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ANALYSIS OF A FLEXIBLE GROUP SERVICE MAP\PH\1 QUEUEING MODEL WITH, IMMEDIATE FEEDBACK, BALKING AND RENEGING
G. Ayyappan, S. Kalaiarasi •
Department of Mathematics Puducherry Technological University Puducherry, India. [email protected], [email protected]
Abstract
Queueing models in which the services are provided in groups (or blocks or batches) have found to be very useful in real-world applications and such queues been extensively analysed in the literature. In this paper we see one such group service queueing model with balking, reneging and immediate feedback. The arrival processes is a Markovian arrival , where, the arriving customer may balk the system while the server is idle and the pool is empty. Customers are provided service in groups of varying size from 1 to the fixed constant, say, N. The service time of a batch follows the phase type distribution corresponding to the each size of the group. A group's service time is taken as the highest of the service times of each customers who make up the group. The group of customers who are dissatisfied with the service then that group will get the service immediately. Here, the feedback of a group is defined as the average of the feedback of each customers who make up the group. During the admission period the customers may renege. We calculated the steady state probabilities by using the matrix geometric method, then, by using it we computed few performance measures. We have studied the busy period and the distribution of waiting time is derived. Results are illustrated with some graphical representations.
Keywords: Markovian Arrival Process, Flexible group service, Phase type Distributions, Immediate Feedback, Balking, Reneging.
1. Introduction
Queueing models with a group service piay a vital roie in many real life and engineering systems. And these queues can be generated physically or simulated by computers. Usually in the group service models, the minimum and maximum size of group are presumed. Baiiey introduced the bulk service queueing model with fixed group size in [4]. Chaudry and Tempieton in [5] studied buik service queues in detaii. A survey paper on buik service queueing modeis by Sasikaia and Indhira [6] is noteworthy. And the authors Banerjee.A, Gupta.U, Chakravarthy.S in [7] derived the significant resuits for queues with group service and many modeis with reai-iife appiications are presented.
Neuts [8] has introduced the generai group service ruie, according to which the server wiii start the service oniy if "a" or more customers are in the queue and the highest service capacity is "b." At the service compietion epoch of a batch, if the number of customers present in the queue is iess than 'a' then the server has to wait untii'a' number of customer arrives. If the number of customers less than or equal to 'b' and greater than or equal to 'a' then the service commences immediately to existing customers. If the number of customers is greater than 'b', then oniy 'b'
customers taken in to service. And in the literature very few papers have dealt with group service with non-exponentiai service times .
The authors Brugno,D'Apice, Dudin, Manzo in [1] have examined a MAP/PH/1 queueing model with flexible group service. A predefined integer, let's say N, is typically used in the analysis of group services, and if there are less costumers in the queue than N, service is not initiated. But in this modei they predefined the batch size as N, and they assumed that the server's idie time is restricted. Even if there are fewer consumers in the queue than N, service wiii stiii start once the idie time runs out. At a service compietion moment, if there are N or more than N number of customers, the server provides service for exactiy N customers. On the other hand, if the number of customers waiting in the queue is iess than N, a admission period starts and its duration follows the PH - distribution. If the number of customers waiting reaches the value N before the admission period expires, the admission period is stopped and the service resumes with N customers. If the admission period expires before the arrival of Nth customer, then the server offers service simultaneously to the group of 'i' customers, where 'i' ranges between 1 to N — 1 or if the admission period was over and when there is no one in queue, a new admissions period begins, and the procedure is repeated.
In [1],[7] for a batch service queueing models, a size 'm' customer group's service time is assumed to be the highest of'm' identical PH- distributions which in turn a PH- distribution. The batch's service wiii be compieted when the service for the iast customer in the batch is compieted. This type of batch service models are studied in cioud and grid computing [12].
Ayyappan and Thiiagavathy in [9] have studied the MAP /PH /1 queueing model with Breakdown, Instantaneous feedback, and server vacation. And Downton in [11] by using random arrivais and a random service time distribution derived the waiting time distribution of buik service queues.
