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Finite buffer vacation models under E-limited with limit variation service and Markovian arrival process
Operations Research Letters 34 (2006) 539 – 547
Operations Research Letters www.elsevier.com/locate/orl
Finite buffer vacation models under E-limited with limit variation service and Markovian arrival process A.D. Banika , U.C. Guptaa,∗ , S.S. Pathakb a Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India b Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology, Kharagpur 721302, India
Received 10 February 2005; accepted 11 August 2005 Available online 27 September 2005
Abstract We consider a finite-buffer single server queue with single (multiple) vacation(s) and Markovian arrival process. The service discipline is E-limited with limit variation (ELV). Several other service disciplines like, Bernoulli scheduling, nonexhaustive and E-limited service can be treated as special cases of the ELV service. © 2005 Elsevier B.V. All rights reserved. Keywords: Finite buffer queue; Markovian arrival process; Vacations; Limited service discipline
1. Introduction Queueing systems with vacations have found wide applications in the modelling and analysis of computer and communication networks, and several other engineering systems. Vacation models are distinguished by their scheduling disciplines, that is, the rules governing when a service stops and a vacation begins. Several service discipline in combination with vacations are possible, e.g., exhaustive, limited, gated, exhaustive limited (E-limited), gated limited (G-limited), etc. In fact, there is an extensive amount of literature ∗ Corresponding author. Tel.: +91 3222 283654.
E-mail addresses: [email protected] (A.D. Banik), [email protected] (U.C. Gupta), [email protected] (S.S. Pathak). 0167-6377/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2005.08.006
available on infinite and finite buffer M/G/1 type vacation models and can be found in [14,15], respectively. Traditional teletraffic analysis using Poisson process is not powerful enough to capture the correlated and bursty nature of traffic arising in the present highspeed networks, e.g., ATM networks where packets or cells of voice, video and data are sent over a common transmission channel on statistical multiplexing basis. The performance analysis of statistical multiplexers whose input consists of superposition of several packetized sources have been done through some analytically tractable arrival process viz., Markovian arrival process (MAP) introduced by Lucantoni et al. [9] where the analysis of MAP/G/1 queue with multiple vacations have been carried out. Further, Kasahara et al. [4] analyzed same queue under N-policy
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with and without vacations. Blondia [1] considered MAP/G/1/N queue with multiple vacations for exhaustive and limited service discipline. A more general study of MAP/G/1/N queue with single (multiple) vacation(s) along with setup and close-down time can be found in [13]. Further, Niu et al. [12] have extended the analysis for BMAP/G/1/N queue. In this paper, we analyze MAP/G/1/N queue where the server serves until either the system empties or a randomly chosen limit of l (0 l L) customers have been served, whichever occurs first. The server then goes for a vacation of random length of time before returning to serve the queue again. Returning from a vacation if the queue is empty then according to the acting vacation policy the server will decide whether to remain dormant and wait for a customer or to take another vacation. This type of service discipline is known as E-limited with limit variation (ELV) and earlier studied by Lamaire [6,5] for the case of M/G/1 finite and infinite buffer queue, respectively. We present an unified approach to analyze both single and multiple vacation models together and for that we define an indicator function: S = 1 gives the results for single vacation policy and S = 0 gives the results for multiple vacation policy. One may note here that the results of M/G/1/N queue with ELV service, earlier studied in [6] and the same queue with limited service studied in [8] can be obtained as special cases of our model. Finally, it may be remarked here that the analysis of MAP/G/1/N queue with limited service discipline is carried out by Gupta et al. [2].
ing a service interval (i.e., the busy period) is determined at the preceding vacation termination instant. The sequence of limits chosen at these instants are independently identically distributed random variables (i.i.d.r.vs.). By suitably choosing pl one can obtain results for several other service disciplines, including Bernoulli scheduling, exhaustive service, pure limited service and E-limited service, e.g., if we take pL = 1 and pl = 0 (0 l < L) then the service system is equivalent to E-limited service. If L = ∞ (=1) in the above assumption, the service system is equivalent to exhaustive service system (=pure limited service). Bernoulli scheduling can be thought of as a special case of ELV service discipline with L = ∞, p0 = 0 and pl = (1 − p)p l−1 , 0 < p < 1 for l = 1, 2, . . . . The input process is MAP(C, D), where C = [cij ], and D = [dij ], 1 i, j m. Let the stationary probability vector be , then (C + D) = 0, e = 1. The fundamental arrival rate of the stationary MAP is given by ∗ = De, see [9]. Let S(x){s(x)}[S ∗ ()] be the distribution function (DF) {probability density function (pdf)}[Laplace–Stieltjes transform (LST)] of the service time S of a typical customer. Similarly, V (x){v(x)}[V ∗ ()] be the DF {pdf} [LST] of a typical vacation time V of the server. The mean service [vacation] time is E(S)[E(V )]. The service, vacation-times are assumed to be i.i.d.r.vs. and each is independent of the arrival process. The traffic intensity is given by = ∗ E(S). Further, let be the probability that the server is busy. The state of the system at time t is described by the r.vs., namely
2. Description of the model Let us consider a MAP/G/1/N queue wherein the server is allowed to serve a maximum of l (0 l L) customers during each visit to the queue, i.e., the server goes for a vacation if either the queue has been emptied or l customers have been served, whichever occur earlier. On return from a vacation if the queue is empty or a limit of zero is chosen, the server immediately goes on another vacation in the case of multiple vacation policy, whereas the server remains dormant and waiting for a customer to serve in the case of single vacation policy. The limit l is a random variable whose mass function is denoted by pl (0 l L). The limit for number of customers to be served dur-
(t) =
⎧ (k, l) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0 d
if the server is serving kth (1 k l) element in the service period consisting of services of l (1 l L) or lesser number of customers, on vacation, on dormancy.
Nq (t) is the number of customers present in the queue excluding the one in service, J (t) the state of the un˜ derlying Markov chain of MAP, S(t) the remaining service time of the customer in service, V˜ (t) the remaining vacation time of the server. We define for 1 i m the joint probability densities of queue length Nq (t), state of the server (t) and
A.D. Banik et al. / Operations Research Letters 34 (2006) 539 – 547
˜ V˜ ), respecthe remaining service (vacation) time S( tively, by [k] i,l (n, x; t)x ˜ < x + x, = P Nq (t) = n, J (t) = i, x < S(t) (t) = (k, l)} ,
0 nN, x 0,
i (n, x; t)x = P Nq (t) = n, J (t) = i, x < V˜ (t) < x + x, (t) = 0} , 0 nN, x 0, i (0; t) = P Nq (t) = 0, J (t) = i, (t) = d .
541
whereas (ti ) = (k, l) (1 l L, 1 k l) indicates the imbedded point is a service completion instant of the kth customer in the present busy period consisting of services of l or lesser number of customers. In limiting case these probability distributions are for 1 j m:
[k] gj,l (n)= lim P Nq (ti )=n, (ti )=(k, l), J (ti )=j , i→∞
0 n N, 1 k l, 1 l L,
fj (n) = lim P Nq (ti ) = n, (ti ) = 0, J (ti ) = j , i→∞
0 n N .
