International Journal Of Advancement In Engineering Technology, Management and
Applied Science (IJAETMAS)
ISSN: 2349-3224 || www.ijaetmas.com || Volume 03 - Issue 09 || September - 2016 || PP. 55-68
1 HETEROGENEOUS NON-EMPTY BULK SERVICE QUEUEING
MODEL WITH SINGLE VACATION
R.Sree parimala1, S.Palaniammal2,
1. Research scholar, 2.Professor& Head,
Department of Science and Humanities
Sri Krishna College of Technology,Kovaipudur,Coimbatore-641 042, Tamil Nadu, India.
Abstract
This paper focus on the two heterogeneous servers M/M(a,b)/(2,1) queueing
system with single vacation. In this model it is assumed that the arrival pattern is Poisson fashion
with parameter λ and the successive service times are assumed to be mutually independent and
exponentially distributed with parameters μ1and μ2 for the fast and slow servers respectively. The
batches are served according to FCFS discipline. In this model, the fast server is always retained
in the system and a single vacation policy for slow server is discussed. The steady state solutions
and the system characteristics are derived and analyzed for this model. The analytical results are
numerically illustrated for different values of the parameters and levels also.
Keywords: Heterogeneous servers, Single vacation, Bulk service
1. INTRODUCTION
In the multi-server queueing models, usually we assume the servers to be identical.
However, this situation is not very realistic and can prevail only when the service process is
mechanically controlled. In the case of human servers we cannot expect them to serve at the
same rate.The situations of this kind in our daily life are at check-out counters in grocery stores
and departmental stores, in banks and many others. In almost any system which uses human
servers will have slow and fast servers.However the more interesting applications are that when
new equipment is introduced as a replacement to old or obsolete service facilities, or when old
equipment is kept as back up to a new facility and the new facility may be a faster
communication channel, a new computing device or a machine. In all these cases one usually has
www.ijaetmas.com
Page 55
International Journal Of Advancement In Engineering Technology, Management and
Applied Science (IJAETMAS)
ISSN: 2349-3224 || www.ijaetmas.com || Volume 03 - Issue 09 || September - 2016 || PP. 55-68
the option of using the old equipment at minimal cost, or at no cost at all. So,in this section we
have discussed the two heterogeneous bulk service with different service rate in which the fast
server is retained in the system always and the slow server is utilized whenever the queue size
becomes more than „a‟ in the system.
Morse (1958) introduced the concept of heterogeneity in service. He discussed the
situation of certain hyper exponential distributions of service times with parallel channels. Saaty
(1961) further discussed Morse problem by assigning the different service rates 1 and  2 to the
two branches respectively. Krishnamoorthi (1963) extended this problem by assigning customers
to the two servers with probabilities  1 and  2 respectively.
Chaudhry et al (1987) have discussed the model M/M(a,b)/2 with heterogeneous
servers and obtained the steady state probabilities and busy period distributions. Later Sitrarasu
(1993) has studied the same model with balking. A bulk queueing model M/M(a,b,c)/2 for nonidentical servers with vacation is studied by Mishra and Pandey (2002).
In the literature described above, customer inter-arrival times and customer service
times are required to follow certain probability distributions with fixed parameters.
The present investigation in this paper, an attempt has been made to analyzetwo
heterogeneous M/M(a,b)/(2,1) queueing system with servers vacation. The study of this queueing
model is organized as follows. The model is described in Section 2. Queueing model is
formulated mathematically along with notations in Section 3. The steady state solutions have
been obtained in Section 4. The performance measures and mean queue length are derived in
Section 5.The numerical results and graphical illustrations are discussed to facilitate the
sensitivity analysis in Section 6 .Concluding remarks and notable features of investigation done
are highlightedin Section 7.
2. PROBLEM DESCRIPTION
www.ijaetmas.com
Page 56
International Journal Of Advancement In Engineering Technology, Management and
Applied Science (IJAETMAS)
ISSN: 2349-3224 || www.ijaetmas.com || Volume 03 - Issue 09 || September - 2016 || PP. 55-68
In this model it is assumed that the arrival pattern is Poisson with parameter λ,
service is done in batches according to the general bulk service rule introduced by Neuts (1967).
The late arrivals are not allowed to join the ongoing service. The successive service times are
assumed to be mutually independent and exponentially distributed with parameters μ 1and μ2 for
the fast and slow servers respectively and 𝜇1 >𝜇2 .
On completing the service if the slow server finds less than „a‟ customers in the
queue and the fast server is busy, he leaves the system for a random period of time called
vacation which is exponentially distributed with parameter θ and on returning from vacation, if
the slow server finds less than „a‟ waiting customers and the fast server is busy or idle in the
system, he stays in the system until he finds at least „a‟ customers. In this system single vacation
for slow server is considered.
