CC BY
68
(M, MAP)/(PH, PH)/1 queue with Nonpremptive Priority, Working Interruption and Protection
1 A. Krishnamoorthy 1,3,2 Divya V 2,4
department of Mathematics CMS College,Kottayam-686001,India. 2 Department of Mathematics N.S.S. College, Cherthala-688556
Abstract
In this paper we consider a (M,MAP)/(PH,PH)/1 queue with nonpremptive priority, working interruption and protection. Two types of priority classes of customers where type I customers arrive according to a Poisson process and type II customers arrive according to Markovian Arrival Process are considered. Service time of both type I and type II customers follow mutually independent phase type distributions. The number of type I customers in the system is restricted to a maximum of L. Also type I customers are assumed to have a non-premptive priority over type II customers. Customer services are subject to interruption by a self induced mechanism. The interruptions occur according to Poisson process. Instead of stopping service completely, the service continues at slower rate during interruption. Also we assume that an interruption occuring while customer is already under interruption will not affect the customer.The server continues to serve at this lower rate until interruption is fixed. The duration of interruption is assumed to be exponentially distributed. A protection mechanism to diminish the effect of interruptions on type I customers service is arranged.The protection for the service of type I customers is provided at the epoch of realization of the clock which starts ticking up the moment a type I customer is taken for service. Type II customers are not provided protection against interruption during their service. Also we assume that type I customers get service at a faster rate starting from the epoch of providing service protection. We analyse the distribution of service time duration of both type I and type II customers and the distribution of a p-cycle. Also we provide LSTs of busy cycle, busy period of type I customers generated during the service time of a type II customer and LSTs of waiting time distributions of type I and type II customers. Also we compute the expected number of interruptions during a type I and a type II service. We perform numerical computations to evaluate important system characteristics and also optimal system cost using a cost function .
Keywords: (M,MAP)/(PH,PH)/1 queue, nonpremptive priority, working interruption, protection
1 Introduction
Queues with interruption play an important role in day to day life. We encounter different kinds of interruptions in various activities like internet browsing, banking, medical check ups, in supermarkets etc. The works so far reported in the literature discuss about interruptions such as server induced, customer induced, enviornment dependent service interruptions, server vacations,
1 Author for correspondence: E-mail:[email protected] 3Emeritus Fellow(EMERITUS-2017-18 GEN 10822(SA-II)), University Grants commission, India
2 E-mail:[email protected]. 4 Research is supported by the University Grants Commission,
Govt. Of India, under Faculty Development Programme( Grant No.F.No.FIP/12th Plan/KLKE008 TF 04 ) in
Department of Mathematics, Cochin University of Science and Technology, Cochin-22
vacation interruptions and arrival of a priority customer. The first reported work on queues with service interruption is by White and Christie in 1958 in which they considered a two-priority single server system with the low priority customer in service pre-empted on arrival of a high priority customer. Even in the case of single class customer system, the customer in service has to wait whenever a system breakdown occurs. The interrupted service starts from the very beginning (repeat) or from where it got interrupted (resumption) on completion of interruption. These two cases are separately considered in Keilson [2], Gaver [4] and by several other researchers. Fiems et al. [3] introduced probability measures for repeat/resumption on completion of interruption without assigning any rule. Krishnamoorthy et al. [6] are the first to give a specific rule for resumption/repetition of service. We refer the review paper by Krishnamoorthy et al. [5]for details on queueing models with system induced service interruption (priority queues not included).
Varghese et al. [12] introduced a new type of interruption called customer induced interruption in which a customer interrupts own service. They considered an infinite capacity queueing system with a single server in which customers arrive according to a Poisson process with the service time following an exponential distribution. The interruptions occur according to a Poisson process and the duration of each interruption follows an exponential distribution. The self-interrupted customers enter into a finite buffer of size K. Any interrupted customer, finding the buffer full, is considered lost. Those interrupted customers who complete their interruptions move into another buffer of same size and are given a nonpreemptive priority over new customers. They evaluated several performance measures. Numerical illustrations of the system behavior are also provided and also discussed an optimization problem through an illustrative example. Krishnamoorthy et al. [7] extended this to a multi-server queueing system. They investigated the behavior of the queueing system, several performance measures are evaluated and numerical illustrations of the system behavior are provided. Also an optimization problem to maximize the revenue with respect to number of servers is employed and optimal buffer size for the self-interrupted customers are discussed through two illustrative examples. Dudin et al. [13] extended these to MMAP/PH(PH)/c queue with negative arrivals. Varghese and Krishnamoorthy [8] considered a single-server retrial queue with infinite capacity of the primary buffer and finite capacity of the orbit to which customers arrive according to a Poisson process, and the service time follows phase-type distribution. The customer-induced interruption occurs according to a Poisson process. The self-interrupted customers enter into orbit. Any interrupted customer, finding the orbit full, is considered lost. The interrupted customers retries for service after the interruption is completed. Several performance measures were evaluated and some numerical illustrations of the system behavior were provided.
In this paper we consider a single server queueing model with two priority classes of customers where the type I customers are assumed to have a non-premptive priority over type II customers.We consider customer induced interruption during own service. Instead of stopping service completely, the service continues at slower rate during interruption. The protection for the service of type I customers is provided at the epoch of realization of the clock which starts at the epoch at which the type I customer is taken for service. The rest of the paper is arranged as follows. The mathematical formulation is given in section 2. Section 3 provides steady state analysis of the model. Waiting time analysis of type I and type II customers are discussed in sections 4. Expected number of interruptions during type I and type II services are discussed in sections 5 and 6 respectively. Some other performance measures are discussed in section 7. A related cost function is discussed in section 8. Some numerical results are discussed in section 9. Proofs of two theorems stated in section 4, are given in appendix.
Notations and abbreviations used in the sequel:
• e(a) = Column vector of 1's of order a
• e =Column vector of 1's of appropriate order.
• CTMC: Continuous time Markov chain.
• Ia = identity matrix of order a.
• ea(b) = column vector of order b with 1 in the ath position and the remaining entries zero.
• MAP: Markovian Arrival Process
• LST: Laplace-Steiltges Transform
• LIQBD: Level independent Quasi-Birth and-Death
• WI: Working Interruption
• Parameters: A-arrival rate of type I customers,/- arrival rate of interruptions, parameter of exponential duration of interruption, 5- parameter of exponential protection clock.
2 Mathematical formulation
We consider a single server queue with two priority classes of customers type I and type II with the former arriving according to a Poisson process of rate A and the latter according to Markovian Arrival Process with representation(D0,D1). Service time of both types follow distinct phase type distributions with representations PH(a,T) of order m1 and PH(S, S) of order m2 respectively. The number of type I customers in the system is restricted to a maximum of L. Also type I customers are assumed to have a non-premptive priority over type II customers. Customer services are subject to interruption by a self induced mechanism. While in interruption arrival of another interruption doesnot affect the customer.The interruptions occur according to Poisson process with rate y. Instead of stopping the service of that customer completely,it continues at slower rate during interruption. That is, the service time of type I and type II ,during an interruption follow phase type distributions with representation PH(a,8T) and PH(fi,8'S), 0<8,8'<1 respectively.Thus ^ = [a(-T)-1e]-1 is the normal service rate and 8^ is the interrupted service rate of type I customers and n' = [&(-S)-1e]-1 and 8'^' are respectively the corresponding rates of normal and interrupted services of type II customers. The server continues to serve at this lower rate until a random clock expires. The duration of interruption is assumed to
be exponentially distributed with parameter A protection mechanism to diminish the effect of
1
interruptions on type I customers service is arranged. An exponential random clock with mean - is started simultaneously with each type I service. The protection for the service of type I customers is provided at the epoch of realization of this clock. Type II customers are not provided protection against interruption during their service. Also we assume that the service time of type I customers on activation of protection clock, follows phase type distribution with representation PH( a,$T), $ > 1 and finite.
Let Q* = D0 + D1 be the generator matrix of the type II arrival process and n* be its stationary probability vector. Hence n is the unique (positive) probability vector satisfying n*Q* = 0, n*e = 1. The constant p* = n*D1e, referred to as fundemental rate, gives the expected number of type II arrivals per unit of time in the stationary version of the MAP. It is assumed that the two arrival processes are mutually independent and are also independent of the service time distributions.