Further, let us denote the row vectors of order 1 ×
[k] m: g[k] l (n) = gj,l (n) , f(n) = fj (n) , 1 j m. Let An (Vn ), n0 denote an m × m matrix whose (i, j )th element represents the conditional probability that n customers have been accepted during a service (vacation) time of a customer and the underlying Markov chain is in phase j at the end of and (0)=[i (0)], 1 i m, the service (vacation) time given that the underlying [k] Markov chain was in phase i at the beginning of the where i,l (n, x) denotes the arbitrary epoch probabilservice (vacation). Further, let us denote An and Vn ity that there are n customers in the queue and the N by An = N state of the arrival process is i when the server is servk=n Ak , Vn = k=n Vk , 0 nN . Observing the system immediately after an imbeding the kth customer whose remaining service time is ded point, we have the transition probability matrix x in the present busy period consisting of the sum of (TPM) P with four block matrices of the form (N+1) L(L+1) m×(N+1) L(L+1) m (N+1) L(L+1) m×(N+1)m 2 2 2 P= . (N+1)m×(N+1) L(L+1) m (N+1)m×(N+1)m (N+1)((L(L+1)/2)+1)m×(N+1)((L(L+1)/2)+1)m
As we shall discuss the model in limiting case i.e., when t → ∞ the above probabilities will be denoted by [k] i,l (n, x), i (n, x), and i (0), respectively. Let us further define the row vectors of order 1 × m [k] [k] (n, x)= (n, x) and (n, x)=[i (n, x)] l i,l
2
the service times of l or lesser number of customers. Similarly, i (n, x) and i (0) can be interpreted. 3. Queue length distributions at various epochs 3.1. Queue length distributions at service completion/vacation termination epochs Let t0 , t1 , t2 , . . . be the time epochs at which either service completion or vacation termination occurs. The state of the system at ti is defined as {Nq (ti ), (ti ), J (ti )} where Nq (ti ), (ti ) and J (ti ) are defined earlier. Therefore, (ti ) = 0 indicates that the imbedded point is a vacation termination instant,
First we will explain these block matrices for multiple vacation model. describes the probability of transitions among the service completion epochs. A typical service completion epoch will be denoted by the triplet {Nq (ti ), (ti ), J (ti )} of which we first consider the change in Nq (ti ), i.e., n (0 n N ) and the second element of (ti ) = (k, l), i.e., l (1 l L) to describe the construction of the first block of the TPM. Then the elements of can be written as follows: ⎧Q j −i+1 (l), 1 i N, ⎪ ⎪ ⎪ i − 1 j N − 1, ⎪ ⎪ ⎨ 2 l = l L, (i,l),(j,l ) = ⎪ Qcj −i+1 (l), 1 i N, j = N, ⎪ ⎪ ⎪ ⎪ 2 l = l L, ⎩ otherwise, 0lm
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where Qr (l), Qcr (l), 0 r N, 2 l L are all matrices of order l × l. 0 is the null matrix of order given in the suffix. The matrices Qr (l) and Qcr (l) are given by (Qr (l))i,j =
Qcr (l) i,j
=
Ar , 2 l L, 1 i l; j = i + 1, 0m otherwise, Ar , 2 l L, 1 i l; j = i + 1, 0m otherwise.
gives the probability of transition from any service completion epoch to the next vacation termination epochs. Vacation termination epochs are classified by the Nq (ti ), (ti ), J (ti ) of which we consider the change in Nq (ti ), i.e., n (0 n N ) and in case of vacation termination epoch (ti ) = 0. The structure of is given by ⎧ i = 0, 1l L, 1 k l, Vj , ⎪ ⎪ ⎪ 0 j N − 1, ⎪ ⎪ ⎪ , ⎪ V i = 0, 1l L, 1 k l, ⎪ j ⎪ ⎪ ⎪ j = N, ⎨ (i,l,k)(j,0) = Vj −i , 1 i N − 1, 1l L, ⎪ ⎪ k = l, i j N − 1, ⎪ ⎪ ⎪ , 1 i N, 1l L, k = l, ⎪ V ⎪ j −i ⎪ ⎪ ⎪ j = N, ⎪ ⎩ 0m otherwise. of TPM gives the probability of transition from every vacation termination epoch to the next service completion epochs. This block is of the form given below: ⎧ pl Aj −i+1 , 1 i N − 1, 1l L, ⎪ ⎪ ⎪ k=1, i−1 j N −1, ⎨ (i,0)(j,l,k) = pl Aj −i+1 , 1 i N, 1l L, ⎪ ⎪ k = 1, j = N, ⎪ ⎩ 0m otherwise. of the TPM describes the probability of transitions among vacation termination epochs. This block matrix is of the form ⎧ V , i = 0, 0 j N, ⎪ ⎨ j p0 Vj −i , 1 i N −1, i j N −1, (i,0)(j,0) = ⎪ ⎩ p0 Vj −i , 1 i N, j = N, 0m otherwise. For single vacation model and remains same as the one described above, however, there will be changes in the entries of the matrices and and
their structure are given by ⎧ pl DAj , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ pl Aj −i+1 , (i,0)(j,l,k) = ⎪ ⎪ pl Aj −i+1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0m
i = 0, 1 l L, k = 1, 0 j N − 1, 1 i N − 1, 1 l L, k=1, i−1 j N −1, 1 i N, 1 l L, k = 1, j = N, otherwise.
and
⎧ p V , i = 0, 0 j N, ⎪ ⎨ 0 j p0 Vj −i , 1 i N −1, i j N −1, (i,0)(j,0) = ⎪ ⎩ p0 Vj −i , 1 i N, j = N, otherwise. 0m Note that the factor D=(−C)−1 D represents the phase transition matrix during an inter-arrival time, for detail see [9]. Now the unknown probability vectors g[k] l (n) and f(n) can be obtained by solving thesystem
[k] of equations: g[k] l (n), f(n) = gl (n), f(n) P using GTH (Grassmann, Taksar and Heyman) algorithm given in [7, p. 123].
3.2. Queue length distribution at departure epoch In this sequel we present queue length distributions at departure epoch through the relations between distributions of number of customers in the queue at service completion and departure epochs. Let ul[k] (n) (0 n N, 1 l L, 1 k l), denotes row vector whose ith element represents steady state probability that there are n customers in the queue and phase of the arrival process is i at departure epoch of the kth customer in the service period consisting of services of l or lesser number of customers. Since u[k] (n) is proportional to g[k] l (n) and N L l l [k] n=0 l=1 k=1 ul (n)e = 1, we get g[k] (n) ul[k] (n) = N L l l i=0
l=1
[k] k=1 gl (n)e
.
(1)
3.3. Queue length distributions at arbitrary epoch To determine queue length distributions at arbitrary epoch we will develop relations between distributions of number of customers in the queue at service completion (vacation termination) and arbitrary epochs.
