Abstract
Full Text
MATHEMATICS
A. A. SHAKHBAZOV
ON A QUEUEING PROBLEM FOR A BATCH FLOW OF CUSTOMERS
(Presented by Academician A. N. Kolmogorov, February 23, 1962)
A batch flow of customers arrives at a service facility. Concerning the flow and the service system we shall make the following assumptions:
-
The arrival epochs of customers (t_1, t_2, \ldots) ((0 < t_1 < t_2 < \cdots)) are such that the differences (x_n = t_n - t_{n-1}), (t_0 = 0), form a sequence of independent random variables having one and the same distribution (A(x)).
-
At each of the epochs (t_n), with probability (\varphi_r), (r) ((r = 1, 2, \ldots)) customers arrive.
-
Customers that find the facility busy join the queue after all previously arrived customers; customers of one group are served one at a time in an arbitrary order.
-
The service time of a customer is a random variable (y) with distribution (B(x)).
-
If customers arrive at a moment when the facility is free, then service begins after a random time (z)*, distributed according to the law (C(x)).
-
The quantities (x_n), (y), and (z) are mutually independent, and
[
a = \int_0^\infty x\,dA(x) < \infty,\qquad
b = \int_0^\infty x\,dB(x) < \infty,\qquad
c = \int_0^\infty x\,dC(x) < \infty.
]
Let (W_r) denote the duration of the wait for the beginning of service of the group of customers arriving at time (t_r), and let (N_s) denote the length of the queue at the moment of departure from the service system of the (s)-th served customer. Further put
[
d = \sum_1^\infty r\varphi_r .
]
Theorem 1. If (bd < a), then the limiting distributions exist
[
F(x) = \lim_{r\to\infty} P{W_r \le x},
]
[
p_i = \lim_{s\to\infty} P{N_s = i},\qquad i = 0,1,2,\ldots
]
The function (F(x)) does not depend on the distribution of (W_1) and (in the class of distribution functions) is the unique solution of the equation
[
F(x) =
\begin{cases}
0, & \text{if } x < 0,\[6pt]
\displaystyle \int_{-\infty}^{x} F(x-y)\,dU(y)
- (1 - C(x))\int_{-\infty}^{0} F(-y)\,dU(y), & \text{if } x \ge 0,
\end{cases}
\tag{1}
]
[
\text{* In applied questions this corresponds to the “warm-up” time of the service facility.}
]
where
[
U(x)=\int_0^\infty B_1(x+y)\,dA(y),\qquad
B_1(x)=\sum_{i=1}^{\infty}\varphi_i B^{*(i)}(x),
]
[
B^{(1)}(x)=B(x),\qquad
B^{(i)}(x)=\int_0^\infty B^{*(i-1)}(x-y)\,dB(y)\quad (i=2,3,\ldots).
]
The proof of this theorem is a simple modification of the arguments of D. Lindley ((^4)) and P. Finch ((^5)).
Introduce the notation:
[
P(z)=\sum_0^\infty p_i z^i,\qquad
\widetilde F(t)=\int_{0-}^{\infty} e^{itx}\,dF(x),
]
[
\widetilde B(t)=\int_{0-}^{\infty} e^{itx}\,dB(x),\qquad
\widetilde C(t)=\int_{0-}^{\infty} e^{itx}\,dC(x),
]
[
\varphi(z)=\sum_1^\infty \varphi_i z^i,\qquad
\psi_i=\sum_{s=i}^{\infty}\frac{\varphi_s}{s},\qquad
\psi(z)=\sum_1^\infty \psi_i z^{i-1}.
]
Theorem 2. If (A(x)=1-e^{-\lambda x}), (\lambda bd<1), then
[
P(z)=\frac{1-\lambda bd}{(1+\lambda c)d}\,
\frac{[\widetilde C(i\lambda-i\lambda\varphi(z))\varphi(z)-1]\,\widetilde B(i\lambda-i\lambda\varphi(z))}
{z-\widetilde B(i\lambda-i\lambda\varphi(z))};
\tag{2}
]
[
\widetilde F(t)=\frac{1-\lambda bd}{1+\lambda c}\,
\frac{(\lambda+it)\widetilde C(t)-\lambda}
{\lambda+it-\lambda\varphi(\widetilde B(t))}.
\tag{3}
]
Proof. The quantities (N_s) and (N_{s+1}) are related by
[
N_{s+1}=
\begin{cases}
N_s+\xi_{s+1}-1, & \text{if } N_s>0,\
\zeta+\xi+\xi_{s+1}-1, & \text{if } N_s=0.
\end{cases}
\tag{4}
]
Here (\xi_{s+1}) is the number of customers arriving during the service time of the ((s+1))-st customer; (\zeta) is the size of the group arriving at the moment when the server was free; (\xi) is the number of customers arriving during the waiting time before service of this group began. The generating function of the compound Poisson flow has the form
(M z^{\nu(t)}=\exp[\lambda t(\varphi(z)-1)]), where (\nu(t)) is the number of customers of the flow that arrived during time (t). Hence
[
M z^{\xi_{s+1}}=\widetilde B(i\lambda-i\lambda\varphi(z)),\qquad
M z^\xi=\widetilde C(i\lambda-i\lambda\varphi(z)).
\tag{5}
]
From (4) it follows that
[
zM z^{N_{s+1}}=\mathbf P{N_s>0}M_{N_s>0}z^{N_s+\xi_{s+1}}
+\mathbf P{N_s=0}M_{N_s=0}z^{\zeta+\xi+\xi_{s+1}}.