2.1 The QBD process
The model described in section 1 can be studied as a LIQBD process. First we introduce the followiing notations: At time t:
N1 (t) : number of type II customers in the system, N2 (t) : number of type I customers in the system
№ =
(0, if the type I customer in service is unprotected/type II customer is in service (.1, if the type I customer in service is protected
K(t) =
0, if the server provides service to type I customer in WI
1, if the server provides service to type II customer in WI
2, if the server provides normal service to type I customer
3, if the server provides normal service to type II customer
S(t): the phase of service when the server is busy
M(t) : the phase of arrival of the type II customer.
It is easy to verify that {(N1(t),N2(t),J(t),K(t),S(t),M(t)):t > 0} is a LIQBD with state
space
1(0) = {(0, k)/1 <k <n}u {(0, i2,0,j2, k1, k2)/1 <i2< L,j2 = 0 or 2,1 < k1 < m1,1 < k2 < n} u {(0, i2,1,2, ki, k2)/1 < i2 < L,1 < k1 < m1,1 < k2 < n}
For i1 > 1,
{(Î1,0,0, j^, k1, k2)/)2 = 1 or 3,1 <k1<m1,1<k2<n}u {(h, Î2,0,j2, k1, k2):1<Î2< L,h =0 or 2,1 < k1 < m1,1 < k2 < n} u {(¿1, i2,0,j2, k1, k2): 1 < i2 < L,j2 = 1 or 3,1 <k1< m2,1 <k2<n}u {(i1, i2,1,2, k1, k2)/1<i-2<L,1<k1<mi,1<k2<n}
The infinitesimal generator of this CTMC is
Qi =
Bo co
B,
A!
a2
Ao
A± Ao
where B0 contains transitions within the level 0; C0 represents transitions from level 0 to level 1; B1 represents transitions from level 1 to level 0; A0 represents transitions from level g to level g + 1 for g >1, A1 represents transitions within the level g for g >1 and A2 represents transitions from level g to g — 1 for g >2. The boundary blocks B0, C0,B1 are of orders n(1 + 3m1L) x n(1 + 3m1L), n(1 + 3m1L) x (2m2n + (3m1 + 2m2)nL), (2m2n + (3m1 + 2m2)nL) x n(1 + 3m1L) respectively. A0,A1,A2 are square matrices of order 2m2n + (3m1 + 2m2)nL. Define the entries of
g(h2,i2J2,k2,h),ç(h2,i2J2,k2,h),g(h2,i2J2,k2,h) as transition submatrices which contains transitions of the
0(hi,li,ii,k1,l1) 0(h1,l1,j1,k1,l1) 1(h1,l1,j1,k1,l1)
form
(0, h-1, Î1,Î1, k1,11) ^ (0, h.2, Î2,Î2, k2,12), (0, K, i1,Ï1, k1,11) ^ (1, Î2,Î2, k2,12) and (1, hu i1,]1, k^ I1) ^ (0, h2,i2,i2,k2,l2) respectively. Define the entries of A(^2'i2'i2'k2'l2), A(h2'i2'Î2'k2'Î2), A(^2'i2'Î2'k2'Î2) as
transition submatrices which contains transitions of the form h,h, k1,11) ^ (g + 1, h2, i2,]2, k2, ¿2X where g > 1, (g, h1, i1,j\, k1,
I1) ^ (g,h2,i2,h,k2,h),
where g > 1,(g,h1,i1,j1,k1,l1) ^ (g — 1,h2,i2,j2,k2,l2), where g >2 respectively. Since none or one event alone could take place in a short interval of time with positive probability, in general, a transition such as (g1,h1,i1,j1,k1,l1) ^ (g2,h2,i2,j2,k2,l2) has positive rate only for exactly one of 91, h1, k,j1, k1,11 different from g2, h2, Î2,)2, k2, h.
B,
0(hiXi,ii,ki,l1)
X(a 0In)
XI,
m^n
6T° 0 in To0In
0 In 8T°a 0 In T0a 0 In $T0a 0 In
SL
m1n
D0 — XIn 9T®D0-(X + v + 5)1,
m^n
T®D0-(X + Y + S)Imin <PT®D0- XImin
hi = 0,h2 = 1; i2 = 0; j2 = 2,1 <k2< mi, li, ¡2<n
hi < L - 1,h2 = hi + 1;ii = i2 = 0;yi = ¡2, 0 or 2;1 < ki, k2 < m1; 1 < l1,l2 <n hi < L - 1,h2 = hi + 1;ii = i2 = 1;ji = j2 = 2; k1, k2 < m.i; 1 < li,l2 < n ■1,h2 = 0; ii = 0; ji = 0;1<ki< mi li, ¡2<n
■ 1,h2 = 0; ii = 0; ji = 2;1 < ki < m^ li, ¡2<n
■ 1,h2 = 0; ii = 1; ji = 2;1 < ki < m^ li, h<n
hi < L,h2 = hi- 1; ii = i2 = 0;y^ = 0,j2 = 2; ki, k2 < m.i; 1 < li,l2 < n hi <L,h2=hi- 1; ii = i2 = 0;ji = J2 = 2; ki, k2 < m.i; 1 < li,l2 < n hi <L,h2=hi- 1; ii = 1,12 = 0; yi = y'2 = 2; ki, k2 < m.i; 1 < li,l2 < n hi <L,hi = h2, ii = i2 = 0;ji = 0,]2 = 2; ki, k2 < m.i; 1 < li,l2 < n hi <L,hi = h2; ii = (2 = 0;ji = 2,]2 = 0; ki, k2 < m.i; 1 < li,l2 < n hi < L,hi = h2; ii = 0,i2 = 1; j-i = 0 or 2,j2 = 2; ki, k2 < m.i; 1 < li,l2 < n ■h2 =0;1 <li,l2 <n hi < L — 1,hi = h2;ii = i2 = 0;
i
1< 1<
]'i = 1<
1< hi = 1< hi = 1< hi = 1< 2< 1< 2< 1< 2< 1< 1< 1< 1< 1< 1< 1< hi = 1<
Ji = 1<
j2 = 0; 1 < ki, k2 < mi; 1 < li,l2 < n hi<L — 1,hi=h2;ii = i2 = 0; ji = j2 = 2; 1 < ki, k2 < mi; 1 < li,l2 < n 1 < hi < L — 1,hi = h2;ii = i2 = 1;ji = h = 2; 1 < ki, k2 < mi; 1 < li,l2 < n
"°(hi,iiji,ki,ii)
(P0Di hi=h2= 0; 12 = 0;j2 = 3; 1<k2 < m2 ;1< li,l2 <n
Irni 0 Di 1<hi<L,hi = h2; ii = 12 = 0;ji = h = 0;1 < ki, k2, < mi
1<li,l2<n
¡mi0Di 1 < hi < L,hi = h2; ii = h = 0;ji = h = 2;1 < ki, k2 < mi
1<li,l2<n
¡mi0Di 1 < hi < L,hi = h2; ii = h = 1;ji = h = 2;1 < ki, k2 < mi
< 1<li,l2<n
V
Q(h2.i2.j2.k2.h) _
i(hi,ii,ji,ki,li)
e's° 0 in
So0In 9'S°a 0 In
S0a 0 In
hi = h2 = 0; ii = 0; ji = 1;1 < ki < m2,1 < li, l2<n hi = h2 = 0; ii = 0;yi = 3;1 < ki < m2,1 < li,l2 < n hi = h2,1 < hi < L; ii = i2 = 0;yi = 1,j2 = 2; ;1 < ki < m2, 1 <k2 <mi;1 < i l2<n
hi = h2,1 < hi < L; ii = i2 = 0;yi = 3,j2 = 2; ;1 < ki < m2, 1 <k2 <mi;1 < i l2<n
J^(h2,i2,j2,k2,l2) _
0(hi,iiJi,ki,li)
Im2 h = i2 = 0-.ii =j2 = 1;1< ki, k2 <m2;1< I1, l2<n
Im2 ii = i2 = 0; ji =j2 = 3;1< kr, k2 <m2,1< k, h<n
Imi 0 D1 1 < h1 < L; h1 = h2; i1 = i2 = 0; j1 = j2 = 0 or 2;1 < k1, k2 < m1; 1<l1,l2<n
Im± 0 D1 1 < h1 < L; h1 = h2; i1 = i2 = 1; j1 = j2 = 2;1 < k1, k2 < m1; 1<l1,l2<n
Im2 0 D1 1 < h1 < L; h1 = h2; i1 = i2 = 0; j1 = j2 = 1 or 3; 1 < k1, k2 < m2; 1<l1,l2<n
A.