A.D. Banik et al. / Operations Research Letters 34 (2006) 539 – 547
Supplementary variable method has been used and for that we relate the states of the system at two consecutive time epochs t and t + t and using probabilistic arguments, we have a set of partial differential equations for each phase i. Taking limit as t → ∞ and using matrices and vector notations, [1](1)
−l
(0, x) = [1] l (0, x)C + pl (1, 0)s(x) + S (0)Ds(x), 1l L,
(2)
[k](1)
[k−1] − l (0, x) = [k] (1, 0)s(x), l (0, x)C + l 2 l L, 2 k l, (3)
− l
(4)
(5)
(6)
− (0, x) = (0, x)C +
l L l=1 k=1
− (n, x) = (n, x)C + (n − 1, x)D L [l] + l (n, 0) + p0 (n, 0) v(x), (1)
l=1
(8)
−(1) (N, x) = (N, x)(C + D) + (N − 1, x)D L [l] + l (N, 0)+p0 (N, 0) v(x), l=1
(9) L l=1
pl (0, 0),
[1] − ∗[1] l (n, ) + l (n, 0)
(13)
[k] − ∗[k] l (n, ) + l (n, 0)
(n + 1, 0)S ∗ (), + [k−1] l
(14)
∗[k] = ∗[k] l (N, )(C + D) + l (N − 1, )D,
(7)
0 = S (0)C + S
(12)
[k] − ∗[k] l (N, ) + l (N, 0)
[k] l (0, 0)
+(1−S )(0, 0)+S p0 (0, 0) v(x),
1 n N − 1,
(11)
∗[k] = ∗[k] l (n, )C + l (n − 1, )D
1l L, 1 k l,
∗ = ∗[1] l (0, )C + pl (1, 0)S () ∗ + S (0)DS (),
∗[1] = ∗[1] l (n, )C + l (n − 1, )D + pl (n + 1, 0)S ∗ (),
[k](1) − l (N, x) = [k] l (N, x)(C + D) + [k] l (N − 1, x)D,
(1)
[1] − ∗[1] l (0, ) + l (0, 0)
[k−1] (1, 0)S ∗ (), = ∗[k] l (0, )C + l
[k](1) [k] − l (n, x) = [k] l (n, x)C + l (n − 1, x)D (n + 1, 0)s(x), + [k−1] l
1n N − 1, 2 l L, 2 k l,
[1](1)
(1) (n, x)=(d/dx)[1] where l l (n, x) and (n, x)= (d/dx)(n, x). Let us define the Laplace trans∗[k] form of [k] l (n, x) and (n, x) as l (n, ) and [k] ∗ (n, ), respectively, so that l (n) ≡ ∗[k] l (n, 0) and (n) ≡ ∗ (n, 0). Multiplying Eqs. (2)–(9) by e−x and integrating w.r.t. x over 0 to ∞, we obtain the corresponding transform equations
[k] − ∗[k] l (0, ) + l (0, 0)
[1](1)
[1] (n, x) = [1] l (n, x)C + l (n − 1, x)D + pl (n + 1, 0)s(x), 1n N − 1, 1l L,
543
(10)
(15)
− ∗ (0, ) + (0, 0) L l [k] = ∗ (0, )C + l (0, 0) l=1 k=1
+(1 − S )(0, 0) + S p0 (0, 0) V ∗ (), (16) − ∗ (n, ) + (n, 0) = ∗ (n, )C + ∗ (n − 1, )D L [l] + l (n, 0) + p0 (n, 0) V ∗ (),
(17)
l=1
− ∗ (N, ) + (N, 0) = ∗ (N, )(C + D) + ∗ (N − 1, )D L [l] + l (N, 0) + p0 (N, 0) V ∗ (). l=1
(18)
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A.D. Banik et al. / Operations Research Letters 34 (2006) 539 – 547
probabilities [k] l (n, 0) and (n, 0) which are given by
Lemma 1. L l−1
[k] l (0, 0)e
l=2 k=1
=S
L
+
L N
n=0 l=1 L N
pl (0, 0)e+
l=1
n=1
[k] −1 gi,l (n) = −1 [k] i,l (n, 0) and fi (n) = i (n, 0), 0 n N, 1 l L, 1 k l, 1 i m, (22)
[l] l (n, 0)e pl (n, 0)e.
(19)
l=1
The left hand side is the mean number of entrances to the vacation states per unit of time and the right hand side is the mean number of departure from the vacation states per unit of time. Proof. Setting = 0 in (11)–(15), then postmultiplying all the equations by e and adding them, using (C + D)e = 0 and (10), after simplification we obtain the result. Theorem 3.1. E(S) =
N l L n=0 l=1 k=1 L l N
[k] l (n, 0)e [k] l (n)e = ,
(20)
n=0 l=1 k=1
E(V ) =
N
n=0 N
(n, 0)e + S (0)e (n)e + S (0)e = 1 − .
(21)
N N L l [k] where = n=0 l=1 k=1 l (n, 0)e + n=0 (n, 0)e. Employing the above relations we will determine arbitrary epoch probabilities in terms of service completion or vacation termination epoch probabilities. Setting = 0 in Eqs. (11)–(14), (16) and (17), using (22), we obtain the following relations: [1] −1 [1] l (0) = pl f(1) − gl (0) + S (0)D (−C) , 1 l L, [k−1] [k] [k] (0) = g (1) − g (0) (−C)−1 , l l l
(23)
2 l L, 2 k l, [1] [1] l (n) = l (n − 1)D + pl f(n + 1) −g[1] (−C)−1 , l (n)
(24)
1 n N − 1, 1 l L, [k] [k] (n) = (n − 1)D + g[k−1] (n + 1) l l l −g[k] (−C)−1 , l (n)
(25)
1 n N − 1, 2 l L, 2 k l,
n=0
N
L l
[k] n=0 l=1 k=1 l (n, 0)e denotes the mean number of service completion per unit of time and multiplying this by E(S) will give . Similarly, the other result can be interpreted.
Proof. Differentiating (11)–(15) w.r.t. , setting = 0 in those equations and post-multiplying by e, adding them, using (C + D)e = 0 and (10), and Lemma 1, after simplification we obtain (20). Similarly, from (16)–(18) we obtain (21). 3.3.1. Relations between queue length distributions at arbitrary and service completion (vacation termination) epochs We first relate the service completion (vacation termination) epoch probabilities, g[k] l (n) and f(n) with the
(0) =
L l
(26)
g[k] l (0) + (1 − S )f(0)
l=1 k=1
+ S p0 f(0) − f(0) (−C)−1 ,
(n) = (n − 1)D +
L
(27)
g[l] l (n)
l=1
+p0 f(n) − f(n) 1 nN − 1.
(−C)−1 , (28)
It may be noted here that we do not have such expression for [k] (N ) and (N ). However, one can l l [k] compute L l=1 k=1 l (N )e and (N )e by using
A.D. Banik et al. / Operations Research Letters 34 (2006) 539 – 547
l [k] Theorem 3.1 and is given by L l=1 k=1 l (N )e = [k] N−1 L l − n=0 l=1 k=1 l (n)e and (N )e = (1 − ) − N−1 n=0 (n)e, respectively. Though the vectors [k] (N ) and (N ) are not obtained componentwise, l L l but l=1 k=1 [k] l (N )e and (N )e are sufficient to determine key performance measures (Section 4).The unknown quantities and involve in the above expressions can be evaluated with the help of the lemma given below. One may note here that in a special case, i.e., if we assume S = 0, m = 1, C = −, D = the model becomes M/G/1/N queue with multiple vacations under varying E-limited service, and then with a little algebraic manipulation the expressions for arbitrary epoch probabilities, (23)–(28) match with those of Lamaire [6, Eqs. (30), (32)–(34), p. 365]. Also our expression for given in the following lemma matches with that of [6, Eq. (18), p. 362]. Lemma 2. (probability that the server is busy) is given by
=
N
q(N ) = −
N−1 n=0
L l
545
[k] l (n) + (n)
0 n N − 1. Let p− (n) be the 1 × m vectors whose jth components are given by pj− (n) which gives the probability that an arrival finds n (0 n N ) customers in the queue and the arrival process is in state j. The vectors p− (n) is given by p− (n) = q(n)D/∗ , 0 n N.