The model leads to the state space
{(i, j, n); i= idle,1,j = 0,1, idle, i+j≠ 2,0 ≤ n ≤ a – 1}U{(1, 1, n);n ≥ 0}U{(1, 0, n); n ≥ a}
In the state (i, j, n), the index i=idle refers to the situation that the fast server is idle
and i=1 means that he is busy and the index j=0 refers to the state that the slow server is on
vacation, j=1 that he is busyj=idle refers to the situation that the slow server is idle in the
system. The index n ≥ 0 refers to the number of waiting customers in the queue.
3. STEADY STATE EQUATIONS
Defining Pi,j,n(t) as the probability that the system is in the state (i, j, n) and
assuming that the steady state probabilities exists, the balance equation in the steady state are
given by
( λ + μ1 ) 𝑃 1 𝑖𝑑𝑙𝑒
𝑛
= λ 𝑃 1 𝑖𝑑𝑙𝑒
( λ + μ1 ) 𝑃 1 𝑖𝑑𝑙𝑒
0
= θ 𝑃 100 + λ 𝑃 1 𝑖𝑑𝑙𝑒
(λ + μ2) 𝑃 𝑖𝑑𝑙𝑒
1𝑛
= λ 𝑃 𝑖𝑑𝑙𝑒
www.ijaetmas.com
𝑛 −1
1𝑛 −1 +
+ θ 𝑃 1 0 𝑛 (0 ≤ n ≤ a-1)
(2)
𝑎 −1
μ1𝑃11𝑛
(1)
(0 ≤ n ≤ a-1)
(3)
Page 57
International Journal Of Advancement In Engineering Technology, Management and
Applied Science (IJAETMAS)
ISSN: 2349-3224 || www.ijaetmas.com || Volume 03 - Issue 09 || September - 2016 || PP. 55-68
(λ + μ2) 𝑃 𝑖𝑑𝑙𝑒
10 =
μ1𝑃 110
(λ + θ) 𝑃 𝑖𝑑𝑙𝑒
00 =
(λ + θ) 𝑃 𝑖𝑑𝑙𝑒
0𝑛
λ 𝑃 𝑖𝑑𝑙𝑒
𝑖𝑑𝑙𝑒 𝑛
= λ 𝑃 𝑖𝑑𝑙𝑒
λ 𝑃 𝑖𝑑𝑙𝑒
𝑖𝑑𝑙𝑒 0 =
(4)
μ1𝑃 100 + μ2𝑃 𝑖𝑑𝑙𝑒
10
= μ1𝑃10𝑛 + μ2𝑃 𝑖𝑑𝑙𝑒
1𝑛
𝑖𝑑𝑙𝑒 𝑛−1 +
0 𝑎 −1 +
+λ 𝑃 𝑖𝑑𝑙𝑒
μ1𝑃 1 𝑖𝑑𝑙𝑒
μ1𝑃 1 𝑖𝑑𝑙𝑒 0 + θ𝑃 𝑖𝑑𝑙𝑒
(λ + μ1+ θ) 𝑃 100 = λ𝑃 𝑖𝑑𝑙𝑒
(5)
𝑛
1 𝑛−1 (0
+ θ𝑃 𝑖𝑑𝑙𝑒
≤ n ≤ a-1)
0𝑛 (0
≤ n ≤ a-1)
(6)
(7)
(8)
00
μ2 𝑃 110 + μ1
𝑏
𝑖=𝑎
𝑃10 𝑖𝑑𝑙𝑒
(9)
(λ + μ1+ θ) 𝑃10𝑛 = λ 𝑃 10𝑛 −1 + μ2𝑃 11𝑛 + μ1𝑃 10𝑛 +𝑏 (1 ≤ n ≤ a-1)
(10)
(λ + μ1+ θ) 𝑃10𝑛 = λ 𝑃10𝑛 −1 + μ1𝑃 10𝑛 +𝑏
(11)
(n ≥ a)
(λ + μ1 + μ2) 𝑃 11𝑛 = λ𝑃 11𝑛 −1 + μ1𝑃 11𝑛 +𝑏 + μ2𝑃 11𝑛 +𝑏 + θ𝑃 10𝑛 +𝑏 (n ≥ 1)
(λ + μ1 + μ2) 𝑃 110 = μ1
𝑏
𝑖=𝑎
𝑃11 𝑖𝑑𝑙𝑒 + μ2
+ λ𝑃 1 𝑖𝑑𝑙𝑒
𝑎 −1 +
𝑏
𝑖=𝑎
λ𝑃 𝑖𝑑𝑙𝑒
𝑃11 𝑖𝑑𝑙𝑒 + θ
𝑏
𝑖=𝑎
(12)
𝑃10 𝑖𝑑𝑙𝑒
1 𝑎 −1
(13)
4. COMPUTATION OF STEADY STATE SOLUTIONS
Let E denote the forward shifting operator defined by E(P10n) = P10n+1. From
equation (11) implies
( 𝜇1 Eb+1 – (𝜆 + 𝜇1 + 𝜃)E + 𝜆) 𝑃10𝑛 = 0 𝑛 ≥ 𝑎 − 1 .