(h-2,12,12,^2,12) 2(hi,ii,ji,ki,li)
(6'S°p 0In h1=h2 = 0; h = i2= 0; = 1,]2 = 3;1< kr, k2<m2, 1<l1,l2<n
S°p0In h1=h2 = 0;i1 = i2= 0;]1=]2 = 3;1< k^ < m2; 1<l1,l2<n
9'S°a 0In 1 < h1 < L,h1 = h2;i1 = i2 = 0; j1 = 1,j2 = 2;1<k1< m2,
1 < k2 <m1;1 < l1, l2<n S0a 0In 1 < h1 < L,h1 = h2; i1 = i2 = 0; j1 = 3,j2 = 2;1<k1< m2, 1 < k2 <m1,1 < 1 l2<n
v.
V
2 J
A1m1n 2 I
A-lm2/n m1n
6T0p 0 in T°P0In 0 ln
9T°a 0 ln T°a 0 ln <pT0a 0 ln V^rn-in tf!m.2n
Y^m2n fi^m-in
e'S®Do-(A + v)Im2n S®D0-(X + y)lm2n
1(hi.hJi.ki,h)
1 < h1 < L — 1,h2 = h1 + 1;i1 = i2 = 0;j1 = j2 = 0 or 2; 1 < k1, k2 < m1; 1 < l1,l2 < n
0 < h1 < L — 1,h2 = h1 + 1;i1 = i2 = 0;j1 = j2 = 1 or 3;
1 < k1,k2 <m2;1< l1, l2 <n
1 < h1 < L — 1,h2 = h1 + 1;i1 = i2 = 1;h = ]2 = 2; 1 < k1, k2 < m1; 1 < l1,l2 < n h1 = 1,h2 = 0; i1 = i2= 0; A = 0,j2 = 3;1 < k1 < m1, 1<k2<m2\1< h,l2<n
h1 = 1,h2 = 0; i1 = i2= 0; j1 = 2,j2 = 3;1 < k1 < m1, 1<k2<m2;1< l1tl2 < n
h1 = 1,h2 = 0; i1 = 1,i2 = 0; j1 = 2,j2 = 3;1 < k1 < m1,
1 < k2 <m2;1 < l1,l2 < n
2<h1<L,h2=h1 — 1;i1 = i2 = 0;j1 = 0,j2 = 2;
1 < k1, k2 < m1; 1 < l1,l2 < n
2<h1<L,h2=h1 — 1;i1 = i2 = 0; A = A = 2;
1 < k1, k2 < m1; 1 < l1,l2 < n
2<h1<L,h2=h1 — 1;i1 = i2 = 1;j1 = j2 = 2;
1 < k1, k2 < m1; 1 < l1,l2 < n
1<h1<L,h1= h2; h = i2= 0;j1 = 0,}2 = 2;
1 < k1, k2 < m1; 1 < l1,l2 < n
0<h1<L,h1= h2; h = i2= 0;j1 = 1,}2 = 3;
1 < k1,k2 <m2;1 < l1, l2 <n
1 < h1 < L,h1 = h2;i1 = i2 = 0;J1 = 2,¡2 = 0;
1 < k1, k2 < m1; 1 < l1,l2 < n
0 < h1 < L,h1 = h2;i1 = i2 = 0;J1 = 3J2 = 1;
1 < k1,k2 <m2;1 < l1, l2 <n
1 < h1 < L,h1 = h2;i1 = 0,i2 = 1; j1 = 0 or 2,j2 = 2; 1 < k1, k2 < m1; 1 < l1,l2 < n
h1 = h2 = 0;i1 = 12 = 0J1 = ]2 = 1,1 < k1,k2 <m.2,1< h.h < n h1 = h2 = 0; 11 = 12 = 0;j1 = j2 = 3,1 < k1, k2 <m.2,1< l2<n 1 < h1 < L — 1,h1 = h2;i1 = i2 = 0,j1 = j2 = 0,1 < k1, k2 < m1,
3 Steady State Analysis
Let n — (n0,n1, ...,nL) denote the steady state probability vector of the generator
A — Ao + Ai + A2 —
Fo Fi
AI
F4 F3 AI
F2 F3
F4 F3 AI
f4 f5
nA — 0,ne — 1.
In the above,
. ie,
(1)
(AI, \ AL
m.2n m2n
Fo(k,l) —
k —1,1 —2 k — 2,1 — 4 otherwise,
(9'S®Do-(A + V)Im2n+Im20Di k — 1,1 — 1
r1Im2n + Q'S0p ®In k — 1,1 — 2
ylm2n k — 2,1 — 1
S®Do-(A + Y)Im2n + S0p®In+Im2®Di k — 2,l — 2
,Fi(k,l) —
0
F2(k,l) —
(9T°I3 ® In T°/S ® In <pT0p®In 0
9T®Do-(A + r1 + S)Imin + Imi ® Di k — 1 — 1,
k — 1,1 — 3
^^min k — 1,1 — 5
e'S ®Do-(A + y1)Im2n + Im2 ® Di k — 2,1 — 2
G'S0a ® In k — 2,1 — 3
k — 2,1 — 4
Y^min k — 3,1 — 1
T e Do -(A + Y + S)Imm + ¡mi ® Di k — 3,1 — 3
^^min k — 3,1 — 5
Y^m2n k — 4,1 — 2
S0a ® In k — 4,1 — 3
SeDo-(A + Y)Im2U + Im2 ® Di k — 4, — 4
cpTeDo- AImin + Imi ® Di k — 5,1 — 5
0 otherwise
k —1,1 —2 k — 3,1 — 2 k — 5,1 — 2 otherwise,
,F3(k,l) —
v.
F4(k,l) —
'8T°a ® In T0a ® In $T0a ® In 0
k —1,1 —3 k — 3,1 — 3 k —5,1 —3 otherwise
,Fs(k,l) —
6T ®D0-(V + S)Imin + Imi 0 D± k = T-H = rH
k = 1,1 = 3
^^min k = 1,1 = 5
ffS ®D0- VIm2n + Im2 0 D± k = 2,1 = 2
9'S0a 0 In k = 2,1 = 3
k = 2,1 = 4
Y^min k = 3, = 1
T®Do-(y + 0)Imin+Imi0D1 k = 3, = 3
^^min k = 3, = 5
Y^m2n k = 4,1 = 2
S0a 0 In k = 4,1 = 3
S®D0- YIm2n + ¡m2 0 Di k = = 4
$T®D0 + Imi 0 Di k = 5, = 5
0 otherwise
v
with dimensions of F0,F1,F2 be 2m2nx2m2n, 2m2n x (3m1 + 2m2)n,(3m1 + 2m2)n x 2m2n respectively. F3, F4 and F5 are square matrices of order (3m1 + 2m2)n.
The LIQBD description of the model indicates that the queueing system is stable (see Neuts [9]) if and only if the left drift exceeds that of right drift. That is,
nA0e < nA2e. (2)
The vector n cannot be obtained directly in terms of the parametres of the model. The inequality (2) is simplified in(5) below. From (1)we get
ni = Ui-iUi-i, 1<i<L (3)
where
U0 = -Fi(F3+UiF4)-1
= (-A(F3 + Ui+1F4)-1 for 1<i<L-2 1 l-AF-1 for i = L-1.
