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(2) Yutaka Baba. 54. the beginning of the current busy period. They also analyzed the virtual waiting time at time t. However, they did not obtain the definite scheme for computing the moments of stationary queue length distribution and sojourn time distribution. Baba [1] derived the recursion formulas to obtain the generating function of stationary queue length distribution given that n customers have been served since the beginning of the current busy period by applying the supplementary variable approach. Furthermore, [1] presented a computationally tractable scheme to recursively determine the moments of queue length distribution and sojourn time distribution. Kengaku and Miyazawa [4] derived a regenerative approach with respect to a busy cycle. Their approach neither needs to assume the steady state nor to use the Kolmogorov differential equations used in [1] and [3]. Furthermore, they studied general structure of characteristics of interests such as the mean waiting time, in particular, how they are affected by the exceptional service. In this paper, we expand the M/G/1 queue with exceptional services studied in [1], [3] and [4] into the M/G/1 queue with multiple vacations and exceptional services. Applying the supplementary variable approach used in [1], we derive the recursion formulas to obtain the generating function of steady state queue distribution given that n customers have been served since the beginning of the current busy period. Furthermore, we present a computationally tractable scheme to recursively determine the moments of queue length distribution and sojourn time distribution. This paper is organized as follows. We describe the model and introduce notation in section 2. In section 3, we derive a set of Laplace-Stieltjes transform equations using supplementary variable approach. In section 4, we derive a numerical algorithm to obtain some steady state probabilities and the moments of queue length distribution and sojourn time distribution. In section 5, we derive a numerical algorithm to obtain the generating function of steady state queue length distribution. Special cases are treated in detail, yielding explicit formulas for generating functions of steady state queue length distribution in section 6.. 2. Description of the model. We assume that customers arrive at the system according to a Poisson process with intensity λ. Let Bn (m) denote the service time of the n + 1st customer served in the mth busy period, where {Bn (m)}m are independent and identically distributed according to a random variable Bn . We denote the distribution function (DF) (Laplace-Stieltjes transform (LST)) of Bn by Bn (x) (Bn∗ (θ)). The service times are assumed to be independent random variables that are independent of the arrival process. Furthermore, we assume that the service time distribution becomes stable after some customers have been served in the current busy period. That is, there is a positive integer N ≥ 1 such that Bn (x) = BN (x) for n ≥ N . Similarly, let V denote a typical vacation time of the server. We denote DF (LST) of V by V (x) (V ∗ (θ)). The vacation times are independent and identically distributed random variables that are independent of the arrival process. The state of the system at time t is described by the following random variables, namely, • ξ(t) = {1}(0) if the server {is busy}(on vacation), • L(t) = number of customers in the system including the one in service, • M (t) = number of customers served since the beginning of the current busy period when the server is busy, ˆ = remaining service time of the customer in service, • B(t) • Vˆ (t) = remaining vacation time of the server.. ˆ We define the joint probability densities of L(t), M (t) (if the server is busy), ξ(t) and B(t) ˆ (V (t)), respectively, by ˆ ≤ x + dx, ξ(t) = 1) (i = 1, 2, . . . ; j = 0, 1, . . . , N − 1) πij (x, t)dx ≡ P (L(t) = i, M (t) = j, x < B(t) ˆ ≤ x + dx, ξ(t) = 1) (i = 1, 2, . . . ) πiN (x, t) ≡ P (L(t) = i, M (t) ≥ N, x < B(t) ωi (x, t)dx ≡ P (L(t) = i, x < Vˆ (t) ≤ x + dx, ξ(t) = 0) (i = 0, 1, . . . ).