4. Performance measures As the state probabilities at various epochs are known, the corresponding mean queue lengths can be easily derived. For example, the average number in the queue at arbitrary epoch =Lq = N i=0 iq(i)e, the average number in the queue when the server L l [k] is busy =Lq 1 = N i=0 i[ l=1 k=1 l (i)]e, the averagenumber in the queue when the server is on
L l
[k] l=1 k=1 gl (n)e . L N L l N −1 E(S) n=0 l=1 k=1 g[k] l=1 pl f(0)(−C) e l (n)e + E(V ) n=0 f(n)e + S
E(S)
n=0
Proof. Let b { i } be the random variable denoting the length of busy{idle} period and b {i } be the mean length of a busy{idle} period, then we have N L l [k] b b n=0 l=1 k=1 l (n)e = and = . N b +i i n=0 (n)e+S (0)e Applying Theorem 3.1, and then dividing numerator and denominator by , using (22), we obtain the desired result. L
Lemma 3. P (server is in dormancy)=(0)e= pl f(0)(−C)−1 e.
l=1
Let q(n) denote the row vector of order 1×m whose ith component is the probability distributions of n customers in the queue at arbitrary epoch and state of the arrival process is i. q(n) is given by q(n) =
l L l=1 k=1
[k] l (n) + (n),
,
l=1 k=1
(29)
vacation =Lq 2 = N i=0 i(i)e. The probability of loss =Ploss = p− (N )e. Now we will obtain the LST of waiting time distri bution. Let Wq (x) = Wq1 (x), Wq2 (x), . . . , Wqm (x) , where Wqj (x) is the probability that the arrival process is in phase j at the arrival epoch of an accepted unit and the waiting time in the queue for that unit is no longer than x. The LST of the actual waiting time distribution in the queue is given by Wq∗ () =
1 S (0)D ∗ (1 − Ploss ) +
L l N−1
∗ n ∗[k] l (n, )D S ()
n=0 l=1 k=1
× V1∗ (, n + k − l) +
N−1 n=0
∗
∗
(n, )D S ()
n
V2∗ (, n + 1)
,
546
A.D. Banik et al. / Operations Research Letters 34 (2006) 539 – 547
where V1∗ (, k) =
∞
i V ∗ () Pr
n=1
i=0
V2∗ (, k) = 1, =
i
n < k
i+1
S are of dimension . Similarly, let V (x) follow a PH-distribution with irreducible representation (, T), where and T are of dimension , then the matrices An and Vn can be computed using the procedure described in [11,3]. We have conducted an experiment on the MMPP/E2 /1/20 queue with multiple vacations (vacation time follows PH-distribution) for the following input parameters: The MMPP with five states whose infinitesimal generator is ⎡ ⎤ −1.75 0.25 0.50 0.75 0.25 0.90 0.15 ⎥ ⎢ 0.60 −2.50 0.85 ⎢ ⎥ R = ⎢ 0.40 0.30 −2.10 0.75 0.65 ⎥ ⎣ ⎦ 0.90 0.44 0.30 −2.90 1.26 1.20 0.25 0.45 0.35 −2.25
n ,
n=1
k 0,
i−1 i i V () Pr n < k n ,
∞
∗
i=0
n=1
n=1
k 1, and { n ; n > 1} is a sequence of the maximum number of customers served in each busy period.
and arrival rate matrix = diag(0.889, 1.55, 0.92, 0.748, 1.19). Its MAP representation is taken as C=R− and D=. Here, m = 5, ∗ =1.0000. PHtype representation of vacation time is taken as = [ 0.7 0.3 ], −1.098 1.099 T= 0.071 −1.832
5. Computational procedure and numerical results In this section we will briefly discuss the necessary steps required for the computation of the matrices An , Vn of TPM P. The evaluation of An (Vn ), in general, for arbitrary service (vacation) time distribution requires numerical integration and can be carried out along the lines proposed by Lucantoni and Ramaswami [10]. However, when the service, vacation-time distributions are of phase type (PHdistribution), these matrices can be evaluated without any numerical integration [11, p. 67–70]. For computational purpose let S(x) follow a PH-distribution with irreducible representation (, S), where and
with E(V ) = 1.242509. For E2 service time, PH-type representation is taken as = [ 1.0 0.0 ], − S= 0.0 − with E(S) = 2.0/ and by suitably varying one can get various values of which is equal to ∗ E(S).
(a)
0.08
0.9 0.8
0.07
0.7
0.06
probability of blocking
probability that the server is busy
1
0.6 0.5 0.4 0.3
Uniform Geometric Constant Bimodal
0.2
0.05 0.04
Uniform Geometric Constant Bimodal
0.03 0.02 0.01
0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 traffic intensity
1
1.1 (b)
0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 traffic intensity
Fig. 1. (a) versus . (b) versus Ploss .
A.D. Banik et al. / Operations Research Letters 34 (2006) 539 – 547
We take the maximum value of the limit L = 5 and the following four limit mass functions are examined: (i) Uniform: pl = 16 , for l = 0, 1, 2, . . . , 5, (ii) Geometric: pl = 13 ( 23 )l−1 , for l = 1, 2, . . . and p0 = 0, (iii) Constant: pl =(l−5), (iv) Bimodal: pl =( 21 )(l− 3) + ( 21 )(l − 5), where (.) denotes Dirac delta function. Figs. 1(a) and (b) shows the effect of on and (Ploss ). It may be remarked here that due to lack of space only limited results have been presented; however, details are available with the authors. Acknowledgements The authors would like to thank the referee for his valuable comments and suggestions. The first author wishes to thank CSIR, New Delhi, India, for their financial support. References [1] C. Blondia, Finite capacity vacation model with non-renewal input, J. Appl. Probab. 28 (1991) 174–197. [2] U.C. Gupta, A.D. Banik, S.S. Pathak, Complete analysis of MAP/G/1/N queue with single (multiple) vacation(s) under limited service discipline, J. Appl. Math. Stochastic Anal. 3 (2005) 353–373. [3] U.C. Gupta, P. Vijaya Laxmi, Analysis of MAP/Ga,b /1/N queue, Queueing Syst. Theory Appl. 38 (2001) 109–124.