\tag{6}
]
But
[
\mathbf P{N_s>0}M_{N_s>0}z^{N_s}=M z^{N_s}-\mathbf P{N_s=0}.
\tag{7}
]
Substituting (5) and (7) into (6) and passing to the limit as (s\to\infty), we obtain
[
P(z)=
\frac{[\widetilde C(i\lambda-i\lambda\varphi(z))\varphi(z)-1]\,
\widetilde B(i\lambda-i\lambda\varphi(z))\,p_0}
{z-\widetilde B(i\lambda-i\lambda\varphi(z))}.
\tag{8}
]
The unknown (p_0) is found from the normalization condition (\sum_0^\infty p_i=1). Applying L’Hôpital’s rule to (8), recalling here that
[
\widetilde A'(0)=ia,\qquad \widetilde B'(0)=ib,\qquad \widetilde C'(0)=ic,\qquad \varphi'(1)=d,
]
we find
[
p_0=\frac{1-\lambda b d}{(1+\lambda c)d}.
]
To solve equation (1), consider the function
[
g(x)=\int_{-\infty}^{x} F(x-y)\,dU(y)\qquad (-\infty<x<\infty),
\tag{9}
]
where
[
U(x)=\lambda\int_0^\infty B_1(x+y)e^{-\lambda y}\,dy.
\tag{10}
]
According to (1) and (10),
[
g(x)=
\begin{cases}
\alpha e^{\lambda x}, & \text{if } x<0,\
\alpha(1-C(x))+F(x), & \text{if } x\ge 0,
\end{cases}
\tag{11}
]
where
[
\alpha=\int_{-\infty}^{0} F(-y)\,dU(y).
]
From (9) and (10),
[
\int_{-\infty}^{\infty} e^{itx}\,dg(x)
=
\frac{\lambda}{\lambda+it}\,\widetilde F(t)\,\varphi(\widetilde B(t)),
]
and from (11),
[
\int_{-\infty}^{\infty} e^{itx}\,dg(x)
=
\frac{\lambda\alpha}{\lambda+it}
-\alpha\widetilde C(t)+\widetilde F(t).
]
Comparing the last two equalities gives
[
\widetilde F(t)=
\frac{(\lambda+it)\widetilde C(t)-\lambda}{\lambda+it-\lambda\varphi(\widetilde B(t))}\,\alpha.
]
Since (\widetilde F(0)=1), we have
[
\alpha=\frac{1-\lambda b d}{1+\lambda c}.
]
Let now (W) be the waiting time until the beginning of service of an arbitrarily chosen customer, and let (W(x)=\mathrm P{W\le x}). If (\nu) is the ordinal number of the service of the customer under consideration within the group to which it belongs, then
[
W=\gamma+\sum_{r=1}^{\nu-1}\gamma_r,
]
where
[
\mathrm P{\gamma<x}=F(x),\qquad \mathrm P{\gamma_r<x}=B(x).
]
By the formula of total probability,
[
\mathrm P\left{\sum_{r=1}^{\nu-1}\gamma_r<x\right}
=
\sum_{i=1}^{\infty}\psi_i B^{*(i-1)}(x),
]
Hence
[
\widetilde W(t)=\widetilde F(t)\,\psi(\widetilde B(t)).
]
Consequently,
[
\widetilde W(t)=\frac{1-\lambda b d}{1+\lambda c}\,
\frac{[(\lambda+it)\widetilde C(t)-\lambda]\psi(\widetilde B(t))}
{\lambda+it-\lambda\varphi(\widetilde B(t))}.
\tag{12}
]
The formulas (2) and (12) obtained by us contain, as special cases, the corresponding results of papers ((^{1-7})). If (\mathrm P{z=0}=1) and (\varphi_i=\delta_{ri},\ i=1,2,\ldots,) then
[
P(z)=\frac{(r^{-1}-\lambda b)(z^r-1)\,\widetilde B(i\lambda-i\lambda z^r)}
{z-\widetilde B(i\lambda-i\lambda z^r)},
]
[
\widetilde W(t)=
\frac{i(r^{-1}-\lambda b)t\displaystyle\sum_{0}^{r-1}(\widetilde B(t))^k}
{\lambda+it-\lambda(\widetilde B(t))^r}.
]
Similar formulas were obtained by F. Foster ((^8)) in the case when (B(x)=1-e^{-x/b}), (A(x)) is an arbitrary distribution function with finite mathematical expectation, and (\varphi_i=\delta_{ri},\ i=1,2,\ldots).
In conclusion the author expresses sincere gratitude to B. V. Gnedenko for posing the problem and supervising the work.
Moscow State University
named after M. V. Lomonosov
Received
23 II 1962
CITED LITERATURE
({}^1) A. Ya. Khinchin, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 49 (1955).
({}^2) A. Ya. Khinchin, Matem. sborn., 39, No. 4 (1932).
({}^3) F. Pollaczek, Math. Zs., 32, pp. 64 and 729 (1930).
({}^4) D. V. Lindley, Proc. Cambridge Phil. Soc., 48, No. 2 (1952).
({}^5) P. D. Finch, Acta Math. Acad. Sci. Hungar., 10, No. 3—4 (1959).
({}^6) R. G. Miller, J. Roy. Stat. Soc., B 21, No. 2 (1959).
({}^7) D. G. Kendall, Ann. Math. Stat., 24, No. 3 (1953).
({}^8) F. G. Foster, Acta Math. Acad. Sci. Hungar., 12, No. 1—2 (1961).