From the normalizing condition ne = 1 we have
noti]-11 n{=0 % + l)e = 1. (4)
The inequality (2) gives the stability condition as
n0[(I(2m2) 0 DJe + Yl{l0> U)=0 Uj(hm1 + 2m2 0 »M < n0[[e1(2)(9'S0p 01) + e2(2)S°p 0 Oje^n) + nj=o 11^(5)9'S0 a 0 I) + e4(5)(S0a®I)]e(m2n)]. (5)
Let x be the steady state probability vector of Q.We partition this vector as x = (x0,x1,x2 ...), where x0 is of dimension n(1 + 3m1L) and xx,x2,... are each of dimension n(2m2 + (3m1 + 2m2)L) . Under the stability condition, we have Xi = x1Rl~1, i > 2, where the matrix R is the minimal nonnegative solution to the matrix quadratic equation
R2A2 + RA1 +A0 = 0 and the vectors x0 and xxare obtained by solving the equations
x0B0 + x1B1 = 0 (6)
x0C0 + x1(A1+RA2) = 0 (7)
subject to the normalizing condition
x0e + x1(I -R)-1e = 1 (8)
3.1 Analysis of service time of a type I customer
The duration of service of a type I customer is a phase type distribution with
representation (a',5i) where the underlying MC has state space {(i,j,k): i — 0,j — 0 or 2,1 <k < mi} u {(i,2,k): i — 1,1 < k < m-i)} u {*} where i denotes the status of the protection clock, j, the status of the server, k, the service phase and *, the absorbing state indicating service completion. The infinitesimal generator is
S, =
Si 0
Ç0 0
Vlmi
T-(y + 5)1, 0
mi
mi
simi
<PT
and 5° =
8T — (v + S)Imi SI,
, where, Si —
0
The initial probability vector is a' — [0 a 0 matrix of order 1xmi.
Thus the service time distribution of a type I customer is Ph(a',Si) of order 3min
9T°-
], , where 0 is a zero
3.2 Analysis of service time of a type II customer
The duration of service of a type II customer turn out to be a phase type distribution (P',S2) where the underlying MC has state space {(i,j):i = 1 or 3,1 < j < m2} u {*} where i denotes the status of the server, j, the service phase and *, the absorbing state indicating service completion. The infinitesimal generator is
S7 =
\S2 V],where,S2 — ^ , and S° — [^fl
[o 0 r , 2 \yim2 s-Yim2 2 [S0 ]
The initial probability vector is P' — [0 a ] , where 0 is a zero
matrix of order 1 xm2. Thus we have the service time distribution of a type II customer is
Ph(fi',S2) of order 2m2n.
4 Waiting time analysis 4.1 Type I Customer
To find the waiting time of a type I customer who joins for service at time t, we have to consider different possibilities depending on the status of server at that time. Let W(t) be the waiting time of a type I customer who arrives at time t and W*(s) be the corresponding LST. Case I
Suppose that Ei denote the event the system is in the state (0,v),1 <v<n when the tagged type I customer arrives. Let W*(s/Ei) denote the corresponding LST.Then
W*(s/Ei) — 1
Case II
E2 be the event that the system is in the state (ni,a, 0,0,u,v),ni > 0,1 < a < L — 1,1 < u < mi, 1 < v < n, when the tagged customer arrives the system. In this case the waiting time is the sum of the residual service time of the type I customer in service when the tagged customer arrives and service time of a — 1 remaining type I customers. Let W*(s/E2) represent the corresponding conditonal LST. Then
w*(s/e2) — (eu(3mi)(sI—SiyiS0)(a'(sI—SiyiS°y-\
Case III
E3 denotes the event: the system is in the state (ni,a,0,2,u,v),ni > 0,1 < a < L — 1,1 < u < mi, 1 < v <n, when the tagged customer arrives the system. In this case the waiting time is the sum of the residual service time of the type I customer in service when the tagged customer arrives and service times of a — 1 remaining type I customers. With W*(s/E3) as the corresponding conditonal LST, we have
W*(s/E3) — (emi+u(3rni)(sl — Si)-iS^)(a'(sI — Si)-iS^)a-i.
Case IV
E4 denotes the event: the system is in the state (n1, a, 1,2,u,v),n1 > 0,1 < a < L — 1,1 < u < m1,1 < v <n, when the tagged customer arrives the system. In this case the waiting time is the sum of the residual service time of the type I customer in service when the tagged customer arrives and service times of a — 1 remaining type I customers. Let W*(s/E4) represent the corresponding conditonal LST. Then
W*(s/E4) = (e2mi+u(3rni)(sl — Si)-1Sl>)(a'(sI — Si)-1S°y-1.
Case V
E5 denotes the event: the system is in the state (n1, a, 0,1, u,v),n1 > 1,0 < a < L — 1,1 < u < m2,1 < v <n, when the tagged customer arrives the system. In this case the waiting time is the sum of the residual service time of the type II customer in service when the tagged customer arrives and service times of a remaining type I customers. Let W*(s/E5) represent the corresponding conditonal LST. Then
W*(s/E5) = (eu(2m2)(sl — S2)-1S°)(a'(sI — S1)—1S0)a.
Case VI
E6 denotes the event: the system is in the state (n1, a, 0,3, u,v),n1 > 1,0 < a < L — 1,1 < u < m2,1 < v <n, when the tagged customer arrives the system. In this case the waiting time is the sum of the residual service time of the type II customer in service when the tagged customer arrives and service times of a remaining type I customers. Let W*(s/E6) represent the corresponding conditonal LST. Then
W*(s/E6) = (em2+u(2m2)(sl — S2)—1S0)(a'(sI — S1)-1S°y. Thus the LST of the waiting time W*(s) =
1 E"=1 x0,v + £n!=0 Yla=1 yu=1 TJv=1 W (SIE2)Xni afifiuv + 2n1=0 YJa=1 yu=1 TJv=1 W (S/E3)
xn1,a,0,2,u,v + yn1=0 Ym=1 yu=1 ^v=1 W (s 1 E4)XniA,1,2,u,v + yn1 = 1 Ea=0 yu=1 yi=1 ^ (s/ Fz)xn1,a,0,1,u,v
+ Yi'TI1 = 1 Yia=0 yu=1 2v=1 W (S1E6)Xni,ao,3,u,v] (9)
where, d =
V" y _L yn yL — 1 ym1 yn „ I yn yL-^ ym1 yn „ I
yv=1 x0,v + yn1=0 ya=1 yu=1 yv=1 ■An1,a,0,0,u,v + yn1=0 ya=1 yu=1 yv=1 ■An1,a,0,2,u,v + yn yL—^ ym1 yn yn1=0 ya=1 yu=1 yv=1
y _i_ yn yL—1 ym2 yn v _i_ yn yL—1 ym2 yn y
Xn1,a,1,2,u,v + yn1 = 1 ya=0 yu=1 yv=1 Xn1,a,0>1>u>v + yn1 = 1 ya=0 yu=1 yv=1 Xn1,a,0,3,u,v
4.2 Type II customer
To find the LST of the waiting time distribution of a type II customer, we have to compute certain distributions. We proceed to such computations.
Definition 1 Consider the duration of time with p type I customers in the system at a service commencement epoch of type I customers until the number of type I customers become zero for the first time, we call this a p-cycle, denoted by Bp.
4.2.1 Distribution of a p-cycle
This is a phase type distribution with representation (yp, T1) where the underlying Markov chain has state space {(i,j, k,l):1 < i < L,j = 0,k = 0 or 2,1 < I < m1} U {(i,j, k,l):1 < i < L,j = 1,k = 2,1 < I < m1} U {*} and i,j, k, I and * respectively denote the number of type I customers in the system, the status of the protection clock, the status of the server, the service phase and the absorbing state indicating that the number of type I customers become zero. The infinitesimal generator T1 of Bp (t) has the form
Ei AIn E2 Ei
AL
E
T =
where
E0 0
0
T =
mi
Ei
E E
mi
GT — (A + v + S)Imi YI 0
0 8T0a 0 0 T0a 0 L0 pT0a 0
Ti Ti0 0 0i
,whereT1 =
vm
T — (A + y + S)Im± 0
E =
SIm± pT — AI.
mi
E =
vm
T—(y + s)imi 0
Slm1 SIm± ( T
GT — (v + S)Im1
ym 0
G T0 and E0 = T0 ( T0
The initial probabilty vector is
- 0 y' 0 ••• 0
where 0 s a zero matrix of order 1 x 3m1, with y' = [ and 0 isa zero matrix of order 1xm1.
