CC BY
29
M/M/ro Queue with Catastrophes and Repairable
Servers
Gulab Singh Bura
Department of Mathematics and Statistics, Banasthali Vidyapith, Rajasthan, INDIA
gulabsingh@banasthali.in
Abstract
An infinite server Markovian queueing system with randomly occurring breakdowns and non zero exponentially distributed repair time is proposed. Upon arrival, a catastrophes deactivate all the servers and system is under catastrophic failure. Immediately, a repair process is started and after successful repair the system is ready to serve the newly arrived customers. Continued fraction techniques have been used to obtain the time dependent probabilities of the studied model. The stationary probability distribution for the number of customers in the system is also derived. Some important stationary as well as transient moments are also determined. Further, The availability and reliability of the system under consideration are investigated. Finally, some graphical results are presented to visualize the model practically.
Keywords: M/M/ro Queue, Server Breakdown, Transient Analysis,Steady State Solution, Confluent Hypergeometric Function, Reliability and Availability.
1. Introduction
Here, we consider a classical M/M/ro queueing model subjected to randomly occurring breakdowns (catastrophes). Upon arrival, a catastrophes deactivate all the servers and system is under disasters breakdown. Immediately a repair process is started and after successful repair the system is again restart their functioning and provide service to a newly arrived customer. We analyze this model and provide steady state and time dependent solution.
During the last four decades the interest in catastrophic queueing model has been increased by a rapid phase. Therefore queueing models in the presence of catastrophes has been analyzed by many researchers.(see e.g., [1],[2],[3], [12],[17],[19],[21][22]). Occurrence of catastrophes destroys all present customers and also breakdown the servers. Some authors assumes that whenever catastrophes occurs, it flush out all present customers and immediately the server is ready for service for a newly arrived customer(see e.g. [1], [3],[8], [9]). And some assumes that the server or system may take a non zero repair time for their re-functioning whenever it is affected by a catastrophic failure(see e.g. [4], [18], [21], [25]).
Infinite servers queueing models are also analyzed by many researchers with the possibility of catastrophes. Gursoy et al. [15] analyzed an infinite server queue with randomly occurring interruption and provide steady state solution . Giorno et al. [26] have discussed the various properties of a bilateral birth-death process, affected by randomly occurring catastrophes. Linton and Purdue[6] have obtained the stationary and transient distribution of the probabilities for an M/G/ro queue with catastrophes. Yechiali [25] considered an M/M/ro queues with catastrophes and studied the impatient behavior of customers when server is down. The transient solution of an infinite servers Markovian queue subjected to catastrophes has been obtained by Krishna Kumar et al. [5] and Gulab Singh Bura [8].
In this work, we present an M/M/ro queuing system with catastrophes, Server breakdown and
non-zero repair time. Although,the operating model has been already analyzed by Sophia and Murali [23]. Then, objective of this paper is to illustrate a different approach and provide some additional important measures of the system under consideration.
The M/M/to queueing system with repairable servers finds its application in telecommunication field. Our system under consideration is a University campus which provide free Wi-Fi service to their students. Within the campus, each mobile is considered as one queueing server. Whenever a breakdown occur i.e. (connectivity loss or signal failure), all the servers gets deactivated and none of them works until that breakdown is repaired. So, an M/M/to queue with system failure and repair is a suitable approximation.
The paper is arranged in the following way. Next section describe the formulation of the model. The transient solution of the model have been obtained in section 3. Under section 4, we have obtained some moments of the model in transient form. Section 5, gives the time independent solution of the model. In Section 6, we have discussed about the availability and reliability of the system. Section 7 presents some graphical illustrations to observe the system performance with the effect of various parameters. Conclusion is given in the last Section.
2. MATHEMATICAL MODEL
An infinite servers Markovian queueing system with server breakdown and repair is in operation. Arrivals occur one by one in a Poisson stream with mean rate A. Service times are exponentially distributed with parameter 0. The system may fails due the disastrous breakdown occurs at a Poisson rate 7. Whenever a catastrophes occur all the servers are deactivated and the system is under disasters breakdown. Immediately a repair process is started and the repair time distribution is exponential with rate After successful repair the system again restart their functioning and provide service to a newly arrived customer. Also, it is assumed that, no customer is allowed to enter in to the system during the repair process of failed servers. Let the random variable C(t) represents the number of customers present in the system at time t and Pn(t) denotes its probability.