(3) Analysis of an M/G/1 Queue with Multiple Vacations and Exceptional Services. 55. As we shall discuss the model in steady state, i.e., t → ∞, the above probabilities will be denoted by πij (x) (i = 1, 2, . . . ; j = 0, 1, . . . , N ) and ωi∫(x) (i = 0, 1, . . . ), respectively. Further we define ∫ the LST’s of πij (x) and ωi (x) by Π∗ij (θ) = respectively.. 3. ∞. ∞. e−θx πij (x)dx and Ωi (θ) =. 0. e−θx ωi (x)dx,. 0. Supplementary variable method. In this section, we obtain LST’s Π∗ij (θ) (i ≥ 1, 0 ≤ j ≤ N ) and Ω∗i (θ) (i ≥ 0), which are the bases of analysis in the following section. Observing the system at an arbitrary time t and focusing on the possible state transitions during (t, t + ∆t], we have the following Markovian equation: π10 (x − ∆t, t + ∆t) = (1 − λ∆t)π10 (x, t) + ω10 (0, t){B0 (x) − B0 (x − ∆t)} + o(∆t). (1). or equivalently π10 (x − ∆t, t + ∆t) = (1 − λ∆t)π10 (x, t) + ω10 (0, t). B0 (x) − B0 (x − ∆t) o(∆t) + ∆t ∆t. (2). Taking the limits of ∆t → 0 as well as t → ∞ yields −. dπ10 (x) dB0 (x) = −λπ10 (x) + ω1 (0) . dx dx. (3). Using the similar arguments as above for the cases of πij (x) and ωi (x), we obtain the following system of differential-difference equations. − − − − − − −. dπi0 (x) dB0 (x) = −λπi0 (x) + λπi−1,0 (x) + ωi (0) (i = 2, 3, . . . ) dx dx π1j (x) dBj (x) = −λπ1j (x) + π2,j−1 (0) dx dx πij (x) dBj (x) = −λπij (x) + λπi−1,j (x) + πi+1,j−1 (0) (i = 2, 3, . . . ; j = 1, 2, . . . , N − 1) dx dx dπ1N (x) dBN (x) = −λπ1N (x) + {π2,N −1 (0) + π2N (0)} dx dx dπiN (x) dBN (x) = −λπiN (x) + λπi−1,N (x) + {πi+1,N −1 (0) + πi+1,N (0)} (i = 2, 3, . . . ) dx dx N ∑ ω0 (x) dV (x) π1j (0) + ω0 (0)} = −λω0 (x) + { dx dx j=0 ωi (x) = −λωi (x) + λωi−1 (x) dx. (i = 1, 2, . . . ). (4) (5) (6) (7) (8) (9) (10). Taking the LST’s of (3)–(10), we have (λ − θ)Π∗10 (θ) = ω1 (0)B0∗ (θ) − π10 (0). (11). (λ −. (13). (λ − (λ − (λ − (λ −. θ)Π∗i0 (θ) = λΠ∗i−1,0 (θ) + ωi (0)B0∗ (θ) − πi0 (0) (i = 2, 3, . . . ) θ)Π∗1j (θ) = π2,j−1 (0)Bj∗ (θ) − π1j (0) (j = 1, 2, . . . , N − 1) θ)Π∗ij (θ) = λΠ∗i−1,j (θ) + πi+1,j−1 (0)Bj∗ (θ) − πij (0) (i = 2, 3, . . . ; j = ∗ θ)Π∗1N (θ) = {π2,N −1 (0) + π2N (0)}BN (θ) − π1N (0) ∗ ∗ ∗ θ)ΠiN (θ) = λΠi−1,N (θ) + {πi+1,N −1 (0) + πi+1,N (0)}BN (θ) − πiN (0) N ∑. (λ − θ)Ω∗0 (θ) = {. j=0. π1j (0) + ω0 (0)}V ∗ (θ) − ω0 (0). (λ − θ)Ω∗i (θ) = λΩ∗i−1 (θ) − ωi (0) (i = 1, 2, . . . ). (12) 1, 2, . . . , N − 1) (14). (15). (i = 2, 3, . . . ). (16) (17) (18).