www.ijaetmas.com
(14)
Page 58
International Journal Of Advancement In Engineering Technology, Management and
Applied Science (IJAETMAS)
ISSN: 2349-3224 || www.ijaetmas.com || Volume 03 - Issue 09 || September - 2016 || PP. 55-68
The characteristic equation of (14) has only one real root inside the circle |Z| =1 by
Rouche‟s theorem when 𝜌 =
𝜆 +𝜃
𝑏𝜇1
is less than 1.If r0 (say) is the root of the above characteristic
n
equation with |r0|<1, thenP10n = A1 r0 , (n ≥ a), is the solution for the homogeneous difference
equation (11), and we get
𝑃10𝑛 = 𝑟0𝑛 −𝑎 +1 𝑃10𝑎 −1 𝑛 ≥ 𝑎
(15)
Using equation (12),((𝜇1 + 𝜇2 ) Eb+1 – (𝜆 + 𝜇1 + 𝜇2 )E + 𝜆) 𝑃11𝑛 = - 𝜃 𝑃10𝑛 +𝑏+1 𝑛 ≥ 0
The characteristic equation of this equation has only one real root r1 by Rouche‟s theorem
which lies in the interval (0,1) , 𝜌< 1, where 𝜌 = 𝑏(𝜇
𝜆
1 +𝜇 2 )
and after simplification
𝑃11𝑛 = (𝐴2 𝑟1𝑛 + 𝑘1 𝑟0𝑛 )𝑃10𝑎 −1 𝑛 ≥ 0
where𝐴2 is a constant and 𝑘1 = 𝜇
(16)
−𝜃 𝜇 1 𝑟0𝑏 −𝑎+2
2
𝜆+𝜃 𝑟0 −𝜆 +𝜃𝑟0 𝜇 1
Equation (10) gives,
(𝜇1 Eb+1 – (𝜆 + 𝜇1 + 𝜃)E + 𝜆) 𝑃10𝑛 = -𝜇2 𝑃11𝑛 +1 for 0 ≤ 𝑛 ≤ 𝑎 − 2 and using (15) and (16) in
the above equation and simplifying, we get,
𝑃10𝑛 = (𝐴3 𝑟2𝑛 +𝑘2 𝑟1𝑛 + 𝑘3 𝑟0𝑛 ) 𝑃10𝑎 −1 1 ≤ 𝑛 ≤ 𝑎 − 1
where𝑘2 =
𝜇 2 (𝜇 1 +𝜇 2 )𝐴2 𝑟1
(𝜆−𝜇 1 −𝜃 𝑟1 −𝜆 )𝜇 2 +𝜃𝑟1 𝜇 1
(17)
and 𝑘3 = - 𝜇2 𝑟0 𝑘1
Solving equation (3) using (16), gives
((𝜆 + 𝜇2 )𝐸 − 𝜆) 𝑃𝑖𝑑𝑙𝑒
𝑃𝑖𝑑𝑙𝑒
1𝑛
1𝑛
= 𝜇1 𝑃11𝑛 +1 1 ≤ 𝑛 ≤ 𝑎 − 1
= (𝐴4 𝑟3𝑛 +𝐴2 𝑘(𝑟1 )𝑟1𝑛 + 𝑘1 𝑘(𝑟0 )𝑟0𝑛 ) 𝑃10𝑎 −1 1 ≤ 𝑛 ≤ 𝑎 − 1
www.ijaetmas.com
(18)
Page 59
International Journal Of Advancement In Engineering Technology, Management and
Applied Science (IJAETMAS)
ISSN: 2349-3224 || www.ijaetmas.com || Volume 03 - Issue 09 || September - 2016 || PP. 55-68
𝜆
𝜇 1𝑥
where𝑟3 = 𝜆+𝜇 and k(x) =
𝜆+𝜇 2 𝑥−𝜆
2
fromequation (11)
𝑃1 𝑖𝑑𝑙𝑒𝑛 =(𝐴5 𝑟4𝑛 +𝐴3 𝐻(𝑟2 )𝑟2𝑛 +𝑘2 𝐻 𝑟1 𝑟1𝑛 + 𝑘3 𝐻(𝑟0 )𝑟0𝑛 )𝑃10𝑎 −1 1 ≤ 𝑛 ≤ 𝑎 − 1 (19)