In this paper we analyse a MAP / PH /1 queueing model with flexible group service, balking, reneging and immediate feedback. Here ,the feedback of a group is defined as the average of the feedback of each customers who make up the group.
The article's next sections are organised as follows. The mathematical model is presented with a graphic depiction in section 2. In section 3 we narrated the model and we formulate the QBD matrix. We derive the Ergodicity (stability) condition and the steady state probability vector in section 4. For this model we computed some performance measures in section 5. In section 6 we did busy period analysis. In section 7 waiting time distribution is derived . In section 8 some numericai resuits with graphicai representations are iiiustrated and the conciusion is given in section 9.
2. The Narration of the Model
In this paper, Markovian arrival process is considered with depiction (D0, D1) of order n with the generator matrix D = Do + D\. Markovian arrival's fundamental rate is defined as A = nD\en, where n is the vector of stationary probability of D. Now we assume that the customers wiii get service in groups of size N, with N > 2 a fixed integer. If there are N — 1 customers in the queue and the server is idie an arriving customer wiii get a immediate service and its PH representation is denoted as (fi(N), S(N)) of order M(N) with s0N) + S(N)e = 0 which implies s0N) = —S(N)e otherwise based on the sequence of their arrival, the arriving customers are placed in the buffer. And at this moment choosing customers from the buffer at the time a service is compiete is defined as follows
Figure 1: Diagram illustrating the current model
• The batch of exactiy N consumers (say Block) begins service if at this point of time N or more customers are present in the buffer with PH representation (fi(N),S(N)) of order M(N).
• We refer to a group of consumers as a pool, if there are fewer than N customers. Then the so-caiied admission period begins at this point of time.The admission period foiiows PH distribution of which the PH representation is denoted by (a, T) of order M(0) with To + Te = 0 which impiies To = — Te. And now
- When the pooi's totai customer count equais N, the server starts providing services.
- or the admission period expires.
- If the admission period passed and the pooi has one or more but fewer than N customers, Then aii the customers in the pooi strats service immediateiy and the service of r, customers 1 < r < N — 1 foiiows PH distribution of which the PH representation is given by (fi(r),S((r)) of order M(r) with Sor) + S((r)e = 0 which impiies
S0r) = —S(r) e.
- A new admission period begins if the admission period passes with the pooi empty, and the fundamentai rate of admission period is defined as q = [a( — T)— 1 e] 1.
Fundamentai rate of service to r customers where 1 < r < N is defined as yr = [fi(r)(—Sr)—1e] 1
During the server is idie and the pooi is empty, with probabiiity b, the customer might quit ( baik) the system . During the admission period the customer may renege, which foiiows exponentiai distribution with parameter 5. We are deaiing the singie server queueing modei with immediate feedback, this indicates whether a batch of customers is satisfied after receiving
service, they leave the system with probability c otherwise the batch of customers wiii be receiving feedback service right away with probability d such that c + d = 1. Here in the group service, we presume that the group's feedback is positive if the average of the individuai feedbacks of the group is positive.
2.1. QBD process of system state Notations for our model
• 0 - the matrix Kronecker product.
• ® - the matrix Kronecker sum .
• Im - an identity matrix of m- dimension.
• e- a column matrix with each entry is 1 of appropriate dimension.
• diag{dk,k G 1,..., n} is the diagonal matrix, whose entries are enclosed in brackets.
• Fk is the row matrix of dimension k each of its entries as 0
Let
£(t) = {N(t), I(t), R(t), J(t)(R(t)), V(t) : t > 0 }
is continuous time Markov chain with state ievei independent Quasi-Birth-and-Death process , where
• N (t) indicates that how many batches are present in the system at time t, which includes the batch in service,
• I(t) indicates that how many customers are there in the pool at time t ,0 < I(t) < N — 1,
• R(t) indicates that how many customers are getting the service at time t. Note that R(t) = 0 if N(t) = 0 , as a result the admission period wiii be ongoing and 1 < R(t) < N — 1 if
N(t) > 1
• J(t)(R(t)) indicates the state of the PH process of customer admission if R(t) = 0 with
1 < J(t)(0) < M(0) or it indicates the state of the PH process of customer service process if 1 < R(t) < N with 1 < J(t)(R(t)) < M(R(t)) .
