate and time independent service rate. Let P tn
( )
= the probability that there are n customers in the system at time t given that there were j customers in the system at time 0. Therefore,Probability that [one arrival in the time interval
(
t t, + ∆t)
] = λ(
1 tT 1)
(
1 uT( )
t)
t 0( )
t ;−
+ − ∆ + ∆
Probability that [one departure in the time interval
(
t t, + ∆t)
] = µ∆ + ∆t 0( )
t . Transient state equations for the finite source M/M/1 queueing system are given by( )
(
)
(
( )
)
( )
(
)
(
( )
)
( )
( )
1
1
1 1
d
1 1
d
1 1 : 0
n T n
T n n
p t tT u t p t
t
tT u t p t p t n
λ µ
λ µ
−
−
− +
= − + − +
+ + − + >
(2)
( )
(
1)
(
( )
)
( )
( )
0 0 1
d
1 1 : 0
dt p t λ tT uT t µ p t µp t n
−
= − + − + + = (3)
3. The Model Solution
The solution of our problem of this queueing system of differential-difference equations under the initial condi-tions
(
0)
, where is the Kronecker delta 1 if 0 ifn jn jn
n j P t
n j
ψ ψ ψ =
= = =
≠
(4)
We define,
( )
( )
d d
dt pn t =µdπ pn π (5)
(
,)
Exp{
1(
,)
}
( )
n n
Q π T = ⋅ π +R π T p π (6)
(
,T)
λ(
1 tT 1)
(
1 uT( )
t)
ρ πµ
−
+ −
=
(7)
(
)
(
)
(
( )
)
1
0
1 1 d
, t
T
tT u t t
R T
λ π
µ
−
+ −
=
∫
(8)
OALibJ | DOI:10.4236/oalib.1100973 3 October 2014 | Volume 1 | e973
( )
1(
,)
( )
(
,)
1( )
1( )
n n n n
p′ π = − + ρ π T p π +ρ π T p− π +p+ π (9)
( )
(
) ( )
( )
0 , 0 1
p′ π = −ρ π T p π +p π (10) Rewritten and differentiating these equations, we have,
(
)
(
)
(
)
{
(
)
}
0 d
, , , Exp 1 ,
dπQ π T = −R π T +ρ π T ⋅ π +R π T (11)
(
)
(
)
{
(
(
)
(
)
)
}
( )
{
(
)
}
d
, , 1 ( , ) , , Exp 1 ,
dπQn π T pn π T Rπ T π R π T ρ π T pn π π R π T
′
= + + + − + ⋅ + (12)
The initial conditions
( )
0 : , 0,1, 2,
n t jn
Q t = =ψ j n=
exist and satisfy the system of linear equations. In order to reduce this differential-difference Equation (11) (12), we introduce the generating function:
(
)
(
) (
)
0 , , , ! n n n zQ z T Q T
n
π
π ∞ π
=
−
=
∑
(13)This function will be analytic in z and continuously differentiable in π. Differentiation with respect to z and π gives, using (11) (12). Restate the differential difference equations in term of d
(
,)
dπ Qn π T , taking summa- tions on both sides with limits n = 0 to n = ∞ as a partial differential equation-w.r.t. π, we get
(
)
(
) (
(
)
)
1(
)
(
)
0 0
, , , ,
1 ! !
n n
n n
n n
z z
Q z T Q T Q T
n n
π π
π π π
π π − ∞ ∞ = = − − ∂ = − + ∂
∂
∑
−∑
∂ (14)(
)
0(
)
(
)
(
) (
(
)
)
10 , , , , , 1 ! n n n z
Q z T Q T T Q T
n
π
π π ρ π π
π + ∞ = − ∂ = +
∂
∑
+ (15)Again partial differential equation for Q z
(
, ,π T)
by using (11) (12) w.r.t. z:(
)
(
) (
)
2
, , , , ,
Q z T T Q z T
z π π ρ π π
∂ =
∂ ∂ (16)
These are given by (11) (12) and (14) (15) which transform in the form of
(
, ,)
0(
,)
z
Q z T Q T
π π π π = ∂ = ∂
(17)
(
)
0(
)
(
) (
)
1 , , , , ! n n z n z z
Q z T Q T Q T
n
π
π
π
π = π ∞ π
= = − = +
∑
(18)Also
(
, 0,)
! j
j
z
Q z T
j
= (19)
The method of the solution to be used here, we first solve for Q z
(
, ,π T)
in terms of the function, we define:(
,)
(
0, ,)
j
f π T Q π T
π
∂ =
∂ (20)
Taking Laplace transform of (12) and (16) using (20) w.r.t. z and we obtain
(
)
(
) (
*)
(
)
* , , , 1
, , j ,
T Q s T
Q s T f T
s s
ρ π π
π π
π
∂ − =
∂ (21)
(
)
( )(
)
( ) 1 , * 1 , 0 1 1, , , e d
e R T s j R T s
Q s T f T c
s
π ϑ ϑ
π π
π ϑ − ϑ
−
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(
)
(
)
( ) ( )
0 0 ( )( )
( ) (
()
)*
0 0 1 1
0 1
, ! !