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[4] S. Kasahara, T. Takine, T. Suda, T. Hasegawa, MAP/G/1 queues under N-policy with and without vacations, J. Oper. Res. Soc. Jpn. 39 (1996) 188–212. [5] R.O. Lamaire, An M/G/1 vacation model of an FDDI station, IEEE J. Sel. Areas Commun. 9 (2) (1991) 257–264. [6] R.O. Lamaire, M/G/1/N vacation model with varying Elimited service discipline, Queueing Syst. 11 (4) (1992) 357–375. [7] G. Latouche, V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modelling, SIAM & ASA, Philadelphia, 1999. [8] T.T. Lee, M/G/1/N queue with vacation time and limited service discipline, Perform. Eval. 9 (3) (1988/1989) 181–190. [9] D.M. Lucantoni, K.S. Meier-Hellstern, M.F. Neuts, A singleserver queue with server vacations and a class of non-renewal process, Adv. Appl. Probab. 22 (1990) 676–705. [10] D.M. Lucantoni, V. Ramaswami, Efficient algorithms for solving non-linear matrix equations arising in phase type queues, Stochastic Models 1 (1985) 29–51. [11] M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, Johns Hopkins University Press, Baltimore, MD, 1981. [12] Z. Niu, T. Shu, Y. Takahashi, A vacation queue with set up and close-down times and batch Markovian arrival processes, Perform. Eval. 54 (2003) 225–248. [13] Z. Niu, Y. Takahashi, A finite-capacity queue with exhaustive vacation/close-down/setup times and Markovian arrival processes, Queueing Syst. 31 (1999) 1–23. [14] H. Takagi, Queueing Analysis—A Foundation of Performance Evaluation, Vacation and Priority Systems, vol. 1, NorthHolland, New York, 1991. [15] H. Takagi, Queueing Analysis—A Foundation of Performance Evaluation, Finite Systems, vol. 2, North-Holland, New York, 1993.
Operations Research Letters www.elsevier.com/locate/orl
Finite buffer vacation models under E-limited with limit variation service and Markovian arrival process A.D. Banika , U.C. Guptaa,∗ , S.S. Pathakb a Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India b Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology, Kharagpur 721302, India
Received 10 February 2005; accepted 11 August 2005 Available online 27 September 2005
Abstract We consider a finite-buffer single server queue with single (multiple) vacation(s) and Markovian arrival process. The service discipline is E-limited with limit variation (ELV). Several other service disciplines like, Bernoulli scheduling, nonexhaustive and E-limited service can be treated as special cases of the ELV service. © 2005 Elsevier B.V. All rights reserved. Keywords: Finite buffer queue; Markovian arrival process; Vacations; Limited service discipline
1. Introduction Queueing systems with vacations have found wide applications in the modelling and analysis of computer and communication networks, and several other engineering systems. Vacation models are distinguished by their scheduling disciplines, that is, the rules governing when a service stops and a vacation begins. Several service discipline in combination with vacations are possible, e.g., exhaustive, limited, gated, exhaustive limited (E-limited), gated limited (G-limited), etc. In fact, there is an extensive amount of literature ∗ Corresponding author. Tel.: +91 3222 283654.
E-mail addresses: [email protected] (A.D. Banik), [email protected] (U.C. Gupta), [email protected] (S.S. Pathak). 0167-6377/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2005.08.006
available on infinite and finite buffer M/G/1 type vacation models and can be found in [14,15], respectively. Traditional teletraffic analysis using Poisson process is not powerful enough to capture the correlated and bursty nature of traffic arising in the present highspeed networks, e.g., ATM networks where packets or cells of voice, video and data are sent over a common transmission channel on statistical multiplexing basis. The performance analysis of statistical multiplexers whose input consists of superposition of several packetized sources have been done through some analytically tractable arrival process viz., Markovian arrival process (MAP) introduced by Lucantoni et al. [9] where the analysis of MAP/G/1 queue with multiple vacations have been carried out. Further, Kasahara et al. [4] analyzed same queue under N-policy
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A.D. Banik et al. / Operations Research Letters 34 (2006) 539 – 547
with and without vacations. Blondia [1] considered MAP/G/1/N queue with multiple vacations for exhaustive and limited service discipline. A more general study of MAP/G/1/N queue with single (multiple) vacation(s) along with setup and close-down time can be found in [13]. Further, Niu et al. [12] have extended the analysis for BMAP/G/1/N queue. In this paper, we analyze MAP/G/1/N queue where the server serves until either the system empties or a randomly chosen limit of l (0 l L) customers have been served, whichever occurs first. The server then goes for a vacation of random length of time before returning to serve the queue again. Returning from a vacation if the queue is empty then according to the acting vacation policy the server will decide whether to remain dormant and wait for a customer or to take another vacation. This type of service discipline is known as E-limited with limit variation (ELV) and earlier studied by Lamaire [6,5] for the case of M/G/1 finite and infinite buffer queue, respectively. We present an unified approach to analyze both single and multiple vacation models together and for that we define an indicator function: S = 1 gives the results for single vacation policy and S = 0 gives the results for multiple vacation policy. One may note here that the results of M/G/1/N queue with ELV service, earlier studied in [6] and the same queue with limited service studied in [8] can be obtained as special cases of our model. Finally, it may be remarked here that the analysis of MAP/G/1/N queue with limited service discipline is carried out by Gupta et al. [2].
ing a service interval (i.e., the busy period) is determined at the preceding vacation termination instant. The sequence of limits chosen at these instants are independently identically distributed random variables (i.i.d.r.vs.). By suitably choosing pl one can obtain results for several other service disciplines, including Bernoulli scheduling, exhaustive service, pure limited service and E-limited service, e.g., if we take pL = 1 and pl = 0 (0 l < L) then the service system is equivalent to E-limited service. If L = ∞ (=1) in the above assumption, the service system is equivalent to exhaustive service system (=pure limited service). Bernoulli scheduling can be thought of as a special case of ELV service discipline with L = ∞, p0 = 0 and pl = (1 − p)p l−1 , 0 < p < 1 for l = 1, 2, . . . . The input process is MAP(C, D), where C = [cij ], and D = [dij ], 1 i, j m. Let the stationary probability vector be , then (C + D) = 0, e = 1. The fundamental arrival rate of the stationary MAP is given by ∗ = De, see [9]. Let S(x){s(x)}[S ∗ ()] be the distribution function (DF) {probability density function (pdf)}[Laplace–Stieltjes transform (LST)] of the service time S of a typical customer. Similarly, V (x){v(x)}[V ∗ ()] be the DF {pdf} [LST] of a typical vacation time V of the server. The mean service [vacation] time is E(S)[E(V )]. The service, vacation-times are assumed to be i.i.d.r.vs. and each is independent of the arrival process. The traffic intensity is given by = ∗ E(S). Further, let be the probability that the server is busy. The state of the system at time t is described by the r.vs., namely
2. Description of the model Let us consider a MAP/G/1/N queue wherein the server is allowed to serve a maximum of l (0 l L) customers during each visit to the queue, i.e., the server goes for a vacation if either the queue has been emptied or l customers have been served, whichever occur earlier. On return from a vacation if the queue is empty or a limit of zero is chosen, the server immediately goes on another vacation in the case of multiple vacation policy, whereas the server remains dormant and waiting for a customer to serve in the case of single vacation policy. The limit l is a random variable whose mass function is denoted by pl (0 l L). The limit for number of customers to be served dur-
(t) =
⎧ (k, l) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0 d
if the server is serving kth (1 k l) element in the service period consisting of services of l (1 l L) or lesser number of customers, on vacation, on dormancy.
Nq (t) is the number of customers present in the queue excluding the one in service, J (t) the state of the un˜ derlying Markov chain of MAP, S(t) the remaining service time of the customer in service, V˜ (t) the remaining vacation time of the server. We define for 1 i m the joint probability densities of queue length Nq (t), state of the server (t) and
A.D. Banik et al. / Operations Research Letters 34 (2006) 539 – 547
˜ V˜ ), respecthe remaining service (vacation) time S( tively, by [k] i,l (n, x; t)x ˜ < x + x, = P Nq (t) = n, J (t) = i, x < S(t) (t) = (k, l)} ,
0 nN, x 0,
i (n, x; t)x = P Nq (t) = n, J (t) = i, x < V˜ (t) < x + x, (t) = 0} , 0 nN, x 0, i (0; t) = P Nq (t) = 0, J (t) = i, (t) = d .