• V(t) shows the state of the Markovian arrival process with 1 < V(t) < n. £(t) has the following state space,
B = l(0) U l(k)
where,
l(0) = {(0,i,0,p,s) : 0 < i < N — 1 ; 1 < p < M(0) ;1 < s < n } and this can be simpiy written as
l(0) = {(0,i) : 0 < i < N — 1 } the PH process of admission period and phases of Markovian arrivai are understood.
For k > 1,
l(k) = {(k,i,r,p,s) : k > 1; 0 < i < N — 1; 1 < r < N ; 1 < p < M(r) ;1 < s < n }
and this can be simply written as
l (k) = {(k, i, r) : k > 1 ; 0 < i < N - 1 ; 1 < r < N }
the PH process service to r number of customers where 1 < r < N and arrival phases are understood.
The QBD process of infinitesimal matrix generation is given by
Q
/R00 B01 0 0 0 0 ■ ■ ■ ..A
B10 A1 A0 0 0 0 ■ ■ ■
0 A2 A1 A0 0 0 ■ ■ ■
0 0 A2 A1 A0 0 ■ ■ ■
0 0 0 A2 A1 A0 •••
\ .
..
The matrix Q's block matrices are given below
(B0011 B0012 0 0 0 0 0
21 22 R00 R00 R 23 R00 0 0 0 0
0 B0032 33 R00 R 34 R00 0 0 0
B00 =
00 i i-1 R00 - ii R00 B00i i+1 0
00 R N- 1 N- 2 R00 - - R00N-.1 N-1
B00n = (T + 70a) ® (D0 + bD1); B00 12 = (1 - b)D1 ® ^m(0) ;
B0021 = 5 In ® iM(0) ; R0022 = T ® (D0 - 5 In);
where,
23
® 1M(0); Booii 1 = 5In ® 1M(0);
B00
Booi i = T ® (D0 - 5i„); B00i i+1 = D1 ® ImW
0\
B01
, where
/0 0 ■ (B01 )1,0
(B01 )2,0 . (B01 )N-1,0 0 '
V
(B01 )i,0 =(Fi-1, T0 ® ® In, FN-i) for 1 < i < N - 2 and (B01 )n-1,0 = (Fn-2, 7 ® j8(n-1) ® In, em(0) ® j8(n) ® D1)
/(A1 )0,0 (A1)0,1 0 0
0 (^1)1,1 (A1 )1,2 0 0 0 (A1 )2,2 (A1 )2,3
Ai
, where
V 0 : ... (A1 )n-1,N-1/
(A1 )i,i = di'flg {S(r)+ dS0(r)p(r) ® D0; 1 < r < N,0 < i < N - 1}
and ( A1)M+1 = diag {IM(r) © Dx; 1 < r < N,0 < i < N - 2}
( 0 0 ■ ■ ■ 0\
Ao
, where (Ao)n-i,o = diag {IM(r) © Di; 1 < r < N}
A2
0
( a0 )n-1,0 0 ••• 0 \ /
((Ai )0,0 0 0 0 (ai)1,1 0 0 0 (ai)i,i
V 0 ••. ( Ai )n-1,N-1/
/0 ••• 0 cS0(1) © p(N) © In\
with (A2)i,i
^0 ••• 0 cS0(N) © P(N) © In)
B1,0 = diag {(B1,0)i,i ; 0 < i < N - 1}, where (B1,0)i,i
(cS0(1) © a © In\ CS0(N) © a © In)
3. Condition for stableness
Let us define the matrix A = Ao + Ai + Ah then
IF F \
A
V
F F
F
/IM(1) ® D1
, where F
(S(1) + dS0(1)0(1) © D0
and F
IM(N) ® DV CS0(1)P(N) © In
... cSo(N-i)0(N) © In
V S(N) + dSo(N)© Do + cSo(N)0(N) ©