3. TRANSIENT ANALYSIS UNDER MARKOVIAN SETUP
This section provides the probability mass function of the random variable C(t). For this, the differential-difference equations are given as:
F'(t)= 7(1 - F(t) - Po(t)) - r)F(t) (1)
P0(t) = 0P1(t) + VF(t) - XPo(t) (2)
K (t) = (n +1)dPn+1(t) + XPn-1 (t) - (X + n0 + 7) Pn(t), n =1, 2, 3,... (3)
Initially, at t = 0,
'1 if n = 0 0 if n = 0.
Taking Laplace transform of Eq.(1),Eq.(2),Eq.(3) and by the use of Eq.(4), we have
Prm = {1 if:=0; (4)
(s + 7 + V) F*(*) = l(- - P*(s)) (5)
s
(s + X) P*(s) = 1 + dP*(s)+ r/F*(s) (6)
(8 + X + nd + 7) P:(s) = (n + 1)dP*+1(s) + XP:_1(s), (7)
After some manipulation, Eq.(7), gives an expression
p: (S) =_I_
P*-1(S) £±I±7 + n) - {n +1) ^ 144
Ai^-iOO = (s + X + 7
0 P* (s) V o
+ n^j -
(n +1) A
+n +1) -
(n + 2) A
(i+^+7 +n + 2) -
Now using the identity given by Lorentzen and Waadeland [13]
iFi(q+1;r+1;z) r (q+1)z (q + 2)z
rewritten as
1F1 (q; r;z) r - z+ r - z + 1+ r - z + 2+
iFi(q;r;z) r-z (q+1)z (q + 2)z
i Fi ( + 1; + 1; ) + - + 1+ - + 2+ by using Eq.(9) in Eq.(8), we have
P*(s) X iFi(q + 1;r+1, z)
Pn-i(s) HS~±1 +n) iFi(q;r;z)'
therefore for n > 1,we have
p: &
(X)
iFi(n +1; !+- + n +1; -f)
oj IIU( ^+j) iFi(1; £±2 + 1; -1)
P: (s) = c (s)P*(s),
P*(s),
where
It is well known that
iFi(n +1; a+l +n +1; -f-)
C (s) W n?=i№ +i)iFi(1; ^ + 1; -1)'
F * (s)^P*(s) = -,
s
n=o
by the use of Eq.(12) and Eq.(5), we get
P5(s) = (1 + -L)
after simplification Eq.(15) reduces to
o + x + ,,)-«(;(>) + ' cw
= + ;) t (, + X-Z,v+i
j=o
Efo - sk o) a (s )
,fc=i
on inversion, we get
1—' fi p (t) = y2(-1)j e-(x+ri)(t-u) (t -uy
i=o Jo
+v E(-1)J
J=0
-(A+r )x ^ J!
X> - h0)Ck(u)
.k=i
—
X> - 5k o) Ck (x)
*j
u
k=i
j
dx
(8)
(9) (10)
(11) (12)
(13)
(14)
(15)
(16)
(17)
Now for Pn(t), consider Eq.(12), which on inversion, gives
Pn(t)= Ut) *P0(t), (18)
where the symbol * denotes the convolution and P (t) given in Eq.(17).
Next we derive the expression for (n(t), where (n(t) represents the inverse Laplace transform of
a
i
o
C( s ).
From Eq.(13)
We known that
C(s)
1F1(n +1;
1F1(n +1;H±l + n + 1; )
OJ №=1 № + i) iFi (1; ^ + 1; ) '
S + 7
e
-K ^ (n + 1)k (-è)
+ n+1;
k==0 (^ + n + 1)uk\
where (b)j. represents the Pochhammor symbol, i.e.