, , 1 1
!
n
k k
n n n
k
n n k
n k
Q T n n k
Q s T c c
n s s
π
π ∞ π − + π − +
= =
−
= − + −
∑
∑
(23)At t = 0, then system is in initial state. Therefore
(
)
(
)
( ) * 1 0 0, ! , 0, ! n n nQ T n
Q s T
n s
∞
+ =
=
∑
(24)The solution of (21)-(23) is
(
)
1 ( , ) ( 1)(
)
1 ( , ) 1 ( , )*
0 1
, , e , e d
R T R T
R T j
s s
s
j
Q s T s f T
s
π π π ϑ ϑ
π π
π ϑ ϑ
−
− +
= +
∫
(25)Taking Laplace inverse (25) and using the modified Bessel function, with boundary conditions. Then, we get
(
)
(
)
{
(
)
}
(
)
(
)
(
)
{
}
(
)
2 2 1 2
1 2
0 0
, , , 2 ,
2 , , , d
j j
j j
j
Q z T R T z I zR T
I R T R T z f T
π
π π π π π
π π ϑ ϑ ϑ ϑ
−
=
+ −
∫
(26)(
)
(
)
0(
) (
)
0
, , 0, , , , , , , d
j j j
Q z T z T z T f T
π
π = ϒ π + ϒ
∫
ϑ π ϑ ϑ (27)(
)
(
)
(
)
2 2{
(
(
)
(
)
)
}
1 2, , , , , n n 2 , ,
n ϑ πz T π πR T ϑ ϑR T z In π πR T ϑ ϑR T z
−
ϒ = − −
(
, , ,)
1(
, , ,)
n z T n z T
z ϑ π − ϑ π
∂ ϒ = ϒ
∂ (28)
Noting that
(
)
(
)
(
)
2 2{
(
(
)
(
)
)
}
1 2, , , , , n n 2 , ,
n ϑ πz T π πR T ϑ ϑR T z In π πR T ϑ ϑR T z
π π
−
∂ ∂
ϒ = − −
∂ ∂ (29)
(
, , ,)
(
,)
1(
, , ,)
n ϑ πz T ρ π T n ϑ πz T
π +
∂ ϒ = ϒ
∂ (30)
Define
(
0, 0, 0,)
0n T ψ n
ϒ = (31)
(
, , ,)
0 : 0n π π π T n
−
ϒ = > (32)
(
)
0 π π π, , ,T 1
ϒ = (33)
Partial differentiation (27) w.r.t. π and using (18) (29) (30), we get
(
)
(
)
(
)
(
) (
)
(
) (
) (
)
1 0
1 0
, , , 0, , , , , , ,
, , , , , d
j j j
z
j
Q z T T T T f T
T T f T
π π
π ρ π π π π π π π
π
ρ π ϑ π π ϑ ϑ
+ =
∂
= ϒ + ϒ
∂
+
∫
ϒ(34-35)
(
)
(
)
(
)
1(
)
(
) (
1) (
)
0
, , , , 0, , , , , , , , d
j j j
f T Q T T T T T f T
π
π = π π −ρ π ϒ + π π −
∫
ρ π ϒ ϑ π π ϑ ϑ (36)Then,
( )
1 ( , )(
)
(
) (
)
0
e R T 0, , , , , , , d
n j n n j
p T T f T
π π π
π − + π π ϑ π π ϑ ϑ
− −
= ϒ + ϒ
OALibJ | DOI:10.4236/oalib.1100973 5 October 2014 | Volume 1 | e973
(
)
(
)
0(
) (
)
0
, 0, , , , , , , d
j j j
f T T T f T
π
π =ϖ π π +
∫
ϖ ϑ π π ϑ ϑ (38)Here,
(
, , ,)
(
, , ,)
(
,)
1(
, , ,)
j T j T T j T
ϖ ϑ π π = ϒ ϑ π π −ρ π ϒ + ϑ π π
Consequently, Equation (37) gives the approximate solution to the single server queue which arrival rate is time dependent and service rate is constant with step function, provided a solution (38) can be found.
4. Conclusion
In this paper we have obtained the analytic transient solution of M/M/1/N queuing system with a time-dependent arrival rate λ
(
1+tT−1)
using unit step function and service rate µ is constant. This model finds its applica-tion in computer-networks, telecommunicaapplica-tions, traffic problems and producapplica-tion systems.References
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