It is clear that A is a square matrix which is an irreducible infinitesimal generator matrix whose order is N M(i) n + N M(2) n + ■ ■ ■ + N M(N) n . The steady-state probability vector of A is indicated by z. And the vector z is denoted as z = (zo, zi, z2, ■ ■ ■ , zN-i), where zi = (zi, z2, ■ ■ ■ , zN) , o < i < N - 1 which satisfies zA = o and ze = 1. The QBD structure exists for the Markov process. Also there exists Ergodicity (stability) criteria for our model and that it should satisfy zAoe < zA2e, which is the if and only if condition for stability of a QBD process. By resolving the following equations, the vector z can be determined. zi(S(1) + dSo(1)^(1) © Do) + zN-i( 1M(i) © Di) = o z?(S(2) + dSo(2)p(2) © Do) + zN-i( 1m(2) © Di) = o
N-1
E Z0cSo(i)?>(i) ® In + zNcSo(N)® In + (S(1) + dS0(1)® Do) + zN-1 ( 1M(n) ® Di) = 0. i=1
Simiiariy, for i ,0 < i < N — 2 we have, z1+1 (S(1) + dSo(1)^(1) ® Do) + z1 (1M(1) ® D1) = 0 z2+1 (S(2) + dSo(2)p(2) ® Do) + z2(1M(2) ® D1) = o
N—1
E zi+1cSo(i)p(i) ® In + zN+1 cSo(N)^(N) ® In + (S(1) + dSo(1)^(1) ® Do) + zN( 1M(n) ® D1) = o.
i=1
Foiiowing some algebraic calculation, the stability condition zAoe < zA2e, which is turns to
be
N N—1 N
E zN—1 (eM(r) ® D1 en) < EE zrcSo(r) ® en.
r=1 i=o r=1
After simplification the Ergodicity condition can be precisely written as \ jN < N.
3.1. Study of the Stationary Probability vector
Let x be the Q's the steady-state probability vector and it is subdivided as x = (xo, x1, x2,, ■ ■ ■). Note that xo's dimension is NMon and dimension of x1, x2, x3, ■ ■ ■, are N(M(1) + M(2) + ■ ■ ■ + M(N))n. Then x satisfies the condition xQ = o and xe = 1. Once the stability condition is met, the subvectors of x, except for xo and x1 are provided by the foiiowing equation, which corresponds to the various ievei states.
xj = x1 Rj—\ j > 2
where R represents the minimum non-negative solution of the matrix quadratic equation as R2A2 + RA1 + Ao = o, as defined by Neuts [3]. Due to the stabiiity of our system and the fact that the row sums of the sum of square matrices Ao, A1, and A2 is zero, R , the rate matrix is a square matrix with order N(M(1) + M(2) + ■ ■ ■ + M(N))n. The R matrix is derived from the above quadratic equation and aiso fuifiis RA2e = Aoe. The foiiowing equations were soived to obtain the sub vectors xo and x1.
xo Boo + x1 B1o = o
xo Bo1 + x1( A1 + RA2) = o conditioned on the normaiising state
xoeNM(o)n + x1(1 — R)—1 eN n(M(1)+M(2) + ... +M(N)) = 1.
hence, the matrix R couid be computed theoriticaiy with the reference of Latouche and Ramaswami [2] using necessary steps in the R's Logarithmic reduction aigorithm.
4. Performance Measures
• The expected number of customer biocks, inciuding the one receiving service EfcZock = Ejt=1 kxfce = x1(1 — R)—2.
• The expected number of biocks of customers exciuding the one in service
EWock = Efc=1 (k — 1)xke = EWock — 1 + xo eo.