(4) Yutaka Baba. 56. 4. Steady state probability. Let us define c≡. N ∑. π1j (0) + ω0 (0). (19). j=0. We can obtain ωi (0) (i = 0, 1, . . . , N ), πi0 (0) (i = 1, 2, . . . , N ) and π1j (0) (j = 1, 2, . . . , N ) in terms of c. We will use these expressions to analyze the moments of queue length distribution and sojourn time distribution in section 5.. 4.1. ωi (0) (i = 0, 1, . . . , N ) in terms of c. We express ωi (0) (i = 0, 1, . . . , N ) in terms of c. Differentiating (17) and (18) n + 1 times and inserting θ = λ, we have (n). − (n + 1)Ω0 (λ) = cV ∗(n+1) (λ) − (n +. (n) 1)Ωi (λ). =. (20). (n+1) λΩi−1 (λ). (i = 1, 2, . . . , N ). (21). Using (20) and (21), we have ωi (0) =. 4.2. (−λ)i ∗(i) cV (λ) i!. (i = 0, 1, . . . , N ). (22). πi0 (0) (i = 1, 2, . . . , N ) in terms of c. We now express πi0 (0) (i = 1, 2, . . . , N ) in terms of c. Substituting θ = λ into (11) and (12), we have π10 (0) = ω1 (0)B0∗ (λ). (23). πi0 (0) = λΠ∗i−1,0 (λ) + ωi (0)B0∗ (λ). (i = 2, 3, . . . , N ). (24). Differentiating (13) and (14) n + 1 times and substituting θ = λ, we have ∗(n). ∗(n+1). − (n + 1)Π10 (λ) = ω1 (0)B0 − (n +. ∗(n) 1)Πi0 (λ). =. ∗(n+1) λΠi−1,0 (λ). (λ). +. (25). ∗(n+1) ωi (0)B0 (λ). (i = 2, 3, . . . , N ). (26). Using (23)–(26), we can calculate πi0 (0) (i = 1, 2, . . . , N ) in terms of c.. 4.3. π1j (0) (j = 1, 2, . . . , N ) in terms of c. We finally express π1j (0) (j = 1, 2, . . . , N ) in terms of c. Substituting θ = λ into (13) and (14), we have π1j (0) = π2,j−1 (0)Bj∗ (λ) πij (0) =. λΠ∗i−1,j (λ). +. (j = 1, 2, . . . , N − 1). πi+1,j−1 (0)Bj∗ (λ). (27). (i = 2, 3, . . . ; j = 1, 2, . . . , N − 1). (28). Differentiating (13) and (14) n + 1 times and substituting θ = λ, we have ∗(n). ∗(n+1). − (n + 1)Π1j (λ) = π2,j−1 (0)Bj ∗(n). ∗(n+1). (λ). (j = 1, 2, . . . , N − 1) ∗(n+1). − (n + 1)Πij (λ) = λΠi−i,j (λ) + πi+1,j−1 (0)Bj. (λ). (29). (i = 2, 3, . . . ; j = 1, 2, . . . , N − j) (30). Using (27)–(30), we can calculate πij (0) (j = 1, 2, . . . , N − 1; i = 1, 2, . . . , N − j) in terms of c by the following numerical algorithm..
(5) Analysis of an M/G/1 Queue with Multiple Vacations and Exceptional Services. 57. Algorithm for j = 1 to N − 1 do for i = 1 to N − j do πij (0) =. i−1 ∑ (−1)k λk. k=0. k!. ∗(k). πi−k+1,j−1 (0)Bj. (λ). (31). Hence, π1j (0) (j = 1, 2, . . . , N − 1) can be obtained from (31). Finally, we have π1N (0) = c −. N −1 ∑ j=0. π1j (0) − ω0 (0). (32). from (19). It immediately follows that we can express π1j (0) (j = 1, 2, . . . , N ) in terms of c from (23), (24), (31) and (32).. 4.4. Generating functions. We define the following generating functions. Pj∗ (x, θ) pj (z) ≡. ≡. Using (11)–(18), we have. Π∗ij (θ)z i. i=1. ∞ ∑. πij (0)z i. j=1. Q∗ (z, θ) ≡ q(z) ≡. ∞ ∑. ∞ ∑ i=0. ∞ ∑. (|z| ≤ 1; j = 0, 1, . . . , N ). Ω∗i (θ)z i. i=0. ωi (0)z i. (|z| ≤ 1). (|z| ≤ 1). (|z| ≤ 1). (λ − λz − θ)P0∗ (z, θ) = {q(z) − ω0 (0)}B0∗ (θ) − p0 (z) { } pj−1 (z) (λ − λz − θ)Pj∗ (z, θ) = − π1,j−1 (0) Bj∗ (θ) − pj (z) z { } pN −1 (z) + pN (z) ∗ (λ − λz − θ)PN∗ (z, θ) = − π1,N −1 (0) − π1N (0) BN (θ) − pN (z) z (λ − λz − θ)Q∗ (z, θ) = {. N ∑ j=0. π1j (0) + ω0 (0)}V ∗ (θ) − q(z) = cV ∗ (θ) − q(z). (33) (34) (35) (36). (37) (38) (39) (40). Substituting θ = λ − λz into (37)–(40), we have p0 (z) = {q(z) − ω0 (0)}B0∗ (λ − λz) } { pj−1 (z) − π1,j−1 (0) Bj∗ (λ − λz) (j = 1, 2, . . . , N − 1) pj (z) = z } { pN −1 (z) + pN (z) ∗ − π1,N −1 (0) − π1N (0) BN (λ − λz) pN (z) = z N ∑ π1j (0) + ω0 (0)}V ∗ (λ − λz) = cV ∗ (λ − λz) q(z) = {. (41) (42) (43) (44). j=0. Rearranging (43), we have pN (z) =. ∗ {pN −1 (z) − π1,N −1 (0)z − π1N (0)z}BN (λ − λz) ∗ z − BN (λ − λz). (45).