𝜆
−𝜃𝑥
where𝑟4 = 𝜆+𝜇 and H(x) =
𝜆+𝜇 1 𝑥−𝜆
1
Solving equation (6) using (15) and (18), we get
𝑃𝑖𝑑𝑙𝑒
𝑛
𝑛
𝑛
0𝑛 =(𝐴6 𝑟5 +𝐴4 𝐺(𝑟3 )𝑟3 +𝐴3 𝐺(𝑟2 )𝑟2 +[𝐴2 𝑘
𝑟1 𝐺 𝑟1 + 𝑘2 𝐵 𝑟1 ] 𝑟1𝑛
+[𝑘1 𝑘 𝑟0 𝐺 𝑟0 + 𝑘3 𝐵(𝑟0 )]𝑟0𝑛 ) 𝑃10𝑎 −1 1 ≤ 𝑛 ≤ 𝑎 − 1
here𝑟5 =
𝜆
𝜆+𝜃
𝑥𝜇 2
, 𝐺(𝑥) =
𝜆+𝜃 𝑥−𝜆
(λ+μ1+μ2)(𝐴2 + 𝑘1 )𝑃10𝑎 −1 =(μ1+μ2) 𝐴2
+𝜃 [𝐴3
𝑟2𝑎 −𝑟2𝑏 +1
1−𝑟2
+𝑘2
𝑟1𝑎 −𝑟1𝑏 +1
1−𝑟1
𝑥𝜇 1
, 𝐵(𝑥) =
Substituting the values of 𝑃1 𝑖𝑑𝑙𝑒 𝑛 , 𝑃𝑖𝑑𝑙𝑒
+ 𝑘3
𝜆+𝜃 𝑥−𝜆
0𝑛 , 𝑃10𝑛
𝑟1𝑎 −𝑟1𝑏 +1
1−𝑟1
𝑟0𝑎 −𝑟0𝑏 +1
1−𝑟0
and𝑃11𝑛 in equation (13),
+ 𝑘1
] 𝑃10𝑎 −1
𝑟0𝑎 −𝑟0𝑏 +1
1−𝑟0
(21)
After simplification, 𝐴3 = 𝐴2 𝐿 𝑟1 + 𝐿 𝑟0 ,
where𝐿 𝑥 =
1−𝑟2
1−𝑟2𝑎
[𝜇
𝜃𝑥
1 +𝜇 2
(1−𝑥)𝜆
(20)
(22)
𝐵 𝑥 𝜆+𝜇 1 (1−𝑥 𝑎 )
(1−𝑥)
]
The value of 𝐴4 can be determined using the equation (18), Substituting n = (a-1) in equation
(18), we get
𝐴2 𝑃10𝑎 −1 = (𝐴4 𝑟3𝑎 −1 +𝐴2 𝑘(𝑟1 )𝑟1𝑎 −1 + 𝑘1 𝑘(𝑟0 )𝑟0𝑎−1 ) 𝑃10𝑎 −1
www.ijaetmas.com
Page 60
International Journal Of Advancement In Engineering Technology, Management and
Applied Science (IJAETMAS)
ISSN: 2349-3224 || www.ijaetmas.com || Volume 03 - Issue 09 || September - 2016 || PP. 55-68
1
This gives, 𝐴4 = 𝑟 𝑎 −1 [𝐴2 (1 − 𝑘(𝑟1 )𝑟1𝑎 −1 ) - 𝑘1 𝑘(𝑟0 )𝑟0𝑎−1 ]
(23)
3
Also, the value of 𝐴5 can be obtained using the equation (19) by substituting n=(a-1),
(𝐴3 + 𝜆)𝑃10𝑎 −1 =(𝐴5 𝑟4𝑎 −1 +𝐴3 𝐻(𝑟2 )𝑟2𝑎−1 +𝑘2 𝐻 𝑟1 𝑟1𝑎−1 + 𝑘3 𝐻(𝑟0 )𝑟0𝑎 −1 )𝑃10𝑎 −1
𝐴5 =
1
𝑟4𝑎 −1
[𝐴3 (1 − 𝐻(𝑟2 )𝑟2𝑎−1 )+ 𝜆 -𝑘2 𝐻 𝑟1 𝑟1𝑎−1 − 𝑘3 𝐻(𝑟0 )𝑟0𝑎 −1 ]
(24)
Also from (20),
1
𝐴6 =𝑟 𝑎 −1 [𝐴4 𝐺(𝑟3 )𝑟3𝑎−1 +𝐴3 𝐺(𝑟2 )𝑟2𝑎−1 +[𝐴2 𝑘 𝑟1 𝐺 𝑟1 + 𝑘2 𝐵 𝑟1 ] 𝑟1𝑎−1
5
+ [𝑘1 𝑘 𝑟0 𝐺 𝑟0 + 𝑘3 𝐵(𝑟0 )]𝑟0𝑎 −1 ] (25)
By adding the equations (2) and (4),
(λ + μ1) 𝑃 1 𝑖𝑑𝑙𝑒 0 + (λ + μ2) 𝑃 𝑖𝑑𝑙𝑒
10
= θ 𝑃 100 + λ 𝑃 1 𝑖𝑑𝑙𝑒
𝑎−1 +
μ1𝑃 110 (26)
Using (17),(18) ,(19) and (16), after simplification the value of 𝐴2 can be determined.