ANALYSIS OF A FLEXIBLE GROUP SERVICE QUEUEING MODEL Volume 18, September 2023
• The expected number of customers in the pool
Epool = Ejl-)1 ixoeoi + Efc! EiN=oi EN=1 ixkime = Ejl-)1 ixoeoi + EN=oi ixi(i - R)-V
where eoi is the column vector of order NM(o)n with (i(M(o))n + i) st to ((i + i)M(o)n) th entries are i and all other entries are zeros.
and Si is the column vector of order (N(M(i) +-----+ M(N))n with (i(M(i) +-----+ M(N))n +
i) st to ((i + i)(M(i) + ■ ■ ■ + M(N))n) th entries are i and all other entries are zeros.
• The expected number of customers in the service
Eservice Efc=1 Ej=o Em=1 mxkime
= ^N=i m(xi(i - R)-1 eoi + xi(i - R)-ieim + ■ ■ ■ + xi(i - R)-%-im)
= EN=i m( EN- xi(i - R)-1 em)
where eim are all column vectors of order (N(M(1) + ■ ■ ■ + M(N))n defined as
for m = 1, ei1 has (m( Ek! Mkn) + 1) st to (i( Ek! Mkn) + M(1)n) th entries are 1 and all
other elements are zeros.
for 2 < m < N - 1, eim has (i( EN=1 Mkn) + (j-1 Mjn) + 1) st to (i( EN=1 Mkn) + (Em=i Mjn)) th entries are 1 and all other entries are zeros.
and for m = N , eiN has (i(EN=1 Mkn) + (EN-1 Mjn) + 1) st to ((i + 1)( EN=1 Mkn)) th entries are 1 and all other entries are zeros.
• The mean size of the system (mean system size or expected system size) at an arbitrary moment including the customers in service
Esystem = Ejt=i ENo1 EN=i (kN + i + m)xkime + E^l^o1 ixoeoi
NEblock + Epool + Eservice
• The mean size of the system at some random time excluding the customers in service
Esystem = Efcli ENo1 EN=i ((k - 1) N + i + m)xkime + EN-1 ixoeeoi
NEblock + Epool + Eservice
• The probability of the server is idle at some random time
Pidle = xo eo
5. Busy Period Analysis
• A busy period is defined as the period of time from when a customer first enters an empty system till the first epoch after that when the system is empty once more. Thus it is the first passing time from level 1 to level o. A busy cycle is defined as the whole first time at level o after visiting a state in any other level at least once.
• We must first define the term "fundamental period" before we can study the busy time. It is the first passing time from level k to level k - 1 for the QBD process under examination. for k > 2.
• The case where k = o, 1 correspond to boundary states must be discussed seperately.
• Note that for each level k, for k > 1, there corresponds N(M(1) + M(2) +-----+ M(N))n
states. The ordered pair (k, j) represents j th state of level k where the states are ordered in the lexicographic order.
• Let Gjj' (v, x) provides the conditional probability of the Quasi - Birth - Death process, this process commences from the state (k,j) at time t = o accesses the level k - 1 within the time x. We can alter the v transition move left and get into the the state (k - 1, j').
To proceed further, we present the combined transform
Gw (z, s) = E / e-sxdGfcfc' (v, x) ; |z| < 1, Re(s) > 0
v=1 70
and the matrix is denoted as
G (z, s) = Gfcfc/ (z, s) (1)
then (1) satisfies the equation
G (z, s) = z(si - Ai )-1 A2 + (si - A1)-1 Ao G2 (z, s)
Now G = Gkk' = G(1,0) handles the first passage timings, with the exception of the boundary states. Using the result
G = -( Ai + RA2)-1 A2.
G matrix could be determine if R matrix is already known. Otherwise G matrix could be
determine using logarithmic reduction algorithm method.
From the above discussions for the boundary levels 1 and 0 we have
G(1,0)(z,s) = z(si - A1 )-1 B10 + (si - A1 )-1 AoG(z,s)G(1,0)(z,s), G(0,0)(z,s) = (si - B00)-1 B01G(z,s)G(1,0)(z,s).