0 a 0
,1<p <L ], 1 < p < L is in the pth position
We can compute the LST of the length of the busy period as Yp(sI — T) 1T1
4.2.2 LST of the busy cycle generated by type I customers arriving during the service time of a type II customer
Theorem 1
The LST of the busy cycle generated by type I customers arriving during the service time of a type II customer is given by
Bc,(s) = P'[(s + A)I — 52]-152) + Yh~=\ YP(sI — T1)-1T10Ar'l3'[(s + A)I — S2]-(v+1)S° +
yl(si — w^T)
p'[A-1((s + A)I — S2)]-l[/ — ¿[(s + A)I — S2]-1r1[(s + W — S2]-1S2)
(10)
Proof.
The proof is given in the appendix.
4.2.3 LST of the busy period of type I customers generated during the service time of a type II customer
Theorem 2
The LST of the busy period generated by type I customers arriving during the service time of a type II customer is given by
Bl(s) = p[M — S2]-1S2) + T,Lp-=\ YP(sI — T1)-1T°^APp'[AI — S2]-(p+1)S0! + yl(s! —
t1)-1t1)
P'[X-1(M — S2)]-l[I — X[M — S2]-1]-1[AI — S2]-1S2) (11)
Proof.
The proof is given in the appendix.
Now, to find the waiting time of a type II customer who joins for service at time t, we have to consider different possibilities depending on the status of server at that time. Let W(t) be the waiting time of a type II customer who arrives at time t and W*(s) be the corresponding LST.
Case I
Suppose that F1 denotes the event the system is in the state (0, v),1 < v < n when the tagged customer arrives. Let W*(s/F1) denote the corresponding LST.Then
W*(s/F1) = 1
Case II
F2 be the event that the system is in one of the states (b, a, 0,0, u, v),b > 0,1 < a < L,1 < u < m1,1 < v <n when the tagged customer arrives. In this case , the waiting time is the length of the busy cycle generated by a type I customers starting from his arrival epoch plus lengths of busy cycles of type I customers generated during service times of each of the b type II customers. Let W*(s/F2) denote the corresponding LST.Then
W'(S/F2) = e(a-1)3m1+u(3Lm1)(sI — T1y1T?(BCL(s))b
Case III
F3 denotes the event the system is in one of the states (b, a, 0,2, u,v),b > 0,1 < a < L,1 < u < m1,1 < v <n when the tagged customer arrives. In this case, the waiting time is the length of the busy cycle generated by a type I customers starting from his arrival epoch plus lengths of busy cycles of type I customers generated during service times of each of the b type II customers. Let W*(s/F3) denote the corresponding LST.Then
W\s/F3) = eia-1)3mi+mi+u(3Lm1)(sI — T1r1T10(BcL(s))b
Case IV
F4 denotes the event the system is in one of the states (b,a, 1,2,u,v),b > 0,1 < a < L,1 < u < m1,1 < v <n when the tagged customer arrives. In this case, the waiting time is the length of the busy cycle generated by a type I customers starting from his arrival epoch plus lengths of busy cycles of type I customers generated during service times of each of the b type II customers. Let W*(s/F4) denote the corresponding LST.Then
W*(s/F4) = e(a-1)3mi+2mi+u(3Lm1)(sI — T1)-1T10(^cL(s))b
Case V
F5 denotes the event the system is in one of the states (b, a, 0,1, u,v),b > 1,0 < a < L,1 < u < m2,1 < v < n when the tagged customer arrives. In this case, the waiting time is the length of residual service time of the type II customer in service plus length of the busy period generated by type I customers arriving during the service time of the type II customer in service plus lengths of busy cycles of type I customers generated during service time of each of the b — 1 type II customers. Let W*(s/F5) denote the corresponding LST.Then
W*(s/Fs) = eu(2m2)(sl — S2)-1S°2BL(s)(BcL(s))b-1
Case VI
F6 denotes the event the system is in one of the states (b,a,0,3,u,v) when the tagged customer arrives. In this case the waiting time is the length of residual service time of the type II customer in service plus the length of the busy period generated by type I customers arriving during the service time of the type II customer in service plus lengths of busy cycles of type I
customers generated during service time of each of the b — 1 type II customers. Let W*(s/F6) denote the corresponding LST.Then
W'(s/F6) = em2+u(2m2)(sl — S2)-1S2lBL(s)(BCL(s))b-1 Thus the LST of the waiting time W*(s) =
YH=1 x0,v + £b=0 Yia=1 £u=1 £v=1 W (S/F2)xba,0,0,uv + £b=0 Yia=1 £u=1 £v=1 W (S/F3)
xb,a,0,2,u,v + ybb=0 £a=1 £u==1 ^v=1 W (S/F4)Xba,1,2,u,v + Yi'b = 1 £a=0 Eu^ ^v=1 ^ (s /
Fs)xb,a,0,1,u,v
+ Hbb=1 lLa=0 K121 12=1 W (s/F6)xb a,0,3,u,v (12)
5 Expected number of interruptions during a single type I service
5.1 Distribution of duration of time till interruptions occur during a single type I service
Consider the Markov process, x1 = (N(t),J(t),K(t)), where N(t) denotes the number of interruptions upto time t, ] (t), status of the server (providing normal or interrupted service) and K(t), the service phase at time t. The state space of the process is given by {(0,2, k)/1 < k < m^ u {(i,j,k)/i > 1,j = 0 or 2,1 < k < m1} u {^ u {*2} where ^ denotes the absorbing state indicating the service completion and *2 denotes the absorbing state indicating the realization of protection. The infinitesimal generator of the process is given by
U =
0 0 0 0 0 0 0
Se(m1) f0 T — (y + S)Im± Ylmi 0 0 0
Se(m1) QT0 0 6T — (r1 + S)Imi Vlmi 0 0
Se(m1) f0 0 0 T — (y + S)imi Ylmi 0
Se(m1) QT0 0 0 0 6T — (r1 + S)Imi Vlmi
5.2 Distribution of number of interruptions during a single type I service
Let yk be the probabaility that the number of interruptions during a single type I service is k. Then yk is the probabilty that the absorption occurs from the level k for the process x1- Hence yk are given by
yo = —a(T—(Y + S)I))-1(T0 + Se)
and for k = 1,2,3,...
yk = a(T — (y + S)I)-1yI((6T — (q + 5)I)-1f]I(T — (y + S)I)-1yI)k-1(6T — (q + S)l)-1
((ST0 + Se) — r/I(T — (y + S)I)-1(T0 + Se)) (13)
Therefore, the expected number of interruptions during any particular type I customer service, b
E(i) = ^kyk= a(T — (y + S)I)-1yI((I — (9T — (q + S)I)-1qI(T — (y + S)I)-1yI))-2
k=0
(6T — (q + S)I)-1((9T0 + Se) — qI(T — (y + S)I)-1(T0 + Se)). (14)
6 Expected number of interruptions during a single type II service 6.1 Distribution of duration of time till interruptions occur during a single type II service
Consider the Markov process, x2 = (N(t),J(t),K(t)), where N(t) denotes the number of
interruptions, J (t), status of the server (providing normal or interrupted service) and K(t), the service phase at time t. The state space of the process of the process is given by {(0,3,k)/1 < k < m2} u {(i,j,k)/i > 1,j = 1 or 3,1 < k < m2} u {*} where * denotes the absorbing state indicating the service completion. The infinitesimal generator of the process is given by
u =
0 0 0 0 0 0
s0 s - Ylm2 Y m2 0 0 0
S'S0 0 S'S - r]Im2 m2 0 0
s0 0 0 s - Y m2 Y m2 0
S'S0 0 0 0 S'S - r]lm2 vm
6.2 Distribution of number of interruptions during a single type II service
Let zk be the probabaility that the number of interruptions during a single type II service is k. Then zk is the probabilty that the absorption occurs from the level k for the process x2 - Hence zk are given by
z0 = -a(S-YI))-1S0
and for k = 1,2,3,...