𝐴2 =
𝐻 𝑟2
−1
1− 𝑟 𝑎
2
1−𝑟 2
θ ρ 1 𝑟1 𝜆
−𝐻
𝜆 +𝜃 (1−𝑟 1 )
+ 𝑘 𝑟1
𝑟1
−1
1− 𝑟 𝑎
1
1−𝑟 1
−1
1− 𝑟 𝑎
1
1−𝑟 1
−
− 𝑘 𝑟0
θ ρ 1𝑟0 𝜆
𝜆 +𝜃 (1−𝑟 0 )
−1
1− 𝑟 𝑎
0
1−𝑟 0
where ρ1 = μ
𝜆θ
1
+ μ2
(27)
All the steady state probabilities 𝑃 𝑖𝑗𝑛 for n ≥ 0, i, j = 0,1 are obtained in terms of 𝑃10𝑎 −1 .
The value of 𝑃10𝑎 −1
𝑎 −1
𝑛 =0( 𝑃10𝑛 +𝑃𝑖𝑑𝑙𝑒 1 𝑛 +
www.ijaetmas.com
can be determined using normalizing condition,
𝑃1 𝑖𝑑𝑙𝑒
𝑛
+ 𝑃𝑖𝑑𝑙𝑒
∞
0 𝑛 )+ 𝑛 =𝑎
𝑃10𝑛 +
∞
𝑛 =0 𝑃11𝑛
=1
(28)
Page 61
International Journal Of Advancement In Engineering Technology, Management and
Applied Science (IJAETMAS)
ISSN: 2349-3224 || www.ijaetmas.com || Volume 03 - Issue 09 || September - 2016 || PP. 55-68
Substitutingfor𝑃10𝑛 , 𝑃𝑖𝑑𝑙𝑒
1 𝑛 ,𝑃1 𝑖𝑑𝑙𝑒 𝑛
𝑎 −1
𝑛 =0( 𝑃10𝑛 +𝑃𝑖𝑑𝑙𝑒 1 𝑛 +𝑃1 𝑖𝑑𝑙𝑒 𝑛
+ 𝑃𝑖𝑑𝑙𝑒
𝑎𝑛𝑑 𝑃𝑖𝑑𝑙𝑒
from equations (15),(18),(19) and (20),
0𝑛
0 𝑛 )=[𝐴6 𝑀(𝑟5 )+𝐴5 𝑀(𝑟4 )+𝐴4 [𝐺(𝑟3 ) 𝑀(𝑟3 )
+K(𝑟3 )𝑁(𝑟3 )]+𝐴3 [𝐺(𝑟2 ) 𝑀(𝑟2 )+K(𝑟2 )𝑁(𝑟2 )]+𝐴2 [𝑘1 𝐺(𝑟1 )𝑀(𝑟1 )+𝑘3 B(𝑟1 )𝐾(𝑟1 )]
+ 𝑘1 𝐾(𝑟0 )H(𝑟0 )+ 𝑘2 B(𝑟0 )] 𝑃10𝑎 −1
where 𝑀(𝑥) =
1−𝑥 𝑎
1−𝑥
μ
1
+ (1−𝜆
)
a
−
1−𝑥
(29)
𝑥(1−𝑥 𝑎 )
(1−𝑥)2
, 𝑁(𝑥) =
1−𝑥 𝑎
μ
2
+ (1−𝜆
)
1−𝑥
a
−
1−𝑥
𝑥(1−𝑥 𝑎 )
(1−𝑥)2
using the equations (15), (16) and (27) in the equation (28), and simplifying
𝑃10𝑎 −1 −1 = [𝐴6 𝑀(𝑟5 ) + 𝐴5 𝑀(𝑟4 ) + 𝐴4 [𝐺(𝑟3 ) 𝑀(𝑟3 ) + K(𝑟3 )𝑁(𝑟3 )]
+𝐴3 [𝐺(𝑟2 ) 𝑀(𝑟2 )+K(𝑟2 )𝑁(𝑟2 )] + 𝐴2 [𝑘1 𝐺(𝑟1 ) 𝑀(𝑟1 )+ 𝑘3 B(𝑟1 )𝐾(𝑟1 )]
+
𝑟1
1−𝑟1
+
B(𝑟1 )
1−𝑟1
𝑟
+𝑘1 𝐾(𝑟0 )H(𝑟0 )+ 𝑘2 B(𝑟0 )] + 1−𝑟0 +
0
B(𝑟0 )
(30)
1−𝑟0
5. PERFORMANCE MEASURES
The efficiency of the queueing system can be proved by finding the performance
measures of the queueing systems under consideration. As the steady-state probabilities are
known, various performance measures of the queue can be easily obtained.