(b)k = {
1 if k = 0;
b(b+1)(b + 2)...(b + k +1) if k = 1, 2, 3,....
Therefore
iFi(n +1; ^ + n +1; )
. c (n+k\ ( \
£
(T) (-$Y
nn+k / s+7
e
rij=i(s+7 +j) k=o uj
Applying partial fraction expansion, the above equation can be written as
iFi(n +1; ^ + n +1; -à)
n"=i (^ +j)
=
CO /
t—n v
k=0 n+k
n + k\ ( X
k V 6
(
Ï (-1) -i
Also
where
i Fi
(j- 1)\(n + k - j)\(s + 7 + j0y
^ + 1; =£) = £ (-x)kdi (s),
k=0
d*k(s) = —û-
nk=i (s + 7 + J0)
1
and d0(s) = 1.
F (1.s+y +1—) =J2(X)kek(s),
iFi(1; — + 1 -T) k=0
where e0(s) = 1, and for k=1,2,3,...
( )=
di (s) 1
d2(s) di(S)
dk-i(s) dk-2(s) dk-3(s) dk(s) dk-i(s) dk-2(s)
di(s) 1 d2(s) di(s)
(19)
(20)
J2(-1)1 -i m (s).
=i
By substituting Eq.(19) and Eq.(20) in Eq.(13), we get
C( * ) = (x)nY^ (-X
CC
'B-xnn + j d:+* (») T,(x)k4 (s)
=0 k=0
k
1
On inversion, we obtain
Cn(t) = (\)n Y,(-xy(n+1 dr+ (*) YsW^ (f), (21)
¿=0 ^ ' k=0
A. (t) =_1_V_(-1)3_e(-j6+i)t k _123
dk (t) = (ef-1^ (k - j)\(j - i)f ,k =12 3—
k 1 ^ (-i)j 1 _
(k - i)\(i - 1)'e =1 - -
k
tk (t)= ^(-1)j-1 dj (t) * ek-j(t), k = 2, 3,4,...; d(t) = di(t)
=1
Now from Eq(5) , we have
F*(*)=.,?.,.. (1 -P*(s))
s + 7 + V \s
On inversion,
F(t)=-y f (1 -P0(z))e-("'+)(t-Z)dz (22)
Jo
0
/ 0
4. TIME DEPENDENT MOMENTS 4.1. MEAN
Let A(t) denote the mean value of the random variable C(t),therefore
A(t) = E(C (t)) = ^nPn (t) (23)
n=i
Initially, at t=0, Eq(23) gives
A(0) = 0,
which implies
—
A'(t) = ^nP^ (t), (24)
n=i
where A'(t) denotes the derivative of A(t). Application of Eq.(3) in Eq.(24),after some calculation gives
A' (t) = -(0 + 7)A(t)+X (25)
which is a linear differential equation in A(t), whose solution gives
A(t) = -+- [1 - e-(e+^] (26)
+
4.2. VARIANCE
An average is not sufficient to understand completely the distribution of the random variable C(t)).Hence, variance is also needed for better understanding. Let Var(C(t)) represents the variance of the random variable C( ), then
Var(C(t)) = E[C(t) - E(C(t))]2
Which may be written as
Var(C(t)) = c(t) - [A(t)]2, (27)
and
—
c(t)=E(C 2(t)) = Y,n2Pn(t),
with
c(0) = 0,
and
c'(t) = (t) (28)
n=i
Substitution of P.n(t) in Eq.(28), after some calculation results in the form of a linear differential equation in ( ) i.e.
c(t) = -(2 9 + ri)c(t) + (2 X + 9) A(t) + X (29)
which after integration gives
(2X + 9)X(9 - e-2e±g)t(39 + 7) + e~(e±^)t(29 + 7))
( )=
(2 9 + ^)9(9 + 7)
+ J2/+V)11 - (30)
subsitutation of Eq.(30) in Eq.(27), gives the expression of Var(C(t)).