zk = a(S - yI)-1yK(9'S - f]I)-1f]I(S - YI)-1YI)k-1(8'S - iI)-1(8'S0 -
f1I((S-Yl)-1S°) (15)
Therefore, the expected number of interruptions during any particular type II customer service, E(i) = ZT=o kZk = a(S - yi)-1yi(i - (8'S - vI)-1vI(S - yi)-1yi)-2
(8'S - iI)-1(8'S0 - iI(S - yi)-1s0). (16)
7 Other Performance Measures
The probability that the server is idle:
Pidle = £v=1 x0,v
Mean number of type I customers in the system:
rn L m1 n rn L m2 n
Ensh = ^^ ^^ ^^ ^^ n2xn1,n2,0,0,u,v + ^^ ^^ ^^ ^^ n2xn1,n2,0,1,u,v +
n1=0 n2 = 1 u=1 v=1 n1 = 1 n2 = 1 U=1 V=1
ro L m1 n ro L m2 n
^ ^ ^ ^ n2xn1,n2,0,2,u,v + ^^ ^^ ^^ ^^ n2xn1,n2,0,3,u,v +
n1=0 n2 = 1 u=1 v=1 n1 = 1 n2 = 1 U=1 V=1
rn L m1 n
^ ^ ^ ^ n2Xn1,n2,1,2,U,V n1 = 0 n2 = 1 U = 1 V=1
Mean number of type II customers in the system:
Ensl = ^ n1Xn1®
n1=0
The fraction of time during which the system is protected:
ro
T
ro L m1 n
-nn
n1 = 0 n2 = 1 U = 1 V=1
The fraction of time the server is providing service to type I customers during WI:
b L mi n
Tih = ^ ^ ^ ^ ^ ^ ^ ^ xni,n2,0,0,u,v ni=0 n2 = 1 U=1 V=1
The fraction of time the server is providing service to type II customers during WI:
b L m2 n
mode:
^H — ^ ^ ^ ^ xn1,n2,0,1,u,v n1 = 1 n2 = 0 U=1 V=1
The fraction of time the server is providing service to type I customers in normal
ro L m1 n
T - Y Y YY x
1 nh / / / / A-ni,n2,0,2,u,v ni = 0 U2 = 1 U=1 V=1
The fraction of time the server provides service to type II customers in normal mode:
ro L m2 n
n1 = 1 n2=0 U=1 V=1
8 Analysis of a cost function
We construct a cost function based on the above performance measures.
Let
Ch: Holding cost for retaining a type I customer
CL: Holding cost for retaining a type II customer
Cp: Unit time cost of providing service with protection
Cih: Unit time cost of providing service when the server is providing service to type I customer in WI
Ca: Unit time cost of providing service when the server is providing service to type II customer in WI
Cnh: Unit time cost of providing service when the server is providing service to type I customer in normal mode
Cnl: Unit time cost of providing service when the server is providing service to type II customer in normal mode
Then the expected cost per unit time,
C = Ensh x Ch + Ensi x Ci + Tp x $Cp + Tih x + Ta x 8'Cu + Tnh x Cnh + Tni x Cni
9 Numerical Results
For the arrival process of type II customers, we consider the following two sets of matrices for D0 and Di:
1. MAP with negetive correlation (MNA)
Dn -
-0.8101 0.8101 0 0 -1.3497 0
L0 0 -40.5065J
, D1-
0 0 0 0.0810 0 1.2687 138.0761 0 2.43041
*n1,n2,1,2,U,V
*n1,n2,0,3,U,V
2. MAP with positive correlation (MPA)
-0.8101 0.8101 0 0 0 0
D0 = 0 -1.3497 0 , D± = 1.2687 0 0.0810
0 0 -40.5065 2.4304 0 38.0761
These two MAP processes are normalized so as to have an arrival rate of 1. The arrival process labeled MNA has correlated arrivals with correlation between two successive interarrival times given by -0.4211 and the arrival process corresponding to the one labelled MPA has a positive correlation with value 0.4211.
6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Ensh 1.3493 1.2748 1.2194 1.1774 1.1450 1.1193 1.0985 1.0815 1.0672 1.0552
Ensl 49.9733 19.8907 13.2051 10.3241 8.7368 7.7382 7.0548 6.5593 6.1843 5.8910
T 0.0334 0.0324 0.0318 0.0313 0.0308 0.0305 0.0302 0.0300 0.0298 0.0296
Tih 0.1298 0.1104 0.0955 0.0838 0.0746 0.0672 0.0611 0.0559 0.0516 0.0479
Tu 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863
Tnh 0.3924 0.3988 0.4032 0.4063 0.4086 0.4103 0.4116 0.4126 0.4134 0.4141
Tnl 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482
c 33.7635 31.2805 30.9839 30.9648 31.0063 31.0595 31.1111 31.1575 31.1982 31.2335
Table 1: Effect of 6:FixL = 3,6' = 0,6,X = 2,i = 0.5,8 = 1,y = 0.6 and <p = 4
Tables 1 to 6 contain the effect of different parameters on various performance measures and on the cost function when the arrival process of type II customer is MNA and tables 7 to 12 contain the effect of different parameters on various performance measures and on the cost function when the arrival process of type II customer is MPA.
Table 1 indicates the effect of the parameter 6 on various performance measures and the cost function.As 6 increases, type I customers get faster service during WI and hence Ensh decreases. Then more number of type II customers also get service and hence Ensl also decreases. Tp and Tih also decreases since the expected number of type I customers during WI decreases. As 6 increases, Ta and Tnl remains fixed due to the diminished effect of 6 on type II customers and Tnh increases due to the fact that the system stays in WI serving type I customers for lesser time and hence it stays more in normal mode serving type I customers. As 6 increases, the system cost first decreases, reach an optimal value(30.9648) corresponding to 6 = 0.4 and then increases.
< 1 1.5 2 2.5 3 3.5 4 4.5 5
Ensh 1.3572 1.1902 1.1112 1.0658 1.0366 1.0162 1.0013 0.9898 0.9808
Ensl 1.1787 x 104 12.1182 7.6530 6.1872 5.4634 5.0334 4.7491 4.5473 4.3968
T 0.1581 0.1087 0.0826 0.0665 0.0557 0.0479 0.0420 0.0374 0.0337
Tih 0.0482 0.0497 0.0504 0.0507 0.0509 0.0511 0.0512 0.0513 0.0513
Tu 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863
Tnh 0.3591 0.3702 0.3751 0.3778 0.3795 0.3806 0.3814 0.3820 0.3824
Tnl 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482
c 1.2112 x 103 34.4147 34.2737 34.2923 34.3216 34.3469 34.3673 34.3837 34.3969
Table 2: Effect of <:FixL = 3,6 = 0.7,6' = 0,6,X = 2,i = 0.5,8 = 1.5 and y = 0.6
Table 2 indicates the effect of the parameter < on various performance measures and the cost function. As < increases, the type I customers in protected mode get faster service and hence
Ensh decreases. As a result, Ensl also decreases. As expected Tp also decreases. As 0 increases, Tih and Tnh increase since Tp decreases. Ta and Tnl remains unchanged since 0 has only a small effect on low priority customers. As 0 increases, the system cost first decreases, reach an optimal value(34.2737) corresponding to $ = 2 and then increases.