5.1
MEAN QUEUE LENGTH
Let 𝐿𝑞 be the expected number of customers in the queue then
𝐿𝑞 =
𝑎 −1
𝑛 =0 𝑛( 𝑃10𝑛
+ 𝑃𝑖𝑑𝑙𝑒
www.ijaetmas.com
1𝑛
+ 𝑃1 𝑖𝑑𝑙𝑒
𝑛
+ 𝑃𝑖𝑑𝑙𝑒
0 𝑛)
+
∞
𝑛 =𝑎
𝑛𝑃10𝑛 +
∞
𝑛 =1 𝑛𝑃11𝑛 (30)
Page 62
International Journal Of Advancement In Engineering Technology, Management and
Applied Science (IJAETMAS)
ISSN: 2349-3224 || www.ijaetmas.com || Volume 03 - Issue 09 || September - 2016 || PP. 55-68
∞
𝑛 =1 𝑛𝑃11𝑛 =
∞
𝑛 =𝑎
𝑎 −1
𝑛 =0 𝑛( 𝑃10𝑛
+ 𝑃𝑖𝑑𝑙𝑒
1𝑛
1
1
(𝐴2 1−𝑟 + 𝑘1 1−𝑟 )𝑃10𝑎 −1
1
𝑟 2
𝑎𝑟
𝑛𝑃10𝑛 = ( 1−𝑟0 + (1−𝑟0
0)
0
+ 𝑃1 𝑖𝑑𝑙𝑒
𝑛
(31)
0
+ 𝑃𝑖𝑑𝑙𝑒
2
) 𝑃10𝑎 −1
(32)
0 𝑛) =
[𝐴6 𝑀1 (𝑟5 )+𝐴5 𝑀1 (𝑟4 )+𝐴4 [𝐺(𝑟3 )𝑀1 (𝑟3 )+K(𝑟3 )𝑁1 (𝑟3 )]
+𝐴3 [𝐺(𝑟2 )𝑀1 (𝑟2 ) +K(𝑟2 )𝑁1 (𝑟2 )] +𝐴2 [𝑘1 𝐺(𝑟1 )𝑀1 (𝑟1 )+𝑘3 B(𝑟1 )𝐾(𝑟1 )]
+𝑘1 𝐾(𝑟0 )H(𝑟0 )+𝑘2 B(𝑟0 )] 𝑃10𝑎 −1
where𝑀1 (𝑥)=
𝑁1 (𝑥)=
𝑥(1−𝑥 𝑎 ) a𝑥 𝑎
(1−𝑥)2
𝑥(1−𝑥 𝑎 ) a𝑥 𝑎
(1−𝑥)2
μ
a(a−1)
1
-1−𝑥 + (1−𝜆)
μ
2
-1−𝑥 + (1−𝜆)
(33)
𝑥(1−𝑥 𝑎 )
x
a𝑥 𝑎
− 1−𝑥 ( (1−𝑥)2 − 1−𝑥 )
2𝜆 (1−𝑥)
a(a−1)
x
𝑥(1−𝑥 𝑎 )
a𝑥 𝑎
− 1−𝑥 ( (1−𝑥)2 − 1−𝑥 )
2𝜆(1−𝑥)
Substituting (31), (32) and (33) in equation (30),
𝐿𝑞 =[𝐴6 𝑀1 (𝑟5 )+𝐴5 𝑀1 (𝑟4 )+𝐴4 [𝐺(𝑟3 )𝑀1 (𝑟3 )+K(𝑟3 )𝑁1 (𝑟3 )]
1
+𝐴3 [𝐺(𝑟2 )𝑀1 (𝑟2 )+K(𝑟2 )𝑁1 (𝑟2 )]+𝐴2 [𝑘1 𝐺(𝑟1 )𝑀1 (𝑟1 )+𝑘3 B(𝑟1 )𝐾(𝑟1 )+1−𝑟 ]
1
1
𝑟 2
𝑎𝑟
+ 𝑘1 [(𝐾(𝑟0 )H(𝑟0 )+1−𝑟 )+𝑘2 B(𝑟0 )+1−𝑟0 + (1−𝑟0
0
www.ijaetmas.com
0
0)
2
]𝑃10𝑎 −1
(34)
Page 63
International Journal Of Advancement In Engineering Technology, Management and
Applied Science (IJAETMAS)
ISSN: 2349-3224 || www.ijaetmas.com || Volume 03 - Issue 09 || September - 2016 || PP. 55-68
5.2
Probability that both the servers are busy (P2B)
𝟏
𝟏
𝑷𝟐𝑩 = (𝑨𝟐 𝟏−𝒓 + 𝒌𝟏 𝟏−𝒓 )𝑷𝟏𝟎𝒂−𝟏
𝟏
5.3
(35)
𝟎
Probability that fast server is busy and the slow server is idle (P1B)
𝑟
𝑟
𝑟
𝑃1𝐵 =[𝐴5 1−𝑟4 +𝐴3 1−𝑟3 (1+ 𝐻(𝑟2 ))+𝑘2 1−𝑟1 (1+ 𝐻 𝑟1 )
4
3
1
𝑟
+ 𝑘3 1−𝑟0 (1+ 𝐻 𝑟0 )]𝑃10𝑎 −1
(36)
0
5.4
𝑃𝑖𝑑𝑙𝑒
Probability that the slow server on vacation and fast server is on idle (Pidle v)
𝑣
= [𝐴6
1−𝑟5𝑛
𝑟5
𝑟
+𝐴4 1−𝑟3 (1+ 𝐺(𝑟3 ))+𝐴3 𝐺(𝑟2 )
1−𝑟2𝑛
3
+𝑘2 𝐵 𝑟1
1−𝑟1𝑛
𝑟1
𝑟2
+𝐴2 𝑘(𝑟1 )
𝑟1
1−𝑟1
(1+ 𝐺(𝑟1 ))
𝑟
+(𝑘1 𝑘(𝑟0 ) (1+ 𝐺(𝑟0 ))+𝑘3 𝐵(𝑟0 ))1−𝑟0 ]𝑃10𝑎 −1 (37)
0
6. NUMERICAL ANALYSIS
For some chosen values of the operational parameters a, b, λ, μ1, μ2 and 𝜃 the
system performance measures are calculated and the numerical values are given in the tables 6.1
-6.3.
