1 generalized processor-sharing queue

Author’s Accepted Manuscript An algorithmic analysis of the BMAP/MSP/ 1 generalized processor-sharing queue Souvik Ghosh, A.D Banik

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S0305-0548(16)30249-0 http://dx.doi.org/10.1016/j.cor.2016.10.001 CAOR4099

To appear in: Computers and Operation Research Received date: 25 August 2015 Revised date: 28 April 2016 Accepted date: 2 October 2016 Cite this article as: Souvik Ghosh and A.D Banik, An algorithmic analysis of the BMAP/MSP/ 1 generalized processor-sharing queue, Computers and Operation Research, http://dx.doi.org/10.1016/j.cor.2016.10.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

An algorithmic analysis of the BM AP/M SP/1 generalized processor-sharing queue Souvik Ghosha , A.D Banika,∗ a

School of Basic Science, Indian Institute of Technology, Bhubaneswar Bhubaneswar, India.

Abstract This article deals with the analysis of the BMAP/MSP/1 generalized processorsharing queue. The analysis is based on RG-factorization technique applied to the Markov chain of the associated quasi birth and death process. The stationary systemlength distribution of the number of customers in the system and the Laplace-Stieltjes transform (LST) of the sojourn time distribution of a tagged customer in the system is obtained in this paper. The mean sojourn time of a tagged customer is derived using the previous LST. The corresponding finite-buffer queueing model is also analyzed and system length distribution is derived using the same technique as stated above. Further, the blocking probabilities for customers with different positions, such as the first-, an arbitrary- and the last-customer of a batch is obtained. The detail computational procedure for these models is discussed. Various numerical results is presented to show the applicability of the results obtained in the study. Keywords: Generalized processor-sharing (GPS) queue; Batch Markovian arrival process (BMAP ); Markovian service process (MSP ); RG-factorization. 1. Introduction Round-robin (RR) is one of the oldest algorithm used in the area of computing to schedule processes fairly. In RR scheduling, a limited amount of CPU time is offered to a job in a first in first out (FIFO) manner. If the CPU-time expires before completion of the job, the CPU is preempted and allocated to the next job waiting in the queue. The preempted job then join at the last of the queue. Processor-sharing (PS) ∗

Corresponding author Email addresses: [email protected], [email protected] (Souvik Ghosh), [email protected], [email protected] (A.D Banik) Preprint submitted to Elsevier

October 4, 2016

was evolved as an approximation of the RR scheduling in computer communication systems, see Kleinrock [8]. In present days, PS is widely being used in the area of communication networks, such as transmission control protocol (TCP) and internet protocol (IP), see Avrachenkov et al. [3]. In Egalitarian PS (EPS), the service capacity is equally distributed among all jobs present in the system. However, more fairness may achieved by idealizing EPS to generalized processor-sharing (GPS). In GPS, the service capacity is shared to different job classes with a class-dependent weight factors, see Alto et al.[1]. Last few decades PS queues have drawn a substantial attention of researchers in different discipline. For a so-called basic M/M/1-PS model, an expression for the mean conditional response time (conditioned on the required service time) was given by O’Donovan [22]. Removing the condition on the required service time, Morrison [17] found a single integral representation of the LST of the response-time distribution. For a MAP/M/1-PS queue, the stationary sojourn time distribution in terms of a recursive formula was developed by Masuyama and Takine [16]. The M/P H/1-PS queueing model was analyzed by Sericola et al. [25] using Uniformization technique. For more detail on PS, interested readers are refered to the excellent survey by Yashkov and Yashkova [26] and the references therein. Very few works can be found on bulk arrival PS queueing model which was first investigated by Kleinrock et al. [9]. They developed an integral equation which describes the derivative of the mean sojourn time conditioned on the job time requirement of an arbitrary customer. They also found that the integral equation was solvable for a special class of service time distributions. Bansal [5] extended Kleinrock’s work by considering hyper-exponential service time distribution. Avrachenkov et al. [3] showed that the integral equation possesses an unique solution and they obtained tight bounds for the expected response time conditioned on the service time. Rommel [23] analyzed response time of a M [X] /G/1-PS queue by modeling parallel processes as fork-join process. Li et al. [12] approximate the mean sojourn time of a BMAP/M/1-GPS queueing model by using RG-factorization technique. Recent trends show that processor-sharing is an important tool to model different type of communication networks. Poisson process is not sufficient to model present days high-speed networks with different characteristics. The system becomes complex if jobs occur in batches and even more if we consider non-renewal inter-arrival times with correlated batch sizes. Neuts [18] introduced N-process to explain the correlation in arrivals, whereas the burstiness of the arrival process was modeled by Lucantoni et al. [15] by proposing Markovian arrival process (MAP ). Later, Lucantoni [14] suggested batch Markovian arrival process (BMAP ) to explain correlated batch arrivals. Markovian service process (MSP) considers correlation among successive service times, see Chaudhry et al. [6]. Li et al. [12] have ana2

lyzed a BMAP/M/1-GPS queueing system. Recently, Samanta et al. [24] studied a BMAP/MSP/1 queue without any processor-sharing of available customers present in the system. Motivated by these works we have analyzed both infinite- and finitebuffer BMAP/MSP/1-GPS queueing model by RG-factorization technique. Neuts [19, 20] has proposed matrix geometric method to analyze GI/M/1- and M/G/1type Markov chain. The key step involve in matrix geometric method is to determine the R matrix for GI/M/1 type Markov chain and G matrix for M/G/1-type Markov chain. Heyman [7] examined the behaviour of a general Markov chain using censoring technique. For some integers i and j, Heyman [7] described Ri,j (i < j) and Gi,j (j < i) matrices as counterparts of R and G matrices. Censoring technique is extensively discussed by Li [11] to study the block-structured Markov chains. Recently Yu and Alfa [27] have computed queue-length distribution at various epochs for a finite DBMAP/G(1,a,b) /1 queue with batch-size-dependent service time. 2. Assumptions and Notations A BMAP/MSP/1-GPS queue is investigated in this paper. In a GPS queue, if at any instant there are n number of customers are present in the system then each customer experiences a service rate f (n) at that instant, where f (·) is a positive function which satisfies 0 < nf (n) ≤ C for all n (≥ 1) and C is a positive constant. In the next subsections the notations of a BMAP/MSP/1 queueing system is briefly introduced. 2.1. Batch Markovian Arrival Process The batch Markovian arrival process (BMAP ) is a generalization of the Poisson arrival process in which “batch” arrivals are allowed. Let Ea = {1, 2, · · · , m1 } be the underlying state space of the Markov process and Ja (t) = {ja : ja ∈ Ea , t ≥ 0} denote the phase of the underlying Markov chain at time t. Let Dk (k ≥ 1) be a matrix of order m1 ×m1 , which denotes the transition rate of Ja (t) for a batch arrival of size k. Similarly, the transition rate for no arrival is denoted by the matrix D 0 of order m1 × m1 . The matrix D 0 is invertible with negative diagonal entries and nonnegative off-diagonal entries, while all the entries of Dk (k ≥ 1) are non-negative. Then the BMAP can be represented by the parameter matrices Dk (k ≥ 0). The irreducible infinitesimal generator of the underlying Markov chain {Ja (t)} for the ∞ BMAP is given by D = k=0 Dk with De = 0, where e is a column vector of ones with suitable size and 0 is a row vector of proper dimension with all zero entries. Let π a = [π a1 , π a2 , · · · , πam1 ] be the stationary probability vector of the underlying Markov process with irreducible infinitesimal generator D, i.e., π a D = 0 with π a e = 1. 3