5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Ensh 1.3590 1.3225 1.2883 1.2562 1.2260 1.1975 1.1706 1.1452 1.1212 1.0985
Ensl 1071.6 57.1361 29.5220 19.9883 15.1618 12.2491 10.3021 8.9100 7.8661 7.0548
TP 0.0035 0.0069 0.0102 0.0133 0.0164 0.0193 0.0222 0.0250 0.0276 0.0302
T,h 0.0865 0.0831 0.0798 0.0767 0.0737 0.0709 0.0683 0.0657 0.0633 0.0611
T„ 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863
Tnh 0.4750 0.4674 0.4599 0.4526 0.4454 0.4384 0.4315 0.4247 0.4181 0.4116
Tnl 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482
c 129.7496 29.2871 27.4764 27.4443 27.8543 28.4282 29.0719 29.7453 30.4286 31.1111
Table 3: Effect of 8: Fix L = 3,6 = 0.7,6' = 0,6, A = 2,q = 0.5, y = 0.6 and <p = 4
Table 3 indicates the effect of the parameter S on various performance measures and the cost function. As 5 increases, protection clock realizes quickly and hence Tp increases, so Tih and Tnh decreases. But Ta and Tnl remains unchanged since 5 has only a small effect on low priority customers. In this case also, as 5 increases, the system cost first decreases, reach an optimal value(27.4443) corresponding to 5 = 0.4 and then increases.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Ensh 1.1161 1.1112 1.1067 1.1025 1.0985 1.0948 1.0913 1.0880 1.0848 1.0819
Ensl 7.7025 7.5160 7.3475 7.1944 7.0548 6.9270 6.8096 6.7013 6.6012 6.5083
Tv 0.0302 0.0302 0.0302 0.0302 0.0302 0.0302 0.0302 0.0303 0.0303 0.0303
Tih 0.0663 0.0649 0.0636 0.0623 0.0611 0.0599 0.0587 0.0576 0.0566 0.0555
Tu 0.0994 0.0958 0.0924 0.0893 0.0863 0.0836 0.0810 0.0785 0.0762 0.0740
Tnh 0.4058 0.4074 0.4089 0.4103 0.4116 0.4129 0.4141 0.4153 0.4165 0.4175
Tm 0.3403 0.3425 0.3445 0.3464 0.3482 0.3499 0.3514 0.3529 0.3543 0.3556
C 31.6461 31.5012 31.3642 31.2344 31.1111 30.9939 30.8823 30.7759 30.6743 30.5772
Table 4: Effect of q: Fix L = 3,6 = 0.7,6' = 0,6,A = 2,-q = 0.5,y = 0.6 and $ = 4
Table 4 indicates the effect of the parameter q on various performance measures and the cost function. As q increases, the server turns to normal mode quickly. Hence Tnh and Tnl increase and Ensh, Ensi, Tih and Ta decrease. q has only a very small effect on Tp. The cost function decreases as q increases.
e' 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
^nsh 1.3367 1.1562 1.0535 0.9894 0.9467 0.9166 0.8945 0.8779 0.8649
Ensl 56.6142 10.2087 6.2053 4.7515 4.0095 3.5628 3.2658 3.0547 2.8973
TP 0.0464 0.0490 0.0504 0.0512 0.0517 0.0521 0.0523 0.0525 0.0527
T,h 0.0369 0.0389 0.0400 0.0406 0.0410 0.0413 0.0415 0.0417 0.0418
Tu 0.2089 0.1576 0.1260 0.1049 0.0898 0.0785 0.0697 0.0627 0.0569
Tnh 0.3182 0.3357 0.3452 0.3508 0.3544 0.3568 0.3586 0.3599 0.3608
Tm 0.3791 0.3685 0.3622 0.3580 0.3551 0.3529 0.3512 0.3499 0.3488
C 38.4471 35.5315 36.0777 36.5074 36.8103 37.0278 37.1887 37.3113 37.4072
Table 5: Effect of 6': Fix L = 3,6 = 0.7, A = 2,q = 0.8,5 = 2,y = 0.6 and $ = 4
Table 5 indicates the effect of the parameter 8' on various performance measures and the cost function. As expected, Ta decreases and hence Ensl and Ensh decrease, Th , Tnh and Tp increase since type I customers have high priority. As a result, Tnl decreases. As 8' increases, the system cost first decreases, reach an optimal value(35.5315) corresponding to 8' = 0.2 and then increases.
y 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Ensh 0.9997 1.0204 1.0407 1.0604 1.0797 1.0985 1.1169 1.1349 1.1525 1.1697
Ensl 4.4562 4.8646 5.3190 5.8279 6.4019 7.0548 7.8046 8.6747 9.6973 10.9167
TP 0.0301 0.0301 0.0302 0.0302 0.0302 0.0302 0.0302 0.0303 0.0303 0.0303
Tih 0.0113 0.0220 0.0324 0.0423 0.0519 0.0611 0.0699 0.0784 0.0866 0.0945
Tu 0.0160 0.0313 0.0459 0.0599 0.0734 0.0863 0.0988 0.1107 0.1222 0.1333
Tnh 0.4586 0.4485 0.4378 0.4293 0.4203 0.4116 0.4033 0.3953 0.3876 0.3801
Tm 0.3904 0.3812 0.3725 0.3640 0.3560 0.3482 0.3407 0.3336 0.3267 0.3200
c 27.0694 27.9294 28.7606 29.5661 30.3486 31.1111 31.8570 32.5901 33.3148 34.0369
Table 6: Effect of y: Fix L = 3,8 = 0.7,A = 2,-q = 0.8,5 = 2,y = 0.6 and < = 4
Table 6 indicates the effect of the parameter y on various performance measures and the cost function. As y increases, more interruptions occur during service and hence both Ensh and Ensl increases. Tp also increases in a slow rate. As y increases Th and Ta increase and Tnh and Tnl decrease since the system stays more time in interruption mode. As y increases, the cost function increases. Note the sharpness in decrease of the value of Ensl is quite pronounced. However the trend is not seen in table 4 which gives the effect of
8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Ensh 1.3471 1.2716 1.2167 1.1761 1.1451 1.1208 1.1013 1.0853 1.0721 1.0609
^nsl 343.0679 141.3074 96.1158 76.4713 65.5556 58.6331 53.8616 50.3784 47.7263 45.6412
T 0.0334 0.0324 0.0318 0.0313 0.0308 0.0305 0.0302 0.0300 0.0298 0.0296
Tih 0.1298 0.1104 0.0955 0.0838 0.0746 0.0672 0.0611 0.0559 0.0516 0.0479
Tu 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863
Tnh 0.3924 0.3988 0.4032 0.4063 0.4086 0.4103 0.4116 0.4126 0.4134 0.4141
Tnl 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482
c 63.0719 43.4206 39.2737 37.5789 36.6882 36.1497 35.7932 35.5413 35.3548 35.2114
Table 7: Effect of 8: Fix L = 3,8' = 0,6, A = 2,-q = 0.5,5 = 1,y = 0.6 and < = 4
Table 7 indicates the effect of the parameter 8 on various performance measures and the cost function. In this case also Ensh and Ensi decreases as 8 increases. But the values of Ensi is much high when the arrival process of type II customer is MPA. All other values are same as in the case of MNA. But the cost function decreases as 8 increases.
< 1 1.5 2 2.5 3 3.5 4 4.5 5
Ensh 1.3571 1.1900 1.1141 1.0711 1.0436 1.0244 1.0104 0.9996 0.9911
Ensl 4.4374 x 104 90.9874 58.6062 47.8674 42.5211 39.3245 37.1995 35.6852 34.5516
Tp 0.1581 0.1087 0.0826 0.0665 0.0557 0.0479 0.0420 0.0374 0.0337
Tih 0.0482 0.0497 0.0504 0.0507 0.0509 0.0511 0.0512 0.0513 0.0513
Til 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863
Tnh 0.3591 0.3702 0.3751 0.3778 0.3795 0.3806 0.3814 0.3820 0.3824
Tnl 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482
c 4.4699 x 103 42.3015 39.3705 38.4629 38.0309 37.7801 37.6169 37.5023 37.4175
Table 8: Effect of 0: Fix L = 3,9 = 0.7,9' = 0,6,X = 2,q = 0.5,8 = 1.5 and y = 0.6
Table 8 indicates the effect of the parameter 0 on various performance measures and the cost function. Both Ensh and Ensi decrease as 0 increases. The cost function and Ensi decreases sharply as 0 increases from 1 to 1.5. However, with further increase in 0 value does not produce that decrease in values of cost function and Ensi.