Table 6.1
λ
θ
www.ijaetmas.com
Mean Queue Length for various values of a, b, μ1=2, μ2= 1
Lq
P1B
Page 64
International Journal Of Advancement In Engineering Technology, Management and
Applied Science (IJAETMAS)
ISSN: 2349-3224 || www.ijaetmas.com || Volume 03 - Issue 09 || September - 2016 || PP. 55-68
a=10,
a=15,
a=20,
a=30,
a=10,
a=15,
a=20,
a=30,
b=30
b=30
b=30
b=50
b=30
b=30
b=30
b=50
0.1
15
20
5
10 0.25
15
3.1279
5.6675
6.3425
7.0759
8.9051
9.1763
13.2451
14.1797
0.3913
0.6585
0.2787
0.47031
0.2107
0.3806
0.1676
0.3141
10.9791
19.8234
3.0645
5.4395
10.0195
11.0000
17.2318
6.0709
6.9866
10.0163
11.5125
17.168
8.5078
9.0726
11.2163
16.1093
19.5651
13.1051
14.0530
14.5556
0.6937
0.8079
0.3427
0.6060
0.7124
0.6161
0.7132
0.2639
0.4335
0.6543
0.5036
0.6522
0.20885
0.3698
0.41295
0.4466
0.5253
0.1526
0.2968
0.4100
20
5
10
15
20
0.5
16.5695
2.9612
5.2492
8.6076
12.6389
15.2099
6.0049
6.8395
9.2109
12.7172
14.8121
8.2097
8.9827
10.6800
13.3497
15.5412
13.0091
13.0198
14.0050
15.1977
0.7409
0.3299
0.5985
0.6858
0.6980
0.6584
0.2569
0.4720
0.5746
0.6060
0.5827
0.1981
0.3014
0.4072
0.5247
0.4847
0.1426
0.2884
0.4035
0.4830
5
10 0.75
15
20
2.5830
5.2606
7.8562
11.1817
5.9152
6.3768
8.8035
11.6001
7.5024
8.6536
10.0030
12.9334
12.5007
12.5805
13.6853
14.2383
0.3098
0.5771
0.6165
0.5537
0.2001
0.4561
0.4999
0.4171
0.0072
0.2559
0.3936
0.2927
0.1426
0.2882
0.3969
0.4523
5
10
The performance measures when μ1= 1.5, μ2= 1, a = 10 and b=30
Table 6.2
𝜆
6
12
18
24
6
12
18
24
6
12
18
24
www.ijaetmas.com
𝜃
0.1
0.2
0.5
𝐿𝑞
𝑃2𝐵
𝑃1𝐵
3.9976
7.7223
14.1211
25.7654
3.5915
7.0983
13.2785
21.2221
2.1418
6.6009
12.7541
19.3546
0.5017
0.4323
0.2948
0.2098
0.6074
0.5040
0.3100
0.2708
0.7024
0.5406
0.3123
0.2777
0.4925
0.6003
0.6946
0.7000
0.3701
0.4945
0.6345
0.7256
0.2921
0.4576
0.6576
0.7276
𝑃𝐼𝑑𝑙𝑒
𝑣
0.0001
0.0015
0.1568
0.1934
0.0008
0.0097
0.0678
0.0500
0.0006
0.0021
0.0524
0.0977
Page 65
International Journal Of Advancement In Engineering Technology, Management and
Applied Science (IJAETMAS)
ISSN: 2349-3224 || www.ijaetmas.com || Volume 03 - Issue 09 || September - 2016 || PP. 55-68
From the above Table 6.2 it is noted that for various values of λ and 𝜃the
normalized condition total probability is 1. Also, when the arrival rate λ is increased, the queue
length also increased.