 Then the fundamental arrival rate of the BMAP is given by λ = π a ( ∞ k=1 kD k )e a and the batch arrival rate is given by λb = −π D 0 e. For details on BMAP , see Lucantoni [14]. 2.2. Markovian Service Process It is assumed that the customers are served according to the Markovian service process (MSP ) with underlying state space Es = {1, 2, · · · , m2 }. The MSP generalizes the Poisson service process where the phase of the underlying Markov chain at time t is denoted as Js (t) = {js : js ∈ Es , t ≥ 0}. Let the matrices L1 and L0 represent the transition rate of Js (t) for a service and for no service, respectively. Note that both the matrices L0 and L1 , are of order m2 × m2 and the matrix L0 is invertible with negative diagonal entries and non-negative off-diagonal entries, while all the entries of L1 are non-negative. It is presumed that a busy period starts with the service phase which is the last attained service phase during the previous busy period. It is also accepted that after completing a busy period the server becomes idle without changing the phase of the service process and during the idle period the phase of the service process remains unchanged, i.e., interruptions occur during idle periods. The irreducible infinitesimal generator of the underlying Markov chain {Js (t)} for the MSP is given by L = L0 + L1 with Le = 0. Let π s = [π s1 , π s2 , · · · , π sm2 ] be the stationary probability vector of the underlying Markov process with the irreducible infinitesimal generator L, i.e., π s L = 0 with π s e = 1. Then the stationary mean service rate of the MSP is given by μ = π s L1 e. For details on the MSP , see Chaudhry et al. [6]. Remark 2.1. If f (n) = system.

1 n

(n ≥ 1), then C = 1 and the system becomes an EPS

Remark 2.2. For the stability of the GPS queue, nf (n) should be a convergent sequence as n increases. Let the sequence converges to Δ as n increases. Then the λ traffic intensity of the system is given by ρ = <1 Δμ 3. Analysis of the BM AP /M SP /1/∞-GPS Model This section includes analysis of the stationary system-length distribution of an arbitrary customer and the mean sojourn time of a tagged customer in the system with infinite buffer capacity.

4

3.1. The Stationary System-length Distribution Let N(t), I(t) and J(t) denote the number of the customers in the system , phase of the BMAP and phase of the MSP at time t (≥ 0), respectively. Then the state of the system at time t can be defined as Y (t) = (N(t), I(t), J(t)). One may note that {Y (t)}t≥0 is a continuous-time level-dependent Markov chain of M/G/1 type on the state space {(n, i, j) : n ≥ 0, 1 ≤ i ≤ m1 , 1 ≤ j ≤ m2 }. For state (n, i, j), n is called the level variable, i is called the BMAP -phase variable and j is called the MSP -phase variable. If we write I r as the identity matrix of order r × r, then the irreducible infinitesimal generator Q of the Markov chain is given by ⎛ ⎞ B 0 A1 A2 A3 A4 · · · ⎜ C 1 B 1 A1 A2 A3 · · · ⎟ ⎜ ⎟ ⎜ C 2 B 2 A1 A2 · · · ⎟ (1) Q=⎜ ⎟, ⎜ ⎟ C B A · · · 3 3 1 ⎝ ⎠ .. .. .. . . . where An B0 Bn Cn

= Dn ⊗ I m2 , n ≥ 1, = D0 ⊗ I m2 , = B 0 + nf (n)I m1 ⊗ L0 , n ≥ 1, = nf (n)I m1 ⊗ L1 , n ≥ 1.

It is clear from the context that the matrices An , B n and C n are of the order m × m, where m = m1 m2 . Further, it may be understood that the matrices An (n ≥ 1) increase the level of the chain by n, C n (n ≥ 1) decrease the level by one and B n (n ≥ 0) keep the chain at the same level n. Evaluation of the stationary system-length of the system needs the determination of the stationary probability vector of the irreducible infinitesimal generator Q. Censoring technique and a UL-type RG-factorization [11] is used to determine the stationary probability vector of Q. Let us re-write the state space of the above Markov chain as Ω = {(n, ζ) : n ≥ 0, 1 ≤ ζ ≤ m}, where n be the level variable and ζ be the phase variable. Under the block structure of the Markov chain the state space can be partitioned as Ω = ∞ · · · , (i, m)}. Let us choose i=0 Li , where Li = {(i, 1), (i, 2), k the censoring set Ek as a subset of Ω such that Ek = i=0 Li , where k ≥ 0. Now according to the subset Ek , we can block-partition the infinitesimal generator Q from level k as

(k)  T H (k) Q= , (2) V (k) W (k) 5

where the matrix T (k) is of order m(k + 1) ×m(k + 1) and the other block-partitioned matrices are of proper order. Lemma 3.1. If Q is irreducible and positive recurrent with V (k) > 0 (k ≥ 0), then ∞ each element of exp(W (k) t)dt = −(W (k) )−1 is finite. 0

Proof. Since Q is irreducible and positive recurrent with V (k) > 0, then the Markov chain with infinitesimal generator W (k) is transient and invertible. We may consider A = {0, 1, · · · , k} be the set of absorbing levels and Ac be the set of transient levels for the Markov chain W (k) so that the expected number of visits to each level in Ac is finite. Hence

∞ exp(W (k) t)dt = (−W (k) )−1 < +∞, 0

this completes the proof. Note that the invertibility of the matrix W (k) is under an infinite-dimensional meaning and may have multiple inverses. Let the minimal non-negative inverse of −W (k)  (k) , where W  (k) (i, j ≥ 1) is a m × m matrix whose (r, r )-th eleis denoted by W i,j ment denotes the expected time spent in the state (k + j, r  ) starting from the state  (k) is the continuous analog of the fundamental ma(k + i, r). One may note that W trix of discrete time Markov chain. Now following the arguments given by Latouche and Ramaswami [10], we define the R-, G- and U-measures of the Markov chain Q. Definition 3.1. Let Rn,n (0 ≤ n < n ) be a matrix of order m × m, whose (r, r )-th element represents the total time spent in the state (n , r  ) before the first visit, i.e., the first passage, to the level (n − 1) of the process, while the initial state of the process is (n, r). Then the matrix sequence {Rn,n } is called the R-measure of the Markov chain Q. Definition 3.2. Let Ψn (n ≥ 0) be a matrix of order m×m, whose (r, r  )-th element represents the total time spent in the state (n, r ) before the first visit, i.e., the first passage, to the level (n − 1), while the process starts from the state (n, r). Then the matrix sequence {Ψn } is called the U-measure of the Markov chain Q. Definition 3.3. Let G(k) (k ≥ 1) be a matrix of order m × m, whose (r, r )-th element represents the probability of first visit, i.e., the first passage, to the state (k − 1, r  ) of the Markov chain, provided that the process initiates from the state (k, r). Then the matrix sequence {G(k) } is called the G-measure of the Markov chain Q. 6

The expressions of R-, G- and U-measures for the BMAP/M/1-GPS queueing model was given by Li et al. [12]. Similarly, for the BMAP/MSP/1-GPS queueing model the R-, G- and U-measures can be given as follows: Rl,k = G(k) =

∞ 

(k−1)

 Ar−l W r−k+1,1 ,

r=k  (k−1) C k , W 1,1

k ≥ 1, 0 ≤ l < k,

k≥1

(3) (4)

and Ψk−1 = B k−1 +

∞ 

 (k−1) C k , Al W l,1

k ≥ 1.

(5)

l=1

Now for k ≥ 1, from Lemma 2.6 given in [11], we have  (k−1) = W l,1

l−1 

 (k−1) , l ≥ 2 G(k+l−p) W 1,1

(6)

p=1

and  (k−1) = −Ψ−1 . W 1,1 k

(7)

Hence, from Equations (3) and (6), it immediately follows that n ∞    (k−1)   Rl,k = Ak−l + Ak+n−l G(k+n+1−p) W 1,1 , n=1

k ≥ 1, 0 ≤ l < k.