541
whereas (ti ) = (k, l) (1 l L, 1 k l) indicates the imbedded point is a service completion instant of the kth customer in the present busy period consisting of services of l or lesser number of customers. In limiting case these probability distributions are for 1 j m:
[k] gj,l (n)= lim P Nq (ti )=n, (ti )=(k, l), J (ti )=j , i→∞
0 n N, 1 k l, 1 l L,
fj (n) = lim P Nq (ti ) = n, (ti ) = 0, J (ti ) = j , i→∞
0 n N .
Further, let us denote the row vectors of order 1 ×
[k] m: g[k] l (n) = gj,l (n) , f(n) = fj (n) , 1 j m. Let An (Vn ), n0 denote an m × m matrix whose (i, j )th element represents the conditional probability that n customers have been accepted during a service (vacation) time of a customer and the underlying Markov chain is in phase j at the end of and (0)=[i (0)], 1 i m, the service (vacation) time given that the underlying [k] Markov chain was in phase i at the beginning of the where i,l (n, x) denotes the arbitrary epoch probabilservice (vacation). Further, let us denote An and Vn ity that there are n customers in the queue and the N by An = N state of the arrival process is i when the server is servk=n Ak , Vn = k=n Vk , 0 nN . Observing the system immediately after an imbeding the kth customer whose remaining service time is ded point, we have the transition probability matrix x in the present busy period consisting of the sum of (TPM) P with four block matrices of the form (N+1) L(L+1) m×(N+1) L(L+1) m (N+1) L(L+1) m×(N+1)m 2 2 2 P= . (N+1)m×(N+1) L(L+1) m (N+1)m×(N+1)m (N+1)((L(L+1)/2)+1)m×(N+1)((L(L+1)/2)+1)m
As we shall discuss the model in limiting case i.e., when t → ∞ the above probabilities will be denoted by [k] i,l (n, x), i (n, x), and i (0), respectively. Let us further define the row vectors of order 1 × m [k] [k] (n, x)= (n, x) and (n, x)=[i (n, x)] l i,l
2
the service times of l or lesser number of customers. Similarly, i (n, x) and i (0) can be interpreted. 3. Queue length distributions at various epochs 3.1. Queue length distributions at service completion/vacation termination epochs Let t0 , t1 , t2 , . . . be the time epochs at which either service completion or vacation termination occurs. The state of the system at ti is defined as {Nq (ti ), (ti ), J (ti )} where Nq (ti ), (ti ) and J (ti ) are defined earlier. Therefore, (ti ) = 0 indicates that the imbedded point is a vacation termination instant,
First we will explain these block matrices for multiple vacation model. describes the probability of transitions among the service completion epochs. A typical service completion epoch will be denoted by the triplet {Nq (ti ), (ti ), J (ti )} of which we first consider the change in Nq (ti ), i.e., n (0 n N ) and the second element of (ti ) = (k, l), i.e., l (1 l L) to describe the construction of the first block of the TPM. Then the elements of can be written as follows: ⎧Q j −i+1 (l), 1 i N, ⎪ ⎪ ⎪ i − 1 j N − 1, ⎪ ⎪ ⎨ 2 l = l L, (i,l),(j,l ) = ⎪ Qcj −i+1 (l), 1 i N, j = N, ⎪ ⎪ ⎪ ⎪ 2 l = l L, ⎩ otherwise, 0lm
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where Qr (l), Qcr (l), 0 r N, 2 l L are all matrices of order l × l. 0 is the null matrix of order given in the suffix. The matrices Qr (l) and Qcr (l) are given by (Qr (l))i,j =
Qcr (l) i,j
=
Ar , 2 l L, 1 i l; j = i + 1, 0m otherwise, Ar , 2 l L, 1 i l; j = i + 1, 0m otherwise.
gives the probability of transition from any service completion epoch to the next vacation termination epochs. Vacation termination epochs are classified by the Nq (ti ), (ti ), J (ti ) of which we consider the change in Nq (ti ), i.e., n (0 n N ) and in case of vacation termination epoch (ti ) = 0. The structure of is given by ⎧ i = 0, 1l L, 1 k l, Vj , ⎪ ⎪ ⎪ 0 j N − 1, ⎪ ⎪ ⎪ , ⎪ V i = 0, 1l L, 1 k l, ⎪ j ⎪ ⎪ ⎪ j = N, ⎨ (i,l,k)(j,0) = Vj −i , 1 i N − 1, 1l L, ⎪ ⎪ k = l, i j N − 1, ⎪ ⎪ ⎪ , 1 i N, 1l L, k = l, ⎪ V ⎪ j −i ⎪ ⎪ ⎪ j = N, ⎪ ⎩ 0m otherwise. of TPM gives the probability of transition from every vacation termination epoch to the next service completion epochs. This block is of the form given below: ⎧ pl Aj −i+1 , 1 i N − 1, 1l L, ⎪ ⎪ ⎪ k=1, i−1 j N −1, ⎨ (i,0)(j,l,k) = pl Aj −i+1 , 1 i N, 1l L, ⎪ ⎪ k = 1, j = N, ⎪ ⎩ 0m otherwise. of the TPM describes the probability of transitions among vacation termination epochs. This block matrix is of the form ⎧ V , i = 0, 0 j N, ⎪ ⎨ j p0 Vj −i , 1 i N −1, i j N −1, (i,0)(j,0) = ⎪ ⎩ p0 Vj −i , 1 i N, j = N, 0m otherwise. For single vacation model and remains same as the one described above, however, there will be changes in the entries of the matrices and and
their structure are given by ⎧ pl DAj , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ pl Aj −i+1 , (i,0)(j,l,k) = ⎪ ⎪ pl Aj −i+1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0m
i = 0, 1 l L, k = 1, 0 j N − 1, 1 i N − 1, 1 l L, k=1, i−1 j N −1, 1 i N, 1 l L, k = 1, j = N, otherwise.
and
⎧ p V , i = 0, 0 j N, ⎪ ⎨ 0 j p0 Vj −i , 1 i N −1, i j N −1, (i,0)(j,0) = ⎪ ⎩ p0 Vj −i , 1 i N, j = N, otherwise. 0m Note that the factor D=(−C)−1 D represents the phase transition matrix during an inter-arrival time, for detail see [9]. Now the unknown probability vectors g[k] l (n) and f(n) can be obtained by solving thesystem
[k] of equations: g[k] l (n), f(n) = gl (n), f(n) P using GTH (Grassmann, Taksar and Heyman) algorithm given in [7, p. 123].
3.2. Queue length distribution at departure epoch In this sequel we present queue length distributions at departure epoch through the relations between distributions of number of customers in the queue at service completion and departure epochs. Let ul[k] (n) (0 n N, 1 l L, 1 k l), denotes row vector whose ith element represents steady state probability that there are n customers in the queue and phase of the arrival process is i at departure epoch of the kth customer in the service period consisting of services of l or lesser number of customers. Since u[k] (n) is proportional to g[k] l (n) and N L l l [k] n=0 l=1 k=1 ul (n)e = 1, we get g[k] (n) ul[k] (n) = N L l l i=0
l=1
[k] k=1 gl (n)e
.