(6) Yutaka Baba. 58. Further substituting θ = 0 into (37)–(40), we obtain {q(z) − ω0 (0)}{1 − B0∗ (λ − λz)} λ(1 − z) ∗ {p (z) − π j−1 1,j−1 (0)z}{1 − Bj (λ − λz)} Pj∗ (z, 0) = (j = 1, 2, . . . , N − 1) λz(1 − z) ∗ {pN −1 (z) − π1,N −1 (0)z − π1N (0)z}{1 − BN (λ − λz)} PN∗ (z, 0) = ∗ λ(1 − z){z − BN (λ − λz)} ∗ c{1 − V (λ − λz)} Q∗ (z, 0) = λ(1 − z) P0∗ (z, 0) =. 4.5. (46) (47) (48) (49). Determination of c. Substituting z = 1 into (41), (42) and (44), we have p0 (1) =. N ∑ j=0. π1j (0) = q(1) − ω0 (0) = c − ω0 (0). (50). pj (1) = pj−1 (1) − π1,j−1 (0) (j = 1, 2, . . . , N − 1). (51). q(1) =. (52). N ∑. π1j (0) + ω0 (0) = c. j=0. Differentiating (44), (41) and (42) and substituting z = 1, we have q (1) (1) = −cλV ∗(1) (0). (53) ∗(1). (1). p0 (1) = q (1) (1) − λ{c − ω0 (0)}B0 (1) pj (1). =. (1) pj−1 (1). (0). (54). − pj−1 (1) − λ{pj−1 (1) −. ∗(1) π1,j−1 (0)}Bj (0). (j = 1, 2, . . . , N − 1). (55). Using L’Hospital’s rule in (45), we have (1). pN (1) = lim pN (z) = z→1−0. pN −1 (1) − π1,N −1 (0) − π1N (0) ∗(1). 1 + λBN (0). (56). Furthermore, using L’Hospital’s rule in (46)–(49), we finally obtain ∗(1). P0∗ (1, 0) = −{c − ω0 (0)}B0. (0). (57) ∗(1). Pj∗ (1, 0) = −{pj−1 (1) − π1,j−1 (0)}Bj PN∗ (1, 0) = −. (1) {pN −1 (1). (0) (j = 1, 2, . . . , N − 1). ∗(1) − π1,N −1 (0) − π1N (0)}BN (0) ∗(1) λ{1 + λBN (0)}. Q∗ (1, 0) = −cV ∗(1) (0). (58) (59) (60). Since ωi (0) (i = 0, 1, . . . , N ) and π1j (0) (j = 0, 1, . . . , N − 1) are expressed in terms of c, we can express Pj∗ (1, 0) (j = 0, 1, . . . , N ) and Q∗ (1, 0) in terms of c using (50)–(60). From the normalization condition, N ∑ Pj∗ (1, 0) + Q(1, 0) = 1 (61) j=0. we can determine c. Therefore, the steady state probabilities immediately follow.. Remark 1. It follows from (56) that this queueing system is stable if and only if ∗(1) 1 + λBN (0) > 0, that is, λE(BN ) < 1..