(8)

p=1

Again using Equations (4) and (6) in the expression (5), the following formulation can be made. l ∞   Ψk−1 = B k−1 + Al G(k+p−l) k ≥ 1. (9) p=1

l=1

Now from Equations (4) and (7) we can write −Ψ−1 k C k = G(k) (k > 1). Then using Equation (5), from the theorem 2.2.2 given in [20] we can conclude that {G(k) } (k > 1) is the minimal non-negative solution to the system of matrix equations C k + B k G(k) +

∞ 

Al

l−1  p=0

l=1

7

G(k+l−p)G(k) = 0.

(10)

The UL-type RG-factorization of M/G/1 type Markov chain Q was derived by Li [11] and can be given as Q = (I − RU )U D (I − GL ), where I is the identity matrix of proper order, ⎛ 0 R0,1 R0,2 R0,3 ⎜ 0 R1,2 R1,3 ⎜ ⎜ 0 R2,3 RU = ⎜ ⎜ 0 ⎝

(11)

⎞ ··· · · ·⎟ ⎟ · · ·⎟ ⎟, · · ·⎟ ⎠ .. .

U D = diag(Ψ0 , Ψ1 , Ψ2 , · · · ) and

⎞

⎛

0 ⎜G(1) 0 ⎜ ⎜ G(2) 0 GL = ⎜ ⎜ G (3) ⎝

0 .. .. . .

⎟ ⎟ ⎟ ⎟. ⎟ ⎠

Let π = [π 0 ,π 1 ,π 2 ,· · · ] be the stationary probability vector of the continuous-time level-dependent Markov chain of Q and z 0 be the stationary probability vector of the censored Markov chain Ψ0 to level 0. Then the expression for π in terms of RG-factorization, as derived by Li et al. [12], can be given by ⎧ if k = 0, ⎨τ z 0 , k−1  (12) πk = ⎩ π l Rl,k , if k ≥ 1, l=0

where τ is a constant and is to be determined by

∞  k=0

π k e = 1.

Let qk (k ≥ 0) be the probability of having k customers in the system in steady state. Then using Equation (12), the stationary system-length distribution can be presented as ⎧ if k = 0, ⎨τ, k−1 qk =  (13) ⎩ π l Rl,k e, if k ≥ 1. l=0

Using Equation (13) one may easily determine the mean and variance of the stationary system-length. 8

3.2. The Sojourn Time Let us define a k-type customer who finds k (≥ 0) other customers in the system k (x, i, j) denotes the probability that the sojourn time of at his arrival epoch. Let W a k-type customer strictly exceeds x (≥ 0), while at the arrival epoch of the tagged customer the phase of the BMAP and MSP are i and j, respectively. Let us write   k (x) = W k (x, 1, 1), W k (x, 1, 2) · · · , W k (x, 1, m2 ), W  k (x, 2, 1), · · · , W ˆ (x, m1 , m2 ) , k ≥ 0. W (14) k

Now applying similar kind of arguments employed by Asmussen [2], we may conclude  k (x)} (k ≥ 0) satisfies the following system of differential that the vector sequence {W equations ∞  d   0 (x) +  n (x), An W W 0 (x) = B 1 W dx n=1

∞  d      k+n (x), An W W k (x) = C k W k−1 (x) + B k+1 W k (x) + dx n=1

(15) k ≥ 1,

(16)

where C k = kf (k + 1)I m1 ⊗ L1 . Note that similar kind of differential equation was derived for MAP/M/1-PS by Masuyama and Takine [16] and for BMAP/M/1-PS by Li et al. [12].  (x) and Q as Now define W  T   (x) = W  (x), W  T (x), W  T (x), · · · T , W 0 1 2 ⎛ ⎞ B 1 A1 A2 A3 · · · ⎜ C  1 B 2 A1 A2 · · · ⎟ ⎜ ⎟  Q=⎜ ⎟. C B A · · · 2 3 1 ⎝ ⎠ .. .. .. . . .

(17) (18)

Then the system of equations (15) and (16) can be modified as d   (x), W (x) = QW dx

(19)

 (0) = 0. Note that one may consider Q as the defective with the initial condition W infinitesimal generator (Masuyama and Takine [16]) of the continuous-time Markov chain with infinite states (n, i, j) (n = 0, 1, 2, · · · , 0 ≤ i ≤ m1 , 0 ≤ j ≤ m2 ). Let at 9

arrival epoch of a k-type customer, the phase of the BMAP is i and the phase of the MSP is j. Then the conditional sojourn time of the k-type customer is the first passage time from state (k, i, j) to an implicit absorbing state. Now using the initial condition, the solution of the equation (19) can be expressed as  (x) = exp(Qx)W  (0). W ∗

(20)

∗

 (s) and W  (s) be the LST of the column vectors W  (x) and W  k (x), Let us denote W k respectively. Then for s > 0, one may write

∞   ∗  (s) =  (0)dx. W exp − (sI − Q)x W (21) 0

 ∗ (s) = [W  ∗T (s), W  ∗T (s), W  ∗T (s), · · · ]T . Now for It may be indicated here that W 0  1  2  −1 s > 0, if the maximal negative inverse of matrix Q−sI is denoted by Q−sI max ,  ∗ (s) can be given by then W     (0).  ∗ (s) = − Q − sI −1 W (22) W max Again the censoring technique −1 and UL-type RG-factorization can be used to determine the matrix (Q − sI max . Similarly from subsection 3.1, R-, G- and U-measure of the matrix (Q − sI) can be defined. Let the sequence of matrices for the R-, G- and U-measure of the matrix (Q−sI) are denoted by {Rl,k (s)} (k ≥ 1, 0 ≤ l < k), {G(k) (s)} (k ≥ 1) and {Ψk (s)} (k ≥ 0), respectively. Then from previous discussions, it can be concluded that the matrix sequence {G(k) (s)} is the minimal non-negative solution to the system of matrix equations C k

+ (B k+1 − s)G(k) (s) +

∞ 

Al

l=1

l−1 

G(k+l−p)(s)G(k) (s) = 0.

(23)

p=0

Further the matrix sequence of the U- and R-measure of the matrix (Q − sI) are given as follows: Ψk (s) = (B k+1 − s) +

∞  l=1

Rl,k (s) =

∞  n=0

Ak+n−l

n−1 

Al

l−1 

G(k+l−p)(s),

k ≥ 0,

(24)

p=0

G(k+n−p)(s)

p=0

10



 − Ψ−1 k (s) ,

0 ≤ l < k.

(25)

Hence from previous argumentation, the matrix (Q − sI) can be factorized as Q − sI = [I − RU (s)][U D (s)][I − GL (s)], where

⎛ 0 R0,1 (s) R0,2 (s) R0,3 (s) ⎜ 0 R1,2 (s) R1,3 (s) ⎜ ⎜ 0 R2,3 (s) RU (s) = ⎜ ⎜ 0 ⎝

(26)

⎞ ··· · · ·⎟ ⎟ · · ·⎟ ⎟, · · ·⎟ ⎠ .. .

U D (s) = diag(Ψ0 (s), Ψ1 (s), Ψ2 (s), · · · ) and

⎛

⎞

0

⎜G(1) (s) 0 ⎜ ⎜ G(2) (s) 0 GL (s) = ⎜ ⎜ G (3) (s) ⎝

0 .. .. . .