𝑳𝒒 for various values of 𝝀, a, when b = 50, 𝜽 = 0.1, μ1= 1.5 and μ2= 1
Table 6.3
6.1
𝜆
a=10
a=20
a=30
a=40
10
5.1299
9.0993
15.2845
18.2809
15
6.9520
11.7587
15.6054
19.9154
20
11.2918
12.7643
16.0769
20.1236
25
13.4311
15.8790
17.9896
21.8732
30
15.9994
19.4367
20.9076
22.8553
35
18.5433
20.0001
22.7654
24.8654
PICTORIAL REPRESENTATION
19
θ = 0.1
17
θ = 0.25
Lq
15
13
θ = 0.5
11
θ = 0.75
9
7
5
5
www.ijaetmas.com
7
9
11
λ
13
15
17
19
Page 66
International Journal Of Advancement In Engineering Technology, Management and
Applied Science (IJAETMAS)
ISSN: 2349-3224 || www.ijaetmas.com || Volume 03 - Issue 09 || September - 2016 || PP. 55-68
Figure 6.1
𝑳𝒒 Vs λ various values of θ when a =15, b = 30, μ1=2 and
μ2 = 1
7.
CONCLUSION
In this present study, a M/M(a,b)/(2,1) queueing model with heterogeneous servers
with single vacation for slow server depends on batch size are considered. The steady state
solutions and the system characteristics are derived and analyzed for these models. The
analytical results are numerically illustrated for different values of the parameters and
levels also. The model proposed here is applied for many real world problems. The two main
objectives of queueing theory as the whole are (i) to reduce the waiting time of customers (ii) to
reduce the queue length. The purposes are satisfied,because the fast server is always retained in
the system.
8. REFERENCES
1. AfthabBegum. M.I,1996, “Queueing models with bulk service and vacation”, Ph.D,
Dissertation, Bharathiar university, Coimbatore, Tamil nadu, India.
2. Bailey, N.T.J, 1954, “On queueing process with bulk service”, Journal of the Royal
Statistics Society: Series B, Vol.16, pp.80-87.
3. Borthakur, A &Gautam Choudhury, 1997, „On A Batch Arrival Poisson Queue with
Generalized Vacation‟, Sankhya: The Indian Journal of Statistics, Series B, Vol.59, pp.369383.
4. Chaudhry.M.L and Easton .G.D 1982,”The queueing systems Ek/M(a,b)/1 and its
numerical analysis”, Computer and operations research,Vol.9,197-205.
5. Cooper, R. B. 1981. “Introduction to Queueing Theory,” 2 nd ed., Elsevier North- Holland,
New York.
6. Doshi, B, T 1986 „Queueing systems with vacations-A survey‟, queueing System, Vol.1,
pp. 29-66.
www.ijaetmas.com
Page 67
International Journal Of Advancement In Engineering Technology, Management and
Applied Science (IJAETMAS)
ISSN: 2349-3224 || www.ijaetmas.com || Volume 03 - Issue 09 || September - 2016 || PP. 55-68
7. Erlang, A.K. 1909. Probability and telephone calls, Nyt. TidsskrKrarupMat. Ser. B. Vol.
20, pp. 33-39.
8. Gross, Donald, and Carl M. Harris. 1998. Fundamentals of Queuing Theory. New York:
JohnWiley& Sons.
9. Kumar, BK & Madheswari, SP 2005, 'An M/M/2 queueing system with
heterogeneous servers and multiple vacations', Mathematical and Computer
Modelling, vol. 41, no. 13, pp. 1415-1429.
10. Medhi, J 2003, „Stochastic Models in Queueing Theory‟, Academic Press, Boston, Second
Edition.
11. Neuts.M.F1967, „A general class of bulk queues with Poisson input‟, Applied
Mathematical and Statistics, Vol.38, 759 – 770.
12. Palaniammal, S 2004, „A study on Markovian queueing models with bulk service and
vacation‟, Ph.D., thesis, Bharathiar University, Coimbatore, India.
13. Singh, VP 1970, 'Two-server Markovian queues with balking: heterogeneous vs.
homogeneous servers', Operations Research, vol. 18, no. 1, pp. 145-159.
14. SreeParimala, R & Palaniammal, S 2015, „M/M(a,d,b)/(2,1) Queueing Model with
servers single and delayed vacation‟, International Journal of Applied Engineering
Research, Research India Publications, vol. 10, no.11, pp.28863-28874.
15. SreeParimala.R, Palaniammal.S 2015, „An analysis of Bulk Service Queueing Model with
servers various vacations‟, International Journal of Advancements in Research &
Technology, Volume 4, Issue 2, February -2015, ISSN 2278-7763.
16. Thaga, K & Sivasamy, R 2015, 'A Poisson queue operated by two heterogeneous
servers', International Journal of Research in Engineering and Applied Sciences, vol.
5, no. 6, pp. 78-90.
17. Thangaraj, V & Vanitha, S 2010, 'M/G/1 queue with two-stage heterogeneous service
compulsory server vacation and random breakdowns', International Journal of
Contemporary Mathematical Sciences, vol. 5, no. 7, pp. 307-322.
www.ijaetmas.com
Page 68