(1)
3.3. Queue length distributions at arbitrary epoch To determine queue length distributions at arbitrary epoch we will develop relations between distributions of number of customers in the queue at service completion (vacation termination) and arbitrary epochs.
A.D. Banik et al. / Operations Research Letters 34 (2006) 539 – 547
Supplementary variable method has been used and for that we relate the states of the system at two consecutive time epochs t and t + t and using probabilistic arguments, we have a set of partial differential equations for each phase i. Taking limit as t → ∞ and using matrices and vector notations, [1](1)
−l
(0, x) = [1] l (0, x)C + pl (1, 0)s(x) + S (0)Ds(x), 1l L,
(2)
[k](1)
[k−1] − l (0, x) = [k] (1, 0)s(x), l (0, x)C + l 2 l L, 2 k l, (3)
− l
(4)
(5)
(6)
− (0, x) = (0, x)C +
l L l=1 k=1
− (n, x) = (n, x)C + (n − 1, x)D L [l] + l (n, 0) + p0 (n, 0) v(x), (1)
l=1
(8)
−(1) (N, x) = (N, x)(C + D) + (N − 1, x)D L [l] + l (N, 0)+p0 (N, 0) v(x), l=1
(9) L l=1
pl (0, 0),
[1] − ∗[1] l (n, ) + l (n, 0)
(13)
[k] − ∗[k] l (n, ) + l (n, 0)
(n + 1, 0)S ∗ (), + [k−1] l
(14)
∗[k] = ∗[k] l (N, )(C + D) + l (N − 1, )D,
(7)
0 = S (0)C + S
(12)
[k] − ∗[k] l (N, ) + l (N, 0)
[k] l (0, 0)
+(1−S )(0, 0)+S p0 (0, 0) v(x),
1 n N − 1,
(11)
∗[k] = ∗[k] l (n, )C + l (n − 1, )D
1l L, 1 k l,
∗ = ∗[1] l (0, )C + pl (1, 0)S () ∗ + S (0)DS (),
∗[1] = ∗[1] l (n, )C + l (n − 1, )D + pl (n + 1, 0)S ∗ (),
[k](1) − l (N, x) = [k] l (N, x)(C + D) + [k] l (N − 1, x)D,
(1)
[1] − ∗[1] l (0, ) + l (0, 0)
[k−1] (1, 0)S ∗ (), = ∗[k] l (0, )C + l
[k](1) [k] − l (n, x) = [k] l (n, x)C + l (n − 1, x)D (n + 1, 0)s(x), + [k−1] l
1n N − 1, 2 l L, 2 k l,
[1](1)
(1) (n, x)=(d/dx)[1] where l l (n, x) and (n, x)= (d/dx)(n, x). Let us define the Laplace trans∗[k] form of [k] l (n, x) and (n, x) as l (n, ) and [k] ∗ (n, ), respectively, so that l (n) ≡ ∗[k] l (n, 0) and (n) ≡ ∗ (n, 0). Multiplying Eqs. (2)–(9) by e−x and integrating w.r.t. x over 0 to ∞, we obtain the corresponding transform equations
[k] − ∗[k] l (0, ) + l (0, 0)
[1](1)
[1] (n, x) = [1] l (n, x)C + l (n − 1, x)D + pl (n + 1, 0)s(x), 1n N − 1, 1l L,
543
(10)
(15)
− ∗ (0, ) + (0, 0) L l [k] = ∗ (0, )C + l (0, 0) l=1 k=1
+(1 − S )(0, 0) + S p0 (0, 0) V ∗ (), (16) − ∗ (n, ) + (n, 0) = ∗ (n, )C + ∗ (n − 1, )D L [l] + l (n, 0) + p0 (n, 0) V ∗ (),
(17)
l=1
− ∗ (N, ) + (N, 0) = ∗ (N, )(C + D) + ∗ (N − 1, )D L [l] + l (N, 0) + p0 (N, 0) V ∗ (). l=1
(18)
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A.D. Banik et al. / Operations Research Letters 34 (2006) 539 – 547
probabilities [k] l (n, 0) and (n, 0) which are given by
Lemma 1. L l−1
[k] l (0, 0)e
l=2 k=1
=S
L
+
L N
n=0 l=1 L N
pl (0, 0)e+
l=1
n=1
[k] −1 gi,l (n) = −1 [k] i,l (n, 0) and fi (n) = i (n, 0), 0 n N, 1 l L, 1 k l, 1 i m, (22)
[l] l (n, 0)e pl (n, 0)e.
(19)
l=1
The left hand side is the mean number of entrances to the vacation states per unit of time and the right hand side is the mean number of departure from the vacation states per unit of time. Proof. Setting = 0 in (11)–(15), then postmultiplying all the equations by e and adding them, using (C + D)e = 0 and (10), after simplification we obtain the result. Theorem 3.1. E(S) =
N l L n=0 l=1 k=1 L l N
[k] l (n, 0)e [k] l (n)e = ,
(20)
n=0 l=1 k=1
E(V ) =
N
n=0 N
(n, 0)e + S (0)e (n)e + S (0)e = 1 − .
(21)
N N L l [k] where = n=0 l=1 k=1 l (n, 0)e + n=0 (n, 0)e. Employing the above relations we will determine arbitrary epoch probabilities in terms of service completion or vacation termination epoch probabilities. Setting = 0 in Eqs. (11)–(14), (16) and (17), using (22), we obtain the following relations: [1] −1 [1] l (0) = pl f(1) − gl (0) + S (0)D (−C) , 1 l L, [k−1] [k] [k] (0) = g (1) − g (0) (−C)−1 , l l l
(23)
2 l L, 2 k l, [1] [1] l (n) = l (n − 1)D + pl f(n + 1) −g[1] (−C)−1 , l (n)
(24)
1 n N − 1, 1 l L, [k] [k] (n) = (n − 1)D + g[k−1] (n + 1) l l l −g[k] (−C)−1 , l (n)
(25)
1 n N − 1, 2 l L, 2 k l,
n=0
N
L l
[k] n=0 l=1 k=1 l (n, 0)e denotes the mean number of service completion per unit of time and multiplying this by E(S) will give . Similarly, the other result can be interpreted.
Proof. Differentiating (11)–(15) w.r.t. , setting = 0 in those equations and post-multiplying by e, adding them, using (C + D)e = 0 and (10), and Lemma 1, after simplification we obtain (20). Similarly, from (16)–(18) we obtain (21). 3.3.1. Relations between queue length distributions at arbitrary and service completion (vacation termination) epochs We first relate the service completion (vacation termination) epoch probabilities, g[k] l (n) and f(n) with the
(0) =
L l
(26)
g[k] l (0) + (1 − S )f(0)
l=1 k=1
+ S p0 f(0) − f(0) (−C)−1 ,
(n) = (n − 1)D +
L
(27)
g[l] l (n)
l=1
+p0 f(n) − f(n) 1 nN − 1.