⎟ ⎟ ⎟ ⎟, ⎟ ⎠

Then the matrix (Q − sI)−1 can be computed as (Q − sI)−1 = [I − GL (s)]−1 [U D (s)]−1 [I − RU (s)]−1 , where

⎛

[I − RU (s)]−1

I X 0,1 (s) X 0,2 (s) X 0,3 (s) ⎜ I X 1,2 (s) X 1,3 (s) ⎜ ⎜ I X 2,3 (s) =⎜ ⎜ I ⎝

X k,k+1(s) = Rk,k+1(s),

⎞ ··· · · ·⎟ ⎟ · · ·⎟ ⎟, · · ·⎟ ⎠ .. .

k ≥ 0,

(27)

(28)

(29)

p−1

X k,k+p(s) =



Rk,k+l (s)X k+l,k+p(s),

k ≥ 0, p ≥ 2,

(30)

l=1

[U D (s)]

−1

−1 −1 = diag(Ψ−1 0 (s), Ψ1 (s), Ψ2 (s), · · · )

11

(31)

and

⎛ [I − GL (s)]−1

⎞

I

⎜Y 1,0 (s) I ⎜ ⎜Y 2,0 (s) Y 2,1 (s) I =⎜ ⎜Y 3,0 (s) Y 3,1 (s) Y 3,2 (s) ⎝ .. .. .. . . .

Y k,k−p(s) =

p−1 

G(k−l) (s),

I .. . . . .

⎟ ⎟ ⎟ ⎟, ⎟ ⎠

k ≥ p ≥ 1.

(32)

(33)

l=0

Now if the initial probability vector of the processor-sharing queue is given by   T  T (0), W  T (0), · · · T = 0,  (0) = W  (0), W W 0 1 2 then following similar arguments given in Theorem 4.1 by Li et al. [12], it is realized that ∞  −1    ∗    W 0 (s) = − Ψ0 (s) W 0 (0) + X 0,k (s)W k (0) (34) k=1

and for n ≥ 1,  ∗ (s) = W n

n−1 

∞    −1   l (0) +  k (0) + W Y n,l (s) − Ψl (s) X l,k (s)W

l=0



− Ψn (s)

k=l+1

−1   n (0) + W

∞ 

  k (0) . X n,k (s)W

(35)

k=n+1

The following theorem provides the sojourn time distribution of a tagged customer. Theorem 3.1. For k ≥ 0, the mean sojourn time of a k-type customer can be evaluated as  ∗ (0). E(Wk ) = eˆW k

(36)

where eˆ is a row vector of ones with proper dimension. Proof. Let for some k ≥ 0, Wk (x) denotes the cumulative distribution function  k (x), (CDF) of sojourn time of the k-type customer. Hence, from the definition of W Wk (x) can be computed as  k (x). Wk (x) = 1 − eˆW 12

(37)

Now using Equation (37) and the fact ∞that the expectation of a random variable X with positive support is given by 0 (1 − F (x))dx, where F is the cumulative distribution function of X, it can be summarized that

∞  k (x)dx eˆW E(Wk ) = 0

 ∗ (0). = eˆW k This completes the proof. Remark 3.1. If the system is in the steady state, then the initial probability vector  (0) = [ˆ is given by W π0, π ˆ 1, π ˆ 2 , · · · ]T , where π ˆ k (k ≥ 0) is the normalized vector of π k , hence it can be written that ∞   −1   ∗  W 0 (s) = − Ψ0 (s) π ˆ0 + X 0,k (s)ˆ πk

(38)

k=1

and for n ≥ 1, ∞    −1  ∗  π ˆn + X n,k (s)ˆ πk + W n (s) = − Ψn (s) k=n+1 n−1 



Y n,l (s) − Ψl (s)

l=0

−1 

π ˆl +

∞ 

 X l,k (s)ˆ πk .

(39)

k=l+1

Remark 3.2. If for some k (≥ 0), the initial probability vector of a stable processorsharing queue is given by   n (0) = α, if n = k, W 0, if n = k, where α is a probability vector of order m with αe = 1. Then  −1  ∗ (s) = − Ψ0 (s) α W 0

(40)

and for n ≥ 1, n−1   −1  −1 ∗  Y n,l (s) − Ψl (s) X l,n (s)α. W n (s) = − Ψn (s) α + l=0

13

(41)

4. Analysis of the BM AP /M SP /1/M -GPS Model In this section we present the stationary system-length distribution of an arbitrary customer and the mean sojourn time of a tagged customer in the system with finite buffer capacity. 4.1. The Stationary System-length Distribution Let the buffer capacity of the system is M, then the state space {Y (t)}t≥0 of the system is given by {(n, i, j) : 0 ≤ n ≤ M, 1 ≤ i ≤ m1 , 1 ≤ j ≤ m2 }. The infinitesimal generator Q of the finite Markov chain is given by ⎞ ⎛ M B 0 A1 A2 ··· AM −2 AM −1 A ⎜ C B A M −1 ⎟ ··· AM −3 AM −2 A ⎟ ⎜ 1 1 1 ⎟ ⎜  ⎜ C 2 B2 ··· AM −4 AM −3 AM −2 ⎟ ⎟ ⎜ .. .. .. .. .. ⎟, (42) Q=⎜ . . . . . ⎟ ⎜ ⎟ ⎜ 2 ⎟ C M −2 B M −2 A1 A ⎜ ⎜ 1 ⎟ ⎠ ⎝ C M −1 B M −1 A  CM A where k = A

∞ 

An ,

for 1 ≤ k ≤ M,

n=k

1 .  = BM + A A M −k increases the level of the chain by (M − k) from For 0 ≤ k < M, the matrices A  k, while A keeps the chain at the level M. For 0 ≤ k < M, let us define M 

[k]

Qi,j = Qi,j +

 [n]  [n] −1 [n] Qi,n − Qn,n Qn,j ,

(43)

n=k+1 [M ]

where Qi,j = Qi,j . Then the R-, U- and G-measures of the finite model can be determined following the arguments described by Li [13, Lemma 2.5] and for 1 ≤ k ≤ M, the matrix sequences of the respective measures can be presented as [k]

Ψk = Qk,k , −1 [k]  Rl,k = Ql,k − U k , 14

(44) 0≤l
(45)

and

 −1 [k] G(k) = − U k Qk,l ,

0 ≤ l < k.

(46)

The stationary probability vector of the finite Markov chain Q is figured out as ⎧ if k = 0, ⎨τ z 0 , k−1  (47) πk = ⎩ π l Rl,k , if 1 ≤ k ≤ M, l=0

where z0 is the stationary probability vector of the censored Markov  chain Ψ0 to level 0 and τ is a constant which can be enumerated by the relation M k=0 π k e = 1. 4.2. Blocking Probabilities For any finite queueing system, one interesting measure is blocking probability. In this subsection we give the expression of the blocking probabilities for different customers, such as the first-, an arbitrary- and the last customer. Blocking Probability of the First Customer in a Batch: Let, an arriving batch finds that M customers are getting service, i.e., the buffer is full. Then the first-customer of the batch as well as the whole batch will be lost. Thus, the blocking probability of the first customer can be expressed as ∞   1 P BLF = πM (D n ⊗ I m2 )e . λb n=1

(48)

Blocking Probability of an Arbitrary Customer in a Batch: Let P k be the matrix of order m×m whose element [P k ]i,j represents the probability that the position of an arbitrary customer in an arriving batch is k with phase changes from state i to j. The expression of the P k was derived by Banik [4] and can be given by ∞  1  Pk = (D n ⊗ I m2 ) . λ n=k

An arbitrary customer will be lost if he finds n (0 ≤ n ≤ M) customers are already getting the service and his position in the arriving batch is k ≥ (M − n + 1). Hence, the blocking probability of an arbitrary customer can be formulated as P BLA =

M 

π(n)

n=0

∞  k=M −n+1

15

P k e.