(7) Analysis of an M/G/1 Queue with Multiple Vacations and Exceptional Services. 5. 59. Moments of queue length and sojourn time. In this section, we derive a computationally tractable scheme which determines the queue length distribution and the sojourn time distribution in steady state. We assume in this paper that the sojourn time is defined as the time between the arrival epoch and the end of the service time of an arbitrary customer.. 5.1. Moments of queue length. We define the generating function of steady state queue length distribution as L(z) ≡ E(z L ) =. N ∑. Pj∗ (z, 0) + Q∗ (z, 0). (62). j=0. where L denotes the steady state queue length. Differentiating (62) n times with respect to z and substituting z = 1, it is shown that nth factorial moment of L is given as E[L(L − 1) · · · (L − n + 1)] ≡ L(n) (1) =. N ∑. ∗(n). Pj. (1, 0) + Q∗(n) (1, 0). (63). j=0. ∗(n). Hence, to obtain the nth factorial moment of L, it suffices to calculate Pj. (1, 0) (j = 0, 1, . . . , N ) ∗(n). and Q (1, 0). We have the following tractable numerical algorithm to calculate Pj (j = 0, 1, . . . , N ) and Q∗(n) (1, 0) by differentiating (41)–(49) and substituting z = 1. ∗(n). Algorithm for k = 0 to n do begin q (k+1) (1) = c(−λ)k+1 V ∗(k+1) (0) (k+1). p0. (k+1). pj. ∗(k+1). (1) = {c − ω0 (0)}(−λ)k+1 B0 (0) ( ) k+1 ∑ k+1 ∗(k+1−m) + q (m) (1)(−λ)k+1−m B0 (0) m m=1 (k). (1). (k+1). pN. end. ∗(k+1). (1) = −(k + 1)pj (1) + {pj−1 (1) − π1,j−1 (0)}(−λ)k+1 Bj ∗(k). + (k + 1){pj−1 (1) − π1,j−1 (0)}(−λ)k Bj (0) ) k ( ∑ k + 1 (m) ∗(k+1−m) + pj−1 (1)(−λ)k+1−m Bj (j = 1, 2, . . . , N − 1) m m=2 [ k+1 ( ∑ k + 1) 1 ∗(m) (k+1−m) = (−λ)m BN (0)pN (1) ∗(1) m (k + 1){1 + λBN (0)} m=2 k+1 ∑ (k + 1) (m) ∗(k+1−m) + pN −1 (1)(−λ)k+1−m BN (0) m m=2 ] (1) ∗(k) +(k + 1){pN −1 (1) − π1,N −1 − π1N (0)}(−λ)k BN (0) (−λ)n ∗(n+1) V (0) n+1 (−λ)n ∗(n+1) ∗(n) P0 (1, 0) = −{c − ω0 (0)} B (0) n+1 0 ( ) n ∑ n + 1 (m) (−λ)n−m ∗(n+1−m) − q (1) B0 (0) m n+1 m=1. Q∗(n) (1, 0) = −. (1, 0).
(8) Yutaka Baba. 60. for k = 1 to n do begin ∗(k). Pj. ∗(k−1). (1, 0) = −kPj. (1, 0) −. 1 ∗(k+1) {pj−1 (1) − π1,j−1 (0)}(−λ)k Bj (0) k+1 ∗(k). (1). ∗(k). PN. − {pj−1 (1) − π1,j−1 (0)}(−λ)k−1 Bj (0) ) k ( 1 ∑ k + 1 (m) ∗(k+1−m) − pj−1 (1)(−λ)k−m Bj (0) k + 1 m=2 m. (j = 1, 2, . . . , N − 1). ∗(k−1). (1, 0) = −kPN (1, 0) 1 ∗(k+1) − {pN −1 (1) + pN (1) − π1,N −1 (0) − π1N (0)}(−λ)k BN (0) k+1 (1). ∗(k+1). (1). − {pN −1 (1) + pN (1) − π1,N −1 (0) − π1N (0)}(−λ)k BN (0) ( ) k 1 ∑ k+1 (m) (m) ∗(k+1−m) {pN −1 (1) + pN (1)}(−λ)k−m BN (0) − k + 1 m=2 m end. 5.2. Moments of sojourn time. Let S and S ∗ (θ) be the steady state sojourn time of an arbitrary customer and its LST, respectively. Since the arrival process is Poissonian and the service discipline is FCFS, the distribution of L and S are related by L(z) = S ∗ (λ − λz) (64) Hence, we have. E[L(L − 1) · · · (L − n + 1)] = L(n) (1) = (−1)n λn S ∗(n) (0) = λn E(S n ). (65). We obtain from (65) that nth moment of S can be calculated from the nth factorial moment of L.. 6. Special cases. In this section, we derive explicit formulas for the generating function of steady state queue length distribution, L(z), and the LST of sojourn time distribution, S ∗ (θ), for special cases that N equals 1 and 2. Substituting z = 1 − λ/θ into (64), we have S ∗ (θ) = L(1 − θ/λ). (66). Hence, if the explicit formula for L(z) is found, then we have the explicit formula for S ∗ (θ) from (66). (i) N = 1 L(z) ≡ P0∗ (z, 0) + P1∗ (z, 0) + Q∗ (z, 0) =. {1 − λE(B1 )}[−V ∗ (λ − λz)B0∗ (λ − λz) + V ∗ (λ){B0∗ (λ − λz) − B1∗ (λ − λz)} + B1∗ (λ − λz)] λ[E(V ) + {1 − V ∗ (λ)}{E(B0 ) − E(B1 )}]{B1∗ (λ − λz) − z}. Remark 2.. If B1∗ (θ) = B0∗ (θ), then L(z) reduces to L(z) =. {1 − λE(B0 )}{1 − V ∗ (λ − λz)}B0∗ (λ − λz) λE(V ){B0∗ (λ − λz) − z}. which is the generating function of steady state queue length distribution of classical M/G/1 queue with multiple vacation (see [6] and [7])..