(−C)−1 , (28)
It may be noted here that we do not have such expression for [k] (N ) and (N ). However, one can l l [k] compute L l=1 k=1 l (N )e and (N )e by using
A.D. Banik et al. / Operations Research Letters 34 (2006) 539 – 547
l [k] Theorem 3.1 and is given by L l=1 k=1 l (N )e = [k] N−1 L l − n=0 l=1 k=1 l (n)e and (N )e = (1 − ) − N−1 n=0 (n)e, respectively. Though the vectors [k] (N ) and (N ) are not obtained componentwise, l L l but l=1 k=1 [k] l (N )e and (N )e are sufficient to determine key performance measures (Section 4).The unknown quantities and involve in the above expressions can be evaluated with the help of the lemma given below. One may note here that in a special case, i.e., if we assume S = 0, m = 1, C = −, D = the model becomes M/G/1/N queue with multiple vacations under varying E-limited service, and then with a little algebraic manipulation the expressions for arbitrary epoch probabilities, (23)–(28) match with those of Lamaire [6, Eqs. (30), (32)–(34), p. 365]. Also our expression for given in the following lemma matches with that of [6, Eq. (18), p. 362]. Lemma 2. (probability that the server is busy) is given by
=
N
q(N ) = −
N−1 n=0
L l
545
[k] l (n) + (n)
0 n N − 1. Let p− (n) be the 1 × m vectors whose jth components are given by pj− (n) which gives the probability that an arrival finds n (0 n N ) customers in the queue and the arrival process is in state j. The vectors p− (n) is given by p− (n) = q(n)D/∗ , 0 n N.
4. Performance measures As the state probabilities at various epochs are known, the corresponding mean queue lengths can be easily derived. For example, the average number in the queue at arbitrary epoch =Lq = N i=0 iq(i)e, the average number in the queue when the server L l [k] is busy =Lq 1 = N i=0 i[ l=1 k=1 l (i)]e, the averagenumber in the queue when the server is on
L l
[k] l=1 k=1 gl (n)e . L N L l N −1 E(S) n=0 l=1 k=1 g[k] l=1 pl f(0)(−C) e l (n)e + E(V ) n=0 f(n)e + S
E(S)
n=0
Proof. Let b { i } be the random variable denoting the length of busy{idle} period and b {i } be the mean length of a busy{idle} period, then we have N L l [k] b b n=0 l=1 k=1 l (n)e = and = . N b +i i n=0 (n)e+S (0)e Applying Theorem 3.1, and then dividing numerator and denominator by , using (22), we obtain the desired result. L
Lemma 3. P (server is in dormancy)=(0)e= pl f(0)(−C)−1 e.
l=1
Let q(n) denote the row vector of order 1×m whose ith component is the probability distributions of n customers in the queue at arbitrary epoch and state of the arrival process is i. q(n) is given by q(n) =
l L l=1 k=1
[k] l (n) + (n),
,
l=1 k=1
(29)
vacation =Lq 2 = N i=0 i(i)e. The probability of loss =Ploss = p− (N )e. Now we will obtain the LST of waiting time distri bution. Let Wq (x) = Wq1 (x), Wq2 (x), . . . , Wqm (x) , where Wqj (x) is the probability that the arrival process is in phase j at the arrival epoch of an accepted unit and the waiting time in the queue for that unit is no longer than x. The LST of the actual waiting time distribution in the queue is given by Wq∗ () =
1 S (0)D ∗ (1 − Ploss ) +
L l N−1
∗ n ∗[k] l (n, )D S ()
n=0 l=1 k=1
× V1∗ (, n + k − l) +
N−1 n=0
∗
∗
(n, )D S ()
n
V2∗ (, n + 1)
,
546
A.D. Banik et al. / Operations Research Letters 34 (2006) 539 – 547
where V1∗ (, k) =
∞
i V ∗ () Pr
n=1
i=0
V2∗ (, k) = 1, =
i
n < k
i+1
S are of dimension . Similarly, let V (x) follow a PH-distribution with irreducible representation (, T), where and T are of dimension , then the matrices An and Vn can be computed using the procedure described in [11,3]. We have conducted an experiment on the MMPP/E2 /1/20 queue with multiple vacations (vacation time follows PH-distribution) for the following input parameters: The MMPP with five states whose infinitesimal generator is ⎡ ⎤ −1.75 0.25 0.50 0.75 0.25 0.90 0.15 ⎥ ⎢ 0.60 −2.50 0.85 ⎢ ⎥ R = ⎢ 0.40 0.30 −2.10 0.75 0.65 ⎥ ⎣ ⎦ 0.90 0.44 0.30 −2.90 1.26 1.20 0.25 0.45 0.35 −2.25
n ,
n=1
k 0,
i−1 i i V () Pr n < k n ,
∞
∗
i=0
n=1
n=1
k 1, and { n ; n > 1} is a sequence of the maximum number of customers served in each busy period.
and arrival rate matrix = diag(0.889, 1.55, 0.92, 0.748, 1.19). Its MAP representation is taken as C=R− and D=. Here, m = 5, ∗ =1.0000. PHtype representation of vacation time is taken as = [ 0.7 0.3 ], −1.098 1.099 T= 0.071 −1.832
5. Computational procedure and numerical results In this section we will briefly discuss the necessary steps required for the computation of the matrices An , Vn of TPM P. The evaluation of An (Vn ), in general, for arbitrary service (vacation) time distribution requires numerical integration and can be carried out along the lines proposed by Lucantoni and Ramaswami [10]. However, when the service, vacation-time distributions are of phase type (PHdistribution), these matrices can be evaluated without any numerical integration [11, p. 67–70]. For computational purpose let S(x) follow a PH-distribution with irreducible representation (, S), where and
with E(V ) = 1.242509. For E2 service time, PH-type representation is taken as = [ 1.0 0.0 ], − S= 0.0 − with E(S) = 2.0/ and by suitably varying one can get various values of which is equal to ∗ E(S).
(a)
0.08
0.9 0.8
0.07
0.7
0.06
probability of blocking
probability that the server is busy
1
0.6 0.5 0.4 0.3
Uniform Geometric Constant Bimodal
0.2
0.05 0.04
Uniform Geometric Constant Bimodal
0.03 0.02 0.01
0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 traffic intensity
1
1.1 (b)
0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 traffic intensity
Fig. 1. (a) versus . (b) versus Ploss .
A.D. Banik et al. / Operations Research Letters 34 (2006) 539 – 547
We take the maximum value of the limit L = 5 and the following four limit mass functions are examined: (i) Uniform: pl = 16 , for l = 0, 1, 2, . . . , 5, (ii) Geometric: pl = 13 ( 23 )l−1 , for l = 1, 2, . . . and p0 = 0, (iii) Constant: pl =(l−5), (iv) Bimodal: pl =( 21 )(l− 3) + ( 21 )(l − 5), where (.) denotes Dirac delta function. Figs. 1(a) and (b) shows the effect of on and (Ploss ). It may be remarked here that due to lack of space only limited results have been presented; however, details are available with the authors. Acknowledgements The authors would like to thank the referee for his valuable comments and suggestions. The first author wishes to thank CSIR, New Delhi, India, for their financial support. References [1] C. Blondia, Finite capacity vacation model with non-renewal input, J. Appl. Probab. 28 (1991) 174–197. [2] U.C. Gupta, A.D. Banik, S.S. Pathak, Complete analysis of MAP/G/1/N queue with single (multiple) vacation(s) under limited service discipline, J. Appl. Math. Stochastic Anal. 3 (2005) 353–373. [3] U.C. Gupta, P. Vijaya Laxmi, Analysis of MAP/Ga,b /1/N queue, Queueing Syst. Theory Appl. 38 (2001) 109–124.
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