Blocking Probability of the Last Customer in a Batch: The last customer of an arriving batch will be lost if he finds n (0 ≤ n ≤ M) customers are already getting the service and his batch size is k ≥ (M − n + 1). So, the blocking probability of the last customer is equated as P BLL =

∞ M   1  πn (D k ⊗ I m2 )e . λb n=0 k=M −n+1

4.3. The Sojourn Time The defective infinitesimal ⎛ B 1 A1 ⎜ C B ⎜ 1 2 ⎜  ⎜ C2 ⎜ ⎜ Q=⎜ ⎜ ⎜ ⎜ ⎝

(49)

generator of the finite Markov chain is given by ⎞ M A2 ··· AM −2 AM −1 A M −1 ⎟ A1 ··· AM −3 AM −2 A ⎟ ⎟  B3 ··· AM −4 AM −3 AM −2 ⎟ ⎟ .. .. .. .. .. ⎟. (50) . . . . . ⎟ ⎟  2 ⎟ C M −2 B M −1 A1 A  1 ⎟ ⎠ C M −1 B M A  C M A

Now for 1 ≤ k ≤ M, the matrix sequences {X i,j }0≤i
l−1 

0 ≤ k ≤ (M − 1),

Rk,k+iX k+i,k+l,

2 ≤ l ≤ M, 0 ≤ k ≤ M − l

(51) (52)

i=1

and Y k,k−l =

l−1 

G(k−n) ,

1 ≤ l ≤ k ≤ M.

(53)

n=0

Now using the Equations (34) and (35), the following results can be obtained. M  −1     0 (0) +   ∗ (0) = − Ψ0 (0) W X (0) W (0) W 0,k k 0 k=1

16

(54)

and for 1 ≤ n < M,  ∗ (0) = W n

n−1 

M  −1     k (0) + W l (0) + Y n,l (0) − Ψl (0) X l,k (0)W



l=0



k=l+1

− Ψn (0)

−1   n (0) + W

M 

  k (0) . X n,k (0)W

(55)

k=n+1

5. Computational Procedure The computation of the stationary probability vector and the mean sojourn time of a tagged customer for the finite model is very straight-forward. Whereas, computational analysis of the infinite model needs some modification of the Markov chain. In this section the computational procedure of the infinite model is given. 5.1. Stationary System-length Distribution of the System The steps involve to calculate the stationary probability vector are discussed below: I) Modification of the Markov chain: As it given in Equation (2), the infinitesimal generator is to be partitioned from the level N, where N is a large positive integer. Then the level-dependent south-east block is to be modified to a level-independent block as follows ⎛ ⎞ B N +1 A1 A2 A3 A4 · · · ⎜ C N +1 B N +1 A1 A2 A3 · · · ⎟ ⎜ ⎟ W (N ) = ⎜ (56) ⎟. C B A A · · · N +1 N +1 1 2 ⎝ ⎠ .. .. .. .. . . . . Then the modified infinitesimal generator can be written as 

(N ) (N ) (N ) H T  Q . = V (N ) W (N )

(57)

II) G- and U-measure of the modified Markov chain: Let G be the minimal non-negative solution to the matrix equation C N +1 + B N +1 G +

∞  l=1

17

Al Gl+1 = 0.

(58)

Now using expression (9), one can define Ψ = B N +1 +

∞ 

Al Gl .

(59)

l=1

 (N ) is formulated as Then the G-measure of the Markov chain Q  (N ) (−Ψ)−1 V 1 , if n = 1, (N ) G(n) = G, if n ≥ 2,

(60)

(N )

where V 1 is the 1st row vector of the matrix V (N ) . The U-measure of the modified Markov chain is expressed as ⎧ ∞ ⎨T (N ) +  H(N ) Gk−1 G(N ) , if n = 0, k (1) ) Ψ(N = (61) k=1 n ⎩ Ψ, if n = 0, (N )

is the k-th column vector of the matrix H (N ) . It should be illustrated (N )  (N ) to here that Ψ0 is the infinitesimal generator of the censored Markov chain Q level 0. where Hk

III) R-measure of the modified Markov chain: For k ≥ 1, let us define (N ) Rk

=

∞ 

Ak+l Gl (−Ψ)−1 .

l=0

Then for k ≥ 1, the R-measure of the Markov chain ⎧∞ ⎨  H(N ) Gl (−Ψ)−1 , k+l (N ) Rn,n+k = l=0 ⎩ (N ) Rk ,

 (N ) can be interpreted by Q if n = 0, if n = 0.

(62)

IV) The stationary probability vector of the modified Markov chain:  = [ π 1 , π 2 ,· · · ] be the stationary probability vector of the modified Markov Let π π0 , (N ) (N ) (N )  0 be the stationary probability vector of Ψ0 . Then z 0 Ψ0 = 0 and z chain Q 0 e = 1. Note that z 0 is a row vector of order m(N +1), while the row vectors π k and z

18

(k ≥ 1) are of order m. Now chasing similar arguments outlined by Li [11, Theorem 2.9], the stationary probability vector can be figured as ⎧ 0 , if k = 0, ⎨τz  k = k−1  π (N ) ⎩ π  l Rl,k , if k ≥ 1.

(63)

l=0

 e = 1. where the constant τ is to be determined by π V) Stationary system-length distribution of the system:  0 = [  0,1 , · · · , π  0,N ], then the stationary probability vector of the levelIf π π 0,0 , π dependent Markov chain Q can be approximately given by   0,n , if 0 ≤ n ≤ N, π πn  (64)  n−N , if n > N. π Hence, the stationary system-length distribution of the system can be evaluated as qn  π n e,

n ≥ 0.

(65)

5.2. Sojourn Time Distribution of a Tagged Customer To determine the mean sojourn time distribution of a tagged customer, one may proceed as previous subsection 5.1 and the steps are given as follows: I) Modification of the Markov chain: It may be observed that the expected sojourn time of a tagged customer depends only on the inverse of the matrix Q. At first it is needed to modify the level-dependent Markov chain to a level-independent Markov chain. Let N be a large positive integer then the modified Markov chain can be formulated as ⎞ ⎛ (N ) (N ) (N ) B(N ) A1 A2 A3 ··· ⎟ ⎜  (N ) ⎟ ⎜ C B A A · · · 1 2 N +1 ⎟ ⎜ (N )   ⎜ (66) Q =⎜ C N B N +1 A1 · · · ⎟ ⎟,  ⎟ ⎜ C N B N +1 · · · ⎠ ⎝ .. .. . . where C

(N )

  = 0, 0, 0 · · · , 0, C  N , 19

(67)

⎛

B(N )

B 1 A1 A2 ⎜ C  1 B 2 A1 ⎜ ⎜ C 2 B3 ⎜ =⎜ C 3 ⎜ ⎜ ⎝

and (N )

Ak

··· ··· ··· ··· .. .

AN −2 AN −3 AN −4 AN −5 .. .

C  N −1

AN −1 AN −2 AN −3 AN −4 .. .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(68)

BN

T  = ATN +k−1 , ATN +k−2 , ATN +k−3 , · · · , ATk .

(69)

II) G- and U-measure of the modified Markov chain: Let G be the minimal non-negative solution to the matrix equation C  N + B N +1 G +

∞ 

Al G l+1 = 0.

(70)

l=1

Now guided by the arguments in subsection 5.1, the following are defined U = B N +1 +

∞ 

Al G l ,

(71)

i=l (N )

G (1) = (−U )−1 C  , ∞  (N ) (N ) Ak G k−1 G (1) . U0 = B +

(72) (73)

k=1

III) R-measure of the modified Markov chain:  (N ) can be defined Following the subsection 5.1, the R-measure of the Markov chain Q as R0,k = Rk =

∞  l=0 ∞ 

(N )

Ak+l G l (−U )−1 ,

k ≥ 1,

(74)

Ak+l G l (−U )−1 ,

k ≥ 1.

(75)

l=0

IV) The inverse of the modified Markov chain: Setting s = 0, Equation (27) can be simplified as  (N ) −1  = [I − G L ]−1 [U D ]−1 [I − RU ]−1 , Q 20

(76)

where

[I − RU ]−1

⎛ I X 0,1 X 0,2 X 0,3 ⎜ I X 1,2 X 1,3 ⎜ ⎜ I X 2,3 =⎜ ⎜ I ⎝

⎞ ··· · · ·⎟ ⎟ · · ·⎟ ⎟, · · ·⎟ ⎠ .. .