(9) Analysis of an M/G/1 Queue with Multiple Vacations and Exceptional Services. (ii) N = 2 L(z) ≡ P0∗ (z, 0) + P1∗ (z, 0) + P2∗ (z, 0) + Q∗ (z, 0) =. {1 − λE(B2 )}A , λz{B2∗ (λ − λz) − z}B. where A = −V ∗ (λ − λz)B0∗ (λ − λz){z − B2∗ (λ − λz) + B1∗ (λ − λz)} + V ∗ (λ)[{B1∗ (λ − λz) − B2∗ (λ − λz)}B0∗ (λ − λz) + {B0∗ (λ − λz) − B2∗ (λ − λz)}]z + λV ∗(1) (λ)B0∗ (λ){B2∗ (λ − λz) − B1∗ (λ − λz)}z + B2∗ (λ − λz)z. B = E(V ) + {1 − V ∗ (λ)}{E(B0 ) + E(B1 ) − 2E(B2 )} + λV ∗(1) (λ)B0∗ (λ){E(B1 ) − E(B2 )} Remark 3.. If B2∗ (λ − λz) = B1∗ (λ − λz), then L(z) reduces to. L(z) = P0∗ (z, 0) + P1∗ (z, 0) + Q∗ (z, 0) =. {1 − λE(B1 )}[−V ∗ (λ − λz)B0∗ (λ − λz) + V ∗ (λ){B0∗ (λ − λz) − B1∗ (λ − λz)} + B1∗ (λ − λz)] , λ[E(V ) + {1 − V ∗ (λ)}{E(B0 ) − E(B1 )}]{B1∗ (λ − λz) − z}. which coincides with the generating function of steady state queue length distribution for N = 1.. References [1] Baba, Y.: On M/G/1 queues with the first N customers of each busy period receiving exceptional services. Journal of the Operations Research Society of Japan, 42 (1999) 490–500. [2] Doshi, B. T.: Queueing systems with vacations – A survey. Queueing Systems, 1 (1986) 29–66. [3] Igaki, N., Sumita, U. and Kowada, M.: On a generalized M/G/1 queue with service degradation/enforcement. Journal of the Operations Research Society of Japan, 41 (1998) 415-429. [4] Kengaku, H. and Miyazawa, M.: A regenerative cycle approach to an M/G/1 queue with exceptional service. Journal of the Operations Research Society of Japan, 43 (2000) 486–504. [5] Li, H., Zhu, Y., Yang, P. and Madharapeddy, S.: On M/M/1 queues with a smart machine. Queueing Systems, 24 (1996) 23–36. [6] Takagi, H.: Queueing Analysis, A Foundation of Performance Evaluation, Vol. 1 : Vacation and Priority Systems (Elsevier, Amsterdam, 1991). [7] Tian, N. and Zhang, Z. G.: Vacation Queueing Models – Theory and Applications (Springer Verlag, New York, 2006). [8] Welch, P. D.: On a generalized M/G/1 queueing process in which the first customer of each busy period receives exceptional services. Operations Research 12, (1964) 736–752.. 61.
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After briefly summarizing basic notation, we present the convergence analysis of the modified Levenberg-Marquardt method in Section 2: Section 2.1 is devoted to its well-posedness
While conducting an experiment regarding fetal move- ments as a result of Pulsed Wave Doppler (PWD) ultrasound, [8] we encountered the severe artifacts in the acquired image2.