5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Ensh 1.3589 1.3214 1.2867 1.2545 1.2244 1.1965 1.1703 1.1458 1.1229 1.1013
Ensl 7.5197 x 103 411.6330 214.5271 146.4434 111.9502 91.1156 77.1729 67.1914 59.6955 53.8616
Tv 0.0035 0.0069 0.0102 0.0133 0.0164 0.0193 0.0222 0.0250 0.0276 0.0302
Tih 0.0865 0.0831 0.0798 0.0767 0.0737 0.0709 0.0683 0.0657 0.0633 0.0611
t„ 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863 0.0863
Tnh 0.4750 0.4674 0.4599 0.4526 0.4454 0.4384 0.4315 0.4247 0.4181 0.4116
Tnl 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482 0.3482
c 774.5584 64.7362 45.9761 40.0889 37.5324 36.3143 35.7588 35.5737 35.6123 35.7932
Table 9: Effect of 5: Fix L = 3,9 = 0.7,9' = 0,6, X = 2,q = 0.5, y = 0.6 and <p = 4
Table 9 indicates the effect of the parameter 5 on various performance measures and the cost function. Both Ensh and Ensl decrease as 5 increases.In this case, as 5 increases, the system cost first decreases, reaches an optimal value(35.5737) corresponding to 5 = 0.8 and then increases. Both Ensl and the cost show sharp decrease in their values when 5 moves from 0.1 to 0.2. Thereafter the decrease is not that pronounced.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
^nsh 1.1184 1.1136 1.1093 1.1051 1.1013 1.0976 1.0942 1.0910 1.0880 1.0851
Ensl 58.6679 57.2868 56.0367 54.8999 53.8616 52.9096 52.0337 51.2250 50.4761 49.7807
Tv 0.0302 0.0302 0.0302 0.0302 0.0302 0.0302 0.0302 0.0303 0.0303 0.0303
T,h 0.0663 0.0649 0.0636 0.0623 0.0611 0.0599 0.0587 0.0576 0.0566 0.0555
Tu 0.0994 0.0958 0.0924 0.0893 0.0863 0.0836 0.0810 0.0785 0.0762 0.0740
Tnh 0.4058 0.4074 0.4089 0.4103 0.4116 0.4129 0.4141 0.4153 0.4165 0.4175
Tnl 0.3403 0.3425 0.3445 0.3464 0.3482 0.3499 0.3514 0.3529 0.3543 0.3556
c 36.7438 36.4795 36.2344 36.0062 35.7932 35.5936 35.4062 35.2298 35.0634 34.9061
Table 10: Effect of q Fix L = 3,9 = 0.7,9' = 0,6, X = 2,q = 0.5, y = 0.6 and (p = 4
Table 10 indicates the effect of the parameter q on various performance measures and the cost function. Both Ensh and Ensi decrease as q increases.The cost fuction decreases as q increases.
e' 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Ensh 1.3380 1.1630 1.0642 1.0027 0.9616 0.9325 0.9111 0.8947 0.8819 0.8716
Ensl 417.6867 79.1550 49.1289 37.9315 32.0828 28.4903 26.0604 24.3078 22.9843 21.9498
TP 0.0464 0.0490 0.0504 0.0512 0.0517 0.0521 0.0523 0.0525 0.0527 0.0528
Tih 0.0369 0.0389 0.0400 0.0406 0.0410 0.0413 0.0415 0.0417 0.0418 0.0419
Tu 0.2089 0.1576 0.1260 0.1049 0.0898 0.0785 0.0697 0.0627 0.0569 0.0521
Tnh 0.3182 0.3357 0.3452 0.3508 0.3544 0.3568 0.3586 0.3599 0.3608 0.3616
Tnl 0.3791 0.3685 0.3622 0.3580 0.3551 0.3529 0.3512 0.3499 0.3488 0.3479
c 74.5558 42.4295 40.3754 39.8320 39.6251 39.5285 39.4764 39.4450 39.4244 39.4098
Table 11: Effect of 9': Fix L = 3,9 = 0.7, A = 2,q = 0.8,5 = 2,y = 0.6 and <p = 4
Table 11 indicates the effect of the parameter 8' on various performance measures and the cost function. Both Ensh and Ensl decrease as 8' increases.The cost fuction decreases as 8' increases, as it is to be expected. However, there is a sharp decrease in value of Ensi when 8' moves from 0.1 to 0.2. For higher values of 8', the initial sharpness in decrease is not seen.
y 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Ensh 1.0050 1.0251 1.0448 1.0640 1.0829 1.1013 1.1193 1.1369 1.1542 1.1711
Ensl 34.0325 37.1338 40.5924 44.4740 48.8618 53.8616 59.6115 66.2942 74.1569 83.5428
T 0.0301 0.0301 0.0302 0.0302 0.0302 0.0302 0.0302 0.0303 0.0303 0.0303
T,h 0.0113 0.0220 0.0324 0.0423 0.0519 0.0611 0.0699 0.0784 0.0866 0.0945
T„ 0.0160 0.0313 0.0459 0.0599 0.0734 0.0863 0.0988 0.1107 0.1222 0.1333
Tnh 0.4586 0.4485 0.4378 0.4293 0.4203 0.4116 0.4033 0.3953 0.3876 0.3801
Tnl 0.3904 0.3812 0.3725 0.3640 0.3560 0.3482 0.3407 0.3336 0.3267 0.3200
c 30.0298 31.1586 32.2900 33.4325 34.5961 35.7932 37.0389 38.3530 39.7616 41.3003
Table 12: Effect of y: Fix L = 3,8 = 0.7, X = 2,y\ = 0.8, S = 2,y = 0.6 and < = 4
Table 12 indicates the effect of the parameter y on various performance measures and the cost function. Both Ensh and Ensl increase as y increases. As expected, the cost increases as y increases.
Conclusion
In this paper, we considered a (M,MAP)/(PH,PH)/1 queue with non premptive priority, exponentially distributed working interruptions and protection. We analysed the distribution of service time of type I and type II customers and the distribution of a p-cycle. Also we provided LSTs of busy cycle, busy period of type I customers generated during the service time of a type II customer. For the waiting time distributions of type I and type II customers, we provided an analysis using LST and the matrix analytic method. We also performed some numerical experiments to evaluate some performance measures and also found optimal values using a cost function. Extension of the model discussed to multi-server is proposed to be taken up in a future study.
References
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Appendix
Proof of Theorem 1
Proof. Let BCL denote the length of the busy cycle generated by type I customers arriving during the service time of a type II customer , BCL (s) the LST of the length of the busy cycle and l the number of type I customers that arrive during service time of type II customer.
Then BCL = X + Bjl + ••• BlL where X denote the service time of the type II customer in service, B[ the busy period generated by jth type I customers that arrive during X, where 1 < j < I.
BCL(s) = E(e_sBc^)
= Co E(e-sB°L/X = x)P(x<X <x + dx)
= Co Zp=° E(e~sB°L/X = x,l = p)P(l = p/X = x)P(x < X < x + dx) = Co Zp=o E(e_sBc^/X = x,l = p)!L^/S'e^S°dx = Co e-(^f]'e^S02dx + Co Vf=\ e_sxYP(sI - r^ri^^ (17)
P'e^SÏdx + C0 zv=l e-sxYL(sI - T1)_1T?e--^f}'e^dx = p[(s + A)I - S2]-1S° + rp~=\ YP(sI - T^TO^C^ xpe-[(s+À)I-s2]xS°dx + i?=l yl(si - T^T^p'C^ xPe[-^>-s2]xS°dx
r xPe^+V'^dx =-- (18)
Jx=0 [(s+^)/_s2]P+1 V I
We have,
Substituting (18) in (17) , its third term
-ip=L
IZ=L Yl(sI - T1)_1Tl'App'[(s + A)I - s2]_<p+1)s°
= Yl(SI - Ti)-1T10P'Y^=l + W - ^2]]_p[(5 + A)i - S2]_1S2)
= Yl(sI - Ti)_1T1^p'[A_1[(s + X)I - 52]]_l U_1[(s + X)I - 52]]
[(s + A)/-52]_152)
(19)
= Yl(sI - T1)_1T1°p'[A_1[(s + A)I - S2]]_l[I - A[(s + A)I - S2]_1]"1
11 [(s + A)/-52]_152)
Substituting (19) in (17) gives
BCL(s) = P'[(s + A)I - S2]_1S2) + YJf=\ Yp(sl - T1)_1T1^AP^[(s + A)I - S2]_(v+1)S° +
yl(s! - T1)_1
T1)P'[A_1[(s + A)I - S2]_l[I - A[(s + A)I - S2]_1]_1[(s + A)I - S2]_1S2) (20)
H
Proof of theorem 2
Proof. Let BL denote the length of the busy period generated by type I customers arriving during the service time of a type II customer , BL(s) the LST of the length of the busy period and l the number of type I customers that arrive during service time of type II customer.
Then BL = Bl + ■■■ BlL ,where B[ denote the busy period generated by jth type I customers that arrive during X, where 1 < j < I. Proceeding as in the above proof, we get the required result.