−1 [U D ]−1 = diag(U −1 , U −1 , · · · ), 0 ,U ⎞ ⎛ I ⎟ ⎜Y 1,0 I ⎟ ⎜ ⎟ ⎜ I [I − G L ]−1 = ⎜Y 2,0 Y 2,1 ⎟, ⎟ ⎜Y 3,0 Y 3,1 Y 3,2 I ⎠ ⎝ .. .. .. .. . . . . . . .

for k ≥ 0,

 R0,1 , if k = 0, X k,k+1 = R1 , if k = 0,

for k ≥ 0 and l ≥ 2,

X k,k+l =

⎧ l−1  ⎪ ⎪ R0,n X n,l , ⎨ ⎪ ⎪ ⎩

for k ≥ l ≥ 1, Y k,k−l

n=1 l−1  n=1

(77)

(78)

(79)

(80)

if k = 0,

Rn X k+n,k+l, if k = 0,

 G l−1 G (1) , if l = k, = Gl, if l < k.

(81)

(82)

V) Mean sojourn time of a tagged customer: For 0 ≤ n ≤ N − 1, we define

  ϕn = 0, 0 · · · , 0, eT , 0, 0 · · · , 0 . ! "# $ ! "# $ n

times

(N −n−1)

times

Then the mean sojourn time of a tagged customer in the steady state can be approximately given by 21

if 0 ≤ n ≤ N − 1, 

E(Wn ) = ϕn − U 0

−1 

π ˆ0 +

∞ 



X 0,k π ˆk ,

(83)

k=1

and if n ≥ N , 

E(Wn ) = e ˆY n−N +1,0 − U 0

−1 

π ˆ0 +

∞ 

 X 0,k π ˆk +

k=1

eˆ

n−N 



Y n−N+1,l − U

l=1

 eˆ − U

−1 

π ˆl +

∞  k=l+1

−1 

π ˆ n−N +1 +

∞ 

 X l,k π ˆk + 

X n−N+1,k π ˆk .

(84)

k=n−N +2

Remark 5.1. Following Chaudhry et al. [6] the expected sojourn time of a k-type (k ≥ 1) customer in a BMAP/MSP/1 queue under FCFS discipline can be given by E(Wk ) =

k  n=0

πn

k      j   n−j −1 −1 I m1 ⊗ − L−1 I L ⊗ − L ⊗ − L L e. I 1 m1 m1 1 0 0 0 j=0

One may note that −L−1 0 L1 e = e as the state transition probability matrix, i.e., −L−1 L is a stochastic matrix. Hence the above equation reduced to 1 0 E(Wk ) =

k  n=0

πn

k  

  j   −1 I m1 ⊗ − L−1 I L ⊗ − L e. 1 m1 0 0

(85)

j=0

Remark 5.2. In a stable GPS system, it may happen that an arriving customer always finds exact k number of customers are getting service. If the initial condition is considered as   n (0) = α, for n = k, W (86) 0, for n = k, n ≥ 0, where α is a probability vector of order m with αe = 1, then the expected sojourn time of the k-type customer is estimated by  −1 E[Wk ] = ϕk − U 0 θ, (87) 22

T  where θ = 0, 0 · · · , 0, αT , 0, 0 · · · , 0 . ! "# $ ! "# $ k

times

(N −k−1)

times

Determination of N and N : It may be noted that G and G is largely dependent on N and N, respectively. Hence it may be written G(N) and G(N ) instead of G and G. The computation error can be made sufficiently small by choosing a suitable value of N and N and can be chosen by trial and error approach proposed by Neuts and Rao [21]. They determined the number using the spectral radius of a matrix. Let η(N) and χ(N ) are the respective spectral radius of the matrices G(N) and G(N ). Then N and N must be so chosen such that the following conditions are satisfied: (a) G(N)  G(N + 1) and G(N )  G(N + 1). (b) η(N + 1) − η(N) < 1 and χ(N + 1) − χ(N ) < 2 , where 1 and 2 are two predetermined small values. 6. Numerical Results and Discussion Based on the expressions obtained in the above discussion, lots of numerical results were generated and few of these results are presented here. The steady-state mean sojourn time of a tagged customer is obtained using the steady-state probability distribution and the two expressions (83) and (84). All the calculations are carried out in a PC having Intel(R) Core i5 processor @3.00 GHz with 4 GB DDR3 RAM using MAPLE 2015. Further, all the calculations are performed in high precision but due to lack of space the numerical values are presented only in 6 decimal places. In Table 1, the steady-state system length of a BMAP/MSP/1/∞-GPS queue with the following parameters is presented. For the experiment a 3-state BMAP and MSP is considered with the positive function, f (n) = (3n + 1)2 /n3 . The BMAP representation is given by ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ −60 2 5 2 6 4 3 8 5 D 0 = ⎣ 3 −55 2 ⎦ , D1 = ⎣ 6 5 3 ⎦ , D2 = ⎣ 3 4 2 ⎦ , 4 1 −75 7 5 4 8 6 7 ⎤ ⎤ ⎤ ⎡ ⎡ 4 7 3 3 4 4 0 0 0 / {0, 1, 2, 3, 5}. D 3 = ⎣ 6 8 2 ⎦ , D5 = ⎣ 2 5 4 ⎦ and D n = ⎣ 0 0 0 ⎦ , for n ∈ 5 9 6 3 4 6 0 0 0 ⎡

23

The 3-state MSP representation is expressed as ⎤ ⎤ ⎡ ⎡ −36 5 7 9 8 7 L0 = ⎣ 8 −48 6 ⎦ and L1 = ⎣ 11 10 13 ⎦ . 2 1 −24 5 8 8 For this model, it is calculated that π a = [0.318744 0.442026 0.239230], π s = [0.281935 0.215484 0.502580], λ = 148.395509, λb = 55.740834, μ = 24.647097 and ρ = 0.668934. It should be remarked here that, for 1 = 0.0001, 20 is a good choice of N for convergence of the matrix G.

n 0 1 2 3 4 5 10 15 20 25 30 35 40 45 50 55 60 65 . . . sum

πn,1 0.046195 0.005893 0.006612 0.006262 0.004562 0.004530 0.001512 0.000595 0.000242 0.000100 0.000042 0.000017 0.000007 0.000003 0.000001 0.000001 0.000000 0.000000 . . . 0.092822

Table 1: System-length distribution for the infinite model

πn,2 0.052097 0.004610 0.005172 0.004869 0.003469 0.003452 0.001112 0.000439 0.000179 0.000074 0.000031 0.000013 0.000005 0.000002 0.000001 0.000000 0.000000 0.000000 . . . 0.087536

πn,3 0.054477 0.010066 0.011314 0.010847 0.008156 0.008106 0.002872 0.001134 0.000461 0.000190 0.000079 0.000033 0.000014 0.000006 0.000002 0.000001 0.000000 0.000000 . . . 0.138386

πn,4 0.062397 0.007913 0.009410 0.009060 0.006555 0.006667 0.002155 0.000846 0.000344 0.000142 0.000059 0.000025 0.000010 0.000004 0.000002 0.000001 0.000000 0.000000 . . . 0.128666

πn,5 0.070372 0.006188 0.007375 0.007069 0.005004 0.005104 0.001585 0.000624 0.000254 0.000105 0.000044 0.000018 0.000008 0.000003 0.000001 0.000001 0.000000 0.000000 . . . 0.120821

πn,6 0.073585 0.013514 0.016049 0.015612 0.011651 0.011851 0.004092 0.001612 0.000655 0.000270 0.000112 0.000047 0.000020 0.000008 0.000004 0.000001 0.000001 0.000000 . . . 0.192539