Since G, GG(1,0)(1,0) and GG(0,0)(1,0) are stochastic, using the above matrices we can calculate the below cases.
(2)
(3)
(4)
(5)
(6) (7)
6. Analysis of Waiting Time Distribution
The first passage time analysis is used in this section to analyse the distribution of a customer's waiting time when they enter the queueing line. Let W(t) be the waiting time distribution function, which takes in to account new customers joining the queue. If there are N — 1 costumers in line and the server is idle, the arriving customer wiii receive service right away, otherwise an arrival has to wait. Let Q be the state space of the absorption time of a Markov chain,
n = (*) u {0,1,2,3,•••}
The absorption state corresponds to the tagged customer wiii be getting service without waiting. The absorption state is defined as follows
(*) = {(0, N — 1)}.
The ievei state 0 is represented as foiiows,
0 = {(0,i);0 < i < N — 2}
H1 = - f<G (z, d s s)|z=1,s= =0e = -[ A1 + A0( i + G)]- 1e
H2 = 9 rt -IT G (z, d z s)|z=1,s= =0e = -[ A1 + A0( i + G)] -1A2 e
H11,0) = 9 r< -5- GG (z, d s s)(1,0) |z= 1,s=0 e = -[ A1 + A0 G]-1 ( A0H1 + e)
H21,0) = 9 ri -—GG (z, d z s)(1,0) lz= 1,s=0 e = -[ A1 + A0 G]-1 ( A0H 2 + B10 e)
H1°,0) = 9 ri - G (z, s)(0,0) |z= 1,s=0 e = -B001[B01H 11,0) + e]
h2°,0) = - ^ (z, d z s)(0,0) lz= 1,s=0 e = -B-1 [B01H 21,0) ]
the level state for p where p > 1 is given by
l(P) = {(P,l,r,k) : p > 1 ; o < l < N - 1; 1 < r < N; 1 < k < M(r) } The absorbing Markov chain's transition matrix Q is given by
\
Q
o o o o o o
Uo Wo o o o o
U1 W2 W1 o o o
o o W3 W1 o o
o o o W3 W1 o
\.
entries of the above matrix are as follows,
Uo
[Fn-2]t
Ui = [[F((N-1)N)]T, cSo(1) © a, ••• , cSo(N) © a]
(T + To a
Wo
SI
m(o)
o
T SI
SI
M(o)
M(o)
o o
T SI
M(o)
oo oo oo
SI
M(o)
T SI
M(o)
o
/(W? )i,i
SIM(o) T - SIM(o)/
W2
o (W?)2,2
o
(W2)N-2,N-2
o
\ o •••
where (W2)i,i for o < i < N - 2, (W2)
W1
((Wi )o,o o •••
o (Wi )i,i ■ ■ ■ o o (W1 )2,2
cSo(1) © a
^cSo(N) © a)
\
, where
\ o o (Wi )n-i,N-1 /
(W1 )i,i = diag {S(r) + dSo(r)j8(r);1 < r < N,o < i < N - 1}
W3
((W3 )o,o o o
o (W3 )i,i o o o (W3 )2,2
\
\ o ... (W3 )n-1,N-1/
(o ... o cSo(1) ©P(N)\
with (W3)i
yo ••• o cSo(N) ©j8(N7
o
o
o
o
o
o
o
We begin by calculating the system's state, stationary probability distribution (that is, how many customers were in the system ) as observed by the tagged client at the moment of arrival. It is denoted by y(o) = (yo (o), y1(o), y2(o), •••) and the system's state conditional probabiiity distribution under tagged customer's arrivai can be used to determine it, using x(o) = (xo(o), x1 (o), x2(o), ■ ■ ■) by the following method
yo (o) = xo(o)( )
Di en
yj (o) = xj (o)( IN2(M(1)+ M(2)+-----+M(N)) ©-J1)
where A indicates the basic (fundamental) rate of Markovian arrival process. Now define y(t) = (y* (t),yi(t),y2(t),y3(t), ■ ■ ■), where yo is of dimension (1 x NM(o)) and yi (t) for i > 1 is of dimension (1 x (NM(1) + NM(2) + ■ ■ ■ + NM(N))). The elements of Y(t) represents the probability of the CTMC with generator Q is in the respective state of ievei i at time t. The probability that the tagged customer is in the absorbing state is given by y* (t). Thus we have W(t) = y*(t), for aii t > o. From the differential equation y'(t) = y(t)Q we have,
1
y* (t) = E yi (t)U
j=o
yo (t)= yo (t)Wo + yi (t)W2
yi(t) = yt(t)Wi + yi+1 (t)W3, for i > 1.