5. STEADY STATE SOLUTION
Here, we derive an expression for the stationary probabilities of the operating model
Theorem 5.1. Stationary probabilities of the system under consideration are given as
F = —L_ (1 - VP1)
7 + v Pn = r/PnPl
Po = VPi
where
/W™ .P.i„xM±,,±l-_
" iFi(n +1; g + n +1; =£)
W n;=i (g +j) iFi(1; g + 1; -I)
Pn J P" .p./vui.
and
pi = £■
- Sn e)Pn
n=i
. (X + r/y±i
3=0
Proof. Multiplying by s on both side of Eq.(16) and taking limit as s ^ 0, and using
lims^o sP*(s) = Po, we get
Po = VPi (31)
where
(-1)
Pi + Vy±i
= 0
- 0)P„
n=i
For n =1, 2,...,
Multiplying by s on both side of Eq.(12) and taking limit as s ^ 0, and using lims^0 sP* (s) = Pn,
we get
Pn = V Pn Pi, (32)
where
Pn =
(X\n iFi(n +1; g + n +1; =f)
n-=i (g +J) iFi(1; g + 1; -I).
The failure distribution is obtained by multiplying on both sides of Eq(5) and using Tauberian theorem after taking the limit as s ^ 0, we get
F = (1 -VPi) (33)
+
■ It is observed that the stationary solution exist only if Pi < 1.
5.1. Mean and Variance
Taking limit as t ^ <x> in Eq.(26) and in Eq.(27) after putting the values of c(t)andA(t), we get directly an expression for steady state mean and variance i.e.
A =7WT-, (34)
{0 + 7)
Var(C) = [(2A + 6)A + A - (29 + 1)A2] (35)
6. RELIABILITY AND AVAILABILITY ANALYSIS
The probability that a system perform well without any failure for a given period of time is known as its reliability. In this section, we derive an expression for availability and reliability of the system. Let Av(t) be the probability that a repairable system is available at a given point of time t. Therefore, from Eq(22), the availability of the system is obtained as
Av(t) = 1 - F(t)
= + -{l+r)t)+7 ! Po(x)e-(r+^)(t-x)dx, (36)
(r + 7) Jo
where P0(t) is given by Eq(17).
Next, we obtain an expression for the average availability of the system i.e.
1 [*
A« (t)* = - A« (y)dy
1 Jo
7 + T^+^ll - e-(r+l)t]) + [ Po(y)[1 - z-{r+l){t-y)]dy, (37)
(r + 7) V (V + 7)t J (r + 7y
If r = 0, then we get from Eq(22)
F(t) = 1 - e yt -7 f Po(x)e r(t-x)dx o
Therefore R(t), the system reliability is obtained as
R(t) = 1 - F(t)
^ (1 + 7 Jo eTXPo(x)dX^j
elXP0(x)dx) (38)
7. NUMERICAL ANALYSIS
Here, some graphical results are presented to study the behavior of the probability P0 and E(C) with various parameters i.e. arrival rate A, catastrophic rate 7 and service rate 9.
In fig.(1 to 2) we have plotted the probability P0 as a function of (A, 7) and (9,7) respectively. We observe that the value of Po is decreasing with increasing value of A and increasing with increasing value of 9. Also, in both the figures P0 increases with increasing 7 i.e. the probability of an empty system increases with the increase in catastrophic rate. Fig.(3 and 4), illustrates that the expected number of customers decreases with the increasing service and catastrophic rates and increases with the corresponding increase in arrival rate.
Gulab Singh Bura RT&A, No 4 (71)
M/M/to Queue with Catastrophes and Repairable servers Volume 17, December 2022
Figure 1: P0 as a function of A for 0 = 10
Figure 2: P0 as a function of 0 for A = 1
Figure 3: E(C) as a function of A for 0 =10
Figure 4: E(C) as a function of 0 for A = 1
8. CONCLUSION
In this paper, we have considered an infinite servers Markovian queueing system with catastrophes and repairable servers. The transient and stationary probabilities are obtained analytically. The system availability and reliability are two important characteristics for those queueing system which are failed and repaired. Therefore, these two measures are also investigated for the system. Some graphical results are also added to visualize the model in practical situations.
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