πn,7 0.029808 0.004499 0.005300 0.004897 0.004171 0.004525 0.001335 0.000516 0.000209 0.000086 0.000036 0.000015 0.000006 0.000003 0.000001 0.000000 0.000000 0.000000 . . . 0.069478

πn,8 0.033577 0.003536 0.004177 0.003830 0.003218 0.003519 0.000980 0.000381 0.000154 0.000064 0.000026 0.000011 0.000005 0.000002 0.000001 0.000000 0.000000 0.000000 . . . 0.063902

πn,9 0.035125 0.007621 0.008963 0.008396 0.007280 0.007862 0.002532 0.000984 0.000398 0.000164 0.000068 0.000028 0.000012 0.000005 0.000002 0.000001 0.000000 0.000000 . . . 0.105850

πn e 0.457633 0.063840 0.074371 0.070841 0.054067 0.055618 0.018175 0.007130 0.002896 0.001193 0.000496 0.000208 0.000087 0.000037 0.000016 0.000007 0.000003 0.000001 . . . 1.000000

Note: If f (n) is set to 1/n, then the system reduces to a BMAP/MSP/1-EPS system and the steady-state probabilities of the processor-sharing queue is same as BMAP/MSP/1 queue, studied by Samanta et al. [24]. The steady-state solution of the EPS system is verified with the result provided by Samanta et al. [24, p. 31] in Table 1 and it is found that the probabilities are matching exactly up to 6 decimal places. For different BMAP and MSP representations numerous results are generated for BMAP/MSP/1/∞-GPS system. In Figure 1, the effect of ρ on the mean-system length at steady state is presented, whereas Figure 2 and 3 describe the effect of ρ on the mean sojourn time of 0-tagged and 10-tagged customer, respectively. The experiments are carried out presuming 2-state correlated and uncorrelated BMAP representations with λ = 20.230769 and 2-state correlated and uncorrelated MSP representations with μ = 34.75. The different values of ρ is generated by taking 20n different positive functions as fk (n) = (5+k)n 2 +5 (k = 1, 2, · · · , 10). The parameters of the BMAP and MSP are illustrated as follows: 24

The 2-state BMAP representation with lag-1 correlation coefficient 0.259915 is taken as * * * ) ) ) 65.50 0.5 35.0 2.50 15.0 1.0 , D1 = , D2 = , D0 = 0.5 −3.5 0.25 1.25 0.25 0.75 * * ) ) 0.0 0.0 10.0 1.50 D3 = , for n ≥ 4. and D n 0.0 0.0 0.0 0.5 The 2-state MSP representation with lag-1 correlation coefficient 0.279256 is considered as * * ) ) −75.0 0.25 74.0 0.75 L0 = and L1 = . 0.25 −15.0 0.25 14.5 The 2-state uncorrelated BMAP representation is given by * * * ) ) ) −12.0 1.5 1.89 1.26 1.89 1.26 , D1 = , D2 = , D0 = 1.35 −10.0 1.557 1.038 1.557 1.038 * * ) ) 2.52 1.68 0.0 0.0 D3 = and Dn = , for n ≥ 4. 2.076 1.384 0.0 0.0 The 2-state uncorrelated MSP representation is assumed as * * ) ) −35.0 3.0 9.6 22.4 and L1 = . L0 = 2.0 −38.1 10.83 25.27

10

mean system−length (Ls) →

9 8

Correlated BMAP correlated MSP Correlated BMAP uncorrelated MSP Uncorrelated BMAP correlated MSP Uncorrelated BMAP uncorrelated MSP

7 6 5 4 3 2 1 0

0.2

0.25

0.3 ρ→

0.35

Figure 1: n versus E(Wn )

25

0.4

0.16 0.14

Correlated BMAP correlated MSP Correlated BMAP uncorrelated MSP Uncorrelated BMAP correlated MSP Uncorrelated BMAP uncorrelated MSP

0.55 0.5 0.45 0.4 E(W10) →

E(W0) →

0.12

Correlated BMAP correlated MSP Correlated BMAP uncorrelated MSP Uncorrelated BMAP correlated MSP Uncorrelated BMAP uncorrelated MSP

0.35

0.1

0.08

0.3

0.25 0.06

0.2

0.04

0.15 0.1

0.02 0.2

0.25

0.3 ρ→

0.35

0.4

0.2

0.25

0.3 ρ→

0.35

0.4

Figure 3: ρ versus E(W10 )

Figure 2: ρ versus E(W0 )

From Figure 1, one can observe that for equivalent model parameters, the mean system-length for correlated arrival and service is higher than the mean systemlength for uncorrelated arrival and service. It is also observable from Figure 2 that the mean sojourn time of the 0-type customer for correlated BMAP and uncorrelated MSP is higher than uncorrelated BMAP and correlated MSP , while the reverse situation is observed for 10-type customer from Figure 3. For 2 = 0.0001, it is checked that when both the BMAP and MSP are correlated the spectral radius of the matrix G(N) converges for N = 149. The value of the spectral radius of the matrix G(N) for different values of N is enlisted in the Table 2. Table 2: Determination of N N 1 10 30 60 90 120 145 146 147 148 149

spectral radius 0.282836 0.90187 0.964575 0.981896 0.98784 0.990846 0.992409 0.992461 0.992511 0.992562 0.992611

It is to be mentioned here that, using uniformization technique the expression to compute the complementary distribution of the sojourn time of an tagged customer in an MAP/M/1-EPS system was formulated by Masuyama and Takine [16, p. 409, 26

Equation 7]. The expected sojourn time of a tagged customer in the MAP/M/1EPS system can be also obtained as a special case of the BMAP/MSP/1-GPS system. Taking the same model parameters as considered by Masuyama and Takine [16, subsection 4.3] along with the initial condition as described in Remark 3.1, the expected sojourn time of the tagged customers are computed using both the uniformization and RG-factorization technique and is presented in Table 3. Table 3: Sojourn time of the tagged customer’s for the M AP/M/1-EPS queue n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Using uniformization technique 2.371660 3.457059 4.546859 5.638030 6.728799 7.817964 8.904579 9.987791 11.066771 12.140675 13.208627 14.269718 15.323007 16.367524 17.402284 18.426294

Using RG-factorization 2.374461 3.462499 4.556171 5.652821 6.751111 7.850341 8.950135 10.050278 11.150646 12.251165 13.351787 14.452481 15.553228 16.654013 17.754826 18.855662

Further it should be remarked here that, assuming the initial condition considered by Li [12, p. 526] and taking the same MAP representation and service rate along with the function f1 (n), the desired results are achievable as a special case of the BMAP/MSP/1-GPS model for E(W0 ) as showed in Figure 1 [12, p. 526] but the negative value of E(W10 ) as presented in Figure 2 [12, p. 526], can not be interpreted. One may note that the second function (f2 (n)) in Example 1 considered by Li [12, p. 526] does not fit to the BMAP/MSP/1-GPS model as nf2 (n) converges to 0 as n → ∞. The expected sojourn time in FCFS, EPS and GPS system of the n-type customers are compared in Figure 4. The figure is generated by considering a 2-state BMAP and 2-state MSP representation. For the GPS system, different traffic intensities, i.e., ρ = 0.1, 0.2 and 0.4, the positive function (f (n)) is taken as n−1.51 , 1.5 1 2n− n−1 2

and

1.5 1 4n− n−2

) D0 =

2

2n

, respectively. The BMAP is represented as

−3.5 0.5 0.25 −2.0

*

) , D2 =

0.5 0.5 0.25 0.25

27

*

) , D4 =

0.5 0.5 0.25 0.25

* ,

) D6 =

0.5 0.5 0.5 0.25

The MSP is expressed as

)

L0 =

*

) and Dn =

−80 4 3 −50

0 0 0 0

*

*

, for n ∈ / {0, 2, 4, 6}. )

and L1 =

75 1 2 45

* .