The row vector ty(s) provides the Lapiace-Steeitjes transform (LST) of the first passage through ievei 1. By Neuts,M.F in [3], we get
TO
#0 = E yr (o)[(si - Wi)-1W3 ]i-1 i=1
We use q>(i,s) to represent the LST of the absorbing time to the state {*} when the process begins at ievei i = o, 1. Using Q we have,
<p(o,s) = (sI - Wo)-1Uo (8)
<p(1,s) = (si - W1 )-1W2^(o,s) + (si - W1 )-1U1. Consequentiy, the LST for the waiting time distribution is
W (s) = yo (o) <p(o, s) + ^(s) <p(1, s). (9)
7. Numerical Results
In this section, we use graphical representations of the numerical values to investigate the model's nature. Where the numerical values for arrival process, admission period and service process were referred by Chakravarthy in [21]. Numerical values for Markovian arrival process,
Exponential Arrival (E-A)
Do = (-1), Di = (1)
Erlang Arrival (Er-A)
Do
-2 2 0 -2
Di
0 0 2 0
Hyper-Exponential Arrival (Hyp-A)
Do
-1.90 0 0 -0.19
Di
MAP-Negative Correlation Arrival (MNC-A)
/-1.00243 1.00243 0 D0 = I 0 -1.00243 0
\ 0 0 - 225.797y
1.710 0.190 0.171 0.019
Di
0
0.01002 v 223.539
0
0.99241 2.258
• MAP-Positive Correlation Arrival (MPC-A)
/-1.00243 1.00243 0 D0 = I 0 -1.00243 0
\ 0 0 - 225.797^
Numerical values for Phase type admission period.
• Exponential admission period (E-AP)
D1
0
0.99241 2.258
« = a), t=(-1)
Erlang admission period (Er-AP)
« =
(-1, 0), T = (-02 -2)
Hyper-Exponential admission period (Hyp-AP)
« = (0* 0*2), T = -»28)
0
0.01002 223.539
We assume that the numerical values for Phase type distributions for service times to m customers where 1 < m < N are all of exponential distributions. That is all of (,8m, S(m)) are exponential distributions irrespective of size. In all the examples we assumed, the arrival rate A = 1, the admission period rate n = 3, the service rate 7 = 6. The numerical value of the service time is taken as
Exponential (E)
ßm = (1), Sm = (-1) V 1 < m < N
Illustrative Example: 8.1.
We have illustrated the effect of the rate of renege in counter to the mean size of the system in the upcoming figures 2 and 3. We assume A = 1, q = 3, 7 = 6, b = 0.5, c = 0.6, d = 0.4 and we amplify the renege rate such that the values leaves the system to be stable. We execute the example for batch size N = 2,3,4.
In Figure 2 we fixed the arrival to follow exponential distribution and we assume the admission periods to foiiow exponential, Eriang and hyper-exponentiai distribution respectively. We observed that by amplifying the renege rate the system size decreases. We also noticed that the system size decreases siowiy in exponential and Eriang whereas quickly in hyper-exponentiai distribution.
In Figure 3 we fixed the arrival to follow Erlang distribution and we assume the admission periods to follow exponential, Erlang and hyper-exponential distribution respectively.