3.5 3

E(Wn) →

2.5

FCFS (ρ=0.15) EPS (ρ=0.15) GPS (ρ=0.1) GPS (ρ=0.2) GPS (ρ=0.4)

2

1.5 1 0.5 0 0

20

40

n→

60

80

100

Figure 4: n versus E(Wn )

From the Figure 4, it is seen that the expected sojourn time of the tagged customers in GPS system increases as the traffic intensity increases. It is also spotted from the figure that the expected sojourn time increases linearly with the tag. On the other hand, it may be concluded from the figure that a customer have to spend more time in a FCFS system than an EPS system, while in a low traffic intensity GPS system a customer can get quick service than both the FCFS and EPS system. In Table 4, the steady-state system length of a BMAP/MSP/1/10-GPS queue 3 with positive function f (n) = 6n− 1 , 3-state BMAP and 2-state MSP is presented. n The results are generated for the following BMAP and MSP representation. ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ −3.5 0.1 0.25 0.25 0.35 0.45 0.35 0.45 0.25 D 0 = ⎣ 0.4 −3.0 0.35 ⎦ , D 1 = ⎣ 0.15 0.25 0.35 ⎦ , D3 = ⎣ 0.25 0.35 0.15 ⎦ , 0.2 0.1 −3.0 0.2 0.3 0.4 0.3 0.4 0.2 ⎤ ⎤ ⎡ ⎡ 0.45 0.25 0.35 0 0 0 ⎦ ⎣ ⎣ / {0, 1, 3, 5}, D 5 = 0.35 0.15 0.25 , D n = 0 0 0 ⎦ , for n ∈ 0.4 0.2 0.3 0 0 0 * * ) ) −65 5 35 25 and L1 = . L0 = 5 −85 45 35 28

For this model, one can compute that π a = [0.314397 0.322203 0.3634], π s = [0.625 0.375], λ = 8.089462, λb = 2.696487, μ = 67.5 and ρ = 0.239683.

n 0 1 2 3 4 5 6 7 8 9 10 sum

Table 4: System-length distribution of the finite model

πn,1 0.133683 0.010071 0.009677 0.010552 0.007194 0.007407 0.002107 0.001533 0.001152 0.000707 0.000506 0.184589

πn,2 0.099209 0.006117 0.005872 0.006409 0.004338 0.004477 0.001210 0.000881 0.000648 0.000399 0.000248 0.129808

πn,3 0.144455 0.010784 0.009179 0.009866 0.004912 0.004681 0.001768 0.001200 0.000869 0.000499 0.000351 0.188565

πn,4 0.107203 0.006548 0.005557 0.005979 0.002926 0.002798 0.001007 0.000685 0.000485 0.000280 0.000170 0.133638

πn,5 0.167273 0.011181 0.008188 0.008559 0.006085 0.006000 0.001832 0.001272 0.000942 0.000580 0.000404 0.212316

πn,6 0.124134 0.006779 0.004933 0.005163 0.003644 0.003605 0.001045 0.000728 0.000529 0.000326 0.000197 0.151084

πe 0.775957 0.051481 0.043407 0.046528 0.029099 0.028968 0.008970 0.006299 0.004626 0.002791 0.001876 1.000000

Note that the performance measures of an BMAP/MSP/1/∞-GPS model can be approximated by corresponding BMAP/MSP/1/M-GPS model for large value of M with low traffic intensity, i.e., ρ << 1. To show this phenomena, the mean systemlengths of a finite-buffer model for different values of M is plotted in Figure 5. For this experiment the following BMAP and MSP representation is considered with 2 the positive function f (n) = 2n− 1 . n

* * ) ) −6 1.5 1.5 0.5 0.5 1.0 , D2 = , D4 = , D0 = 1 −7 1.5 1.0 0.5 1.5 * * ) ) 0.5 0.5 0 0 D6 = , Dn = , for n ∈ / {0, 2, 4, 6}, 1.0 0.5 0 0 ⎤ ⎤ ⎡ ⎡ −50 6 5 13 11 15 L0 = ⎣ 3 −45 2 ⎦ and L1 = ⎣ 15 12 13 ⎦ . 6 4 −44 13 10 11 )

*

29

2.6

mean system-length (LS ) →

2.4 Finite model Infinite model

2.2 2 1.8 1.6 1.4 1.2 1 0.8 0

10

20

30

40

50

60

70

M→

Figure 5: M versus LS

In Figure 6, the blocking probabilities of an arbitrary customer in a BMAP/MSP/1/101 GPS system with the positive function fk (n) = 2(k+1)n , where k is an integer, is presented. The following 3-state BMAP and 3-state MSP representation is taken for the experiment. ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ −5.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 D 0 = ⎣ 0.5 −5.5 0.5 ⎦ , D1 = ⎣ 0.5 0.5 0.5 ⎦ , D2 = ⎣ 0.5 0.5 0.5 ⎦ , 0.5 0.5 −5.5 0.5 0.5 0.5 0.5 0.5 0.5 ⎤ ⎤ ⎡ ⎡ 0.5 0.5 0.5 0 0 0 D 4 = ⎣ 0.5 0.5 0.5 ⎦ , Dn = ⎣ 0 0 0 ⎦ , for n ∈ / {0, 2, 4, 6}. 0.5 0.5 0.5 0 0 0 ⎤ ⎤ ⎡ ⎡ −214 2 2 70 70 70 −214 2 ⎦ and L1 = ⎣ 70 70 70 ⎦ . L0 = ⎣ 2 2 2 −214 70 70 70. 0.5 0.45 0.4 0.35

PBL A

0.3 0.25 0.2 0.15 0.1 0.05 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ρ→

Figure 6: ρ versus P BLA

30

1.8

2

7. Conclusion and Future Research Scope In this paper we have analyzed the BMAP/MSP/1-GPS queue for both finiteand infinite-buffer capacity. Several performance measures, like mean system-length, expected waiting time of a tagged customer and the blocking probabilities have been derived through the use of RG-factorization technique. Similar kind of analysis can be done for the BMAP/MSP/1-SIRO (service in random order) queueing model. One may also be interested to study a BMAP/MSP/1-GPS queue for state-dependent arrival and/or service rates. These problems are left for future investigation. Acknowledgment Authors are thankful to the editors and the referee for their valuable comments and suggestions which help to improve the paper in its present form. This research work received partial financial support from the Department of Science and Technology, New Delhi, India, under the research grant SR/FTP/MS-003/2012. References [1] Aalto, S., Ayesta, U., Borst, S., Misra, V., N´un ˜ ez-Queija, R., 2007. Beyond processor sharing. ACM SIGMETRICS Performance Evaluation Review 34 (4), 36–43. [2] Asmussen, S., 2003. Applied probability and queues. Vol. 51. Springer Science & Business Media. [3] Avrachenkov, K., Ayesta, U., Brown, P., 2005. Batch arrival processor-sharing with application to multi-level processor-sharing scheduling. Queueing Systems 50 (4), 459–480. [4] Banik, A. D., 2009. Queueing analysis and optimal control of BMAP/G(a,b) /1/N and BMAP/MSP (a,b) /1/N systems. Computers & Industrial Engineering 57 (3), 748–761. [5] Bansal, N., 2003. Analysis of the M/G/1 processor-sharing queue with bulk arrivals. Operations Research Letters 31 (5), 401–405. [6] Chaudhry, M., Banik, A., Pacheco, A., 2016. A simple analysis of the batch arrival queue with infinite-buffer and Markovian service process using roots method: GI [X] /C − MSP/1/∞. Annals of Operations Research (Article in press), 1–39. 31

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