We observed that by amplifying the renege rate the system size decreases. We also noticed that the system size decreases moderately in aii three admission periods.
Illustrative Example: 8.2.
We have analysed the hyper-exponentiai arrival with exponential admission period case in the following figures 4 and 5. We assume A = 1, b = 0.5, c = 0.6, d = 0.4 and increase the renege rate, admission period rate and service rate such that the values leaves the system to be stable. We execute the example for batch size N = 2,3,4.
In Figure 4 we fixed the service rate as 7 = 6 and we amplify both the admission period rate and the renege rate against the mean system size. We observed that by amplifying the renege rate and admission period rate the mean system size decreases gradually.
In Figure 5 we fixed the admission period rate as q = 3 and we amplify both the rate of service and the renege against the mean size of the system. We observed that by amplifying the rate of renege and service the mean size of the system decreases and it faiis down rapidiy.
Illustrative Example: 8.3.
We have anaiysed the MAP-Positive Correiation Arrivai with exponentiai admission period case in the following figures 6 and 7. We assume A = 1, b = 0.5, c = 0.6, d = 0.4 and increase the renege rate, admission period rate and service rate such that the values leaves the system to be stabie. We execute the exampie for batch size N = 2, 3, 4.
In Figure 6 we fixed the service rate as 7 = 6 and we amplify both the admission period rate and the renege rate against the mean system size. We observed that by ampiifying the renege rate and admission period rate the mean system size decreases siowiy.
In Figure 7 we fixed the admission period rate as q = 3 and we amplify both the rate of service and renege against the mean size of the system. We observed that by ampiifying the rate of renege and service, the mean size of the system decreases and it fails down moderately.
1 -•- N = 4
0.8 \ -m- N = 3
-•- N =2
0.6 \\ -
0.4 vvx -
0.2 1 -
5 10 15 Renege rate
(a) E-A and E-AP
20
25
a
te
^
fy
^
e
x;
0.8
0.6
0.4
0.2
-•- N =4
-m- N =3
-•- N =2
5 10 15 Renege rate
(b) E-A and Er-AP
20
25
1
0
0
a
iy
<u £
o
• Eh 'ft
a
to
5 10 15 Renege rate
(c) E-A and Hyp-AP
i —N = 4
0.8 ■ \ N = 3
—•— N = 2
0.6 \\ -
0.4 v\V -
0.2 i -
20
25
Figure 2: Renege rate (vs) Expected system size -Exponential Arrival
0
N =4
N =3
N =2
5 10 15 Renege rate
(a) Er-A and E-AP
20
25
N =4
N =3
N =2
5 10 15 Renege rate
(b) Er-A and Er-AP
20
25
0
0
Renege rate (c) Er-A and Hyp-AP
Figure 3: Renege rate (vs) Expected system size -Erlang Arrival
Figure 4: (Renege rate(i) and Admission period rate(n) (vs) Esystem)
Figure 5: (Renege rate(i) and Service rate(Y) (vs) Esystem) [Hyper exponential arrival with Exponential Admission period]
Figure 6: (Renege rate(S) and Admission period rate(y) (vs) Esystem)
service rate o renege rata
Figure 7: (Renege rate(S) and Service rate(y) (vs) Esystem) [MAP Positive correlation Arrival with Exponential Admission period]
8. The conclusion
In this paper we studied a group service queueing model with arrivals happen according to a Markovian arrival process in which arrivals may baik or renege the system. The service follows Phase-type distributions in which size of the group may vary and on depending the size of the group, that is, the number of customers getting service, the service time owns different Phase-type distribution representations. If any group of customers wouid iike to receive feedback service, they wiii receive it immediately. The busy period analysis was done and waiting time distribution was computed. Using the Numericai vaiues of arrivai and service times, we compared the mean size of the system counter to renege rate with different batch sizes , which is represented graphically. This model can be extended with various catastrophes on servers, which is currently being probed.
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