Knessl, C.

Matkowsky, B.J.

Schuss, Z.

Tier, C.

Elsevier - Publisher Connector 

A state dependent M/M/1 queue: a production-inventory model 

A&. Mafh. L/N. Vol. I, No. 3, pp. 235-239, 1988 0893-9659/88 $3.00 + 0.00 
Printed in Great Britain Pergamon Press plc 
A STATE DEPENDENT M/M/l QUEUE: 
A PRODUCTION-INVENTORY MODEL 
C. KNESSL~, B. J. MATKOWSKY~, Z. SCHUSS~, C.TIER~ 
lDept. of Mathematics, Statistics and Computer Science, 
University of Illinois at Chicago 
2Dept. of Engineering Sciences and Applied Mathematics, 
Northwestern University 
3Dept. of Mathematical Sciences, Tel Aviv University 
1. INTRODUCTION 
We consider an extension of a model of production and inventory control given in [1,2]. 
The model incorporates a continuous-level production rate with a base-stock level inven- 
tory policy subject to fluctuating demand. The inventory level at time t is denoted by 
Y(t) and is the time (or effort) required to produce the items currently in stock given a 
production rate of unity. Negative inventory levels Y(t) < 0, reflect orders that cannot 
be satisfied by current inventory and are backlogged. We assume that the inventory is 
produced at a rate which depends on the current inventory level and may differ from 
unity. Thus the production rate is controlled by the producer, which introduces a feed- 
back mechanism for inventory control. We define the base-stock level M as the level of 
inventory at which production stops. Production only restarts when Y(t) < M. Orders 
or demands arrive randomly with an inter-arrival time that is exponentially distributed 
with a parameter that depends on the current inventory level. The size of the orders is 
exponentially distributed. The dependence of the arrival rate on the current inventory 
level can have several interpretations. For example, a company might reject orders or 
discourage arrivals when the backlog of orders becomes sufficiently large or alternatively 
customers may not place orders if there is a backlog. As discussed in [l], this model is 
appropriate if production is highly automated and easily controlled. 
The deficit below the base-stock level is defined by U(t) = M - Y(t) and is the quantity 
of interest here [1,2]. In queueing theory, U(t) is the unfinished work (virtual waiting 
time or buffer content) in the system at time t. In terms of U(t) the inventory control 
model described above is an M/M/l queue with state-dependent arrival and service 
rate. This state-dependence is the new feature introduced into the model described in 
[1,2]. We note that the methods used to analyze state-independent systems, e.g. Laplace 
transforms and generating functions, cannot in general be extended to state-dependent 
systems. Such systems also arise in other applications such as storage and dam theory 
and delays in computer and communications systems [3-61. 
We now formulate our model in terms of the deficit level U(t) including the state- 
dependence of the arrival and service rates on V(t). The state-dependent arrival rate of 
orders is defined by 
Pr{order arrives in (t, t + At)IV(t) = w} = X(w)At + o(At). 
The instantaneous production rate also depends on the current deficit level. 
U(t) = w, the rate at which the deficit level is reduced at time t is defined by 
production rate = r(w), 
w>o 
6, w=o 
(1.1) 
Thus, if 
(1.2) 
Typ-et by AM-W 
235 
236 C. KNESSL et al. 
where we assume that r(w) > 0 if UJ > 0. Thus, the deficit level is decreased (or the 
inventory level is increased) at a rate of r(w) units per unit time when U(t) = w (or 
Y(t) = M - w). When U(t) = 0, th e inventory level is Y(t) = M and production stops. 
The order size is exponentially distributed with mean l/p. We define the instantaneous 
inventory turnover intensity (traffic intensity) by 
o(W) = %!UE = expected total demand due to orders 
r(w) production rate 
(1.3) 
We consider only systems with o(w) < 1 if w > wo for some wo. In terms of the inventory 
level, this means that the production capacity is not overtaken by the demand due to 
orders. 
Our objective is to compute several long run operating characteristics (performance mea 
sures) for the above production facility. Specifically, we compute the stationary density 
of U(t) including the facility utilization and the mean time for U(t) to reach or exceed 
a given value. In addition, we analyze the production busy period of the facility. We 
compute the mean length of a production period and the mean time for a production 
busy period to end, given the current value of U(t) ( i.e. the residual production period). 
We also compute the mean residual total production until the end of a production period 
given the current value of U(t) and the mean total production during a production pe- 
riod, i.e. the integral of U(t) d uring a busy period. Finally, we compute the distribution 
of the maximum of U(t) d uring a production period, i.e. the distribution of the minimum 
inventory level. 
2. PERFORMANCE MEASURES 
We now formulate and solve problems for the performance measures described in Section 
1 for the case when r(w) and A(w) are smooth functions of w. 
i) Stationary Density. We define the density function of U(t) by 
p(w) = &l_ $Jr{U(t) 5 w}, A = t’imm Pr(U(t) = 0). (24 
We expect the stationary limits to exist if the turnover intensity o(w) 5 1 - 6 < 1 for 
some 6 > 0 for all w sufficiently large. Then p(w) and A satisfy the forward equation [7] 
(r(w)p(w))’ - X(w)p(w) + /,” X(w - z)p(w - z)~c-~*dz = -AA(O)fi’ (2.2) 
and the boundary and normalization conditions 
X(O)A = r(O+)p(O+), /mP(w)dw + A = 1 
0 
(2.3) 
where r(O+) = lim,io r(w) > 0. We note that the integrodifferential equation (2.2) is a 
generalization of the Takacs equation in queueing theory (2,7]. By multiplying (2.2) by 
~9” and differentiating we obtain 
(r(w)p(w))” + 1&(w) - X(w))P(w)l’ = 0 
with the boundary condition at w = 0 
~lo[(r(w)p(w))’ + (Lcr(w) - 4w)Mw)l = 0. 
(24 
P-5) 
A state dependent M/M/l queue 
The solution of (2.4)-(2.5) with the normalization condition in (2.3) is 
231 
X(O)-4 _
P(W) = y&J-e 
WefiJoW 4aW 
where 
00 e--c(zecI so* 4s)ds -’ 
+I 
dz . 
1 
(2.6) 
P-7) 
We observe that p(w) has local maxima at points wo where (lr(we) = 1 and a’(we) < 0. 
Thus, in contrast to the state-independent situation, the density of p(w) may have, e.g., 
a bimodal character. 
ii) Mean Time to Reach the Value K. We compute the mean time for the deficit level to 
reach or exceed K, given w units of deficit level. This is the mean time for the inventory 
level Y(t) to reach A4 - K given Y(0) = M - w. First, we define the stopping times 
ti = inf[t : U(t) 2 K]. (24 
The mean time, n(w), to reach or exceed K is 
n(w) = E[fipIqO) = w] (2.9) 
where E denotes expectation. The mean time n(w) satisfies [7] 
-r(w)n’(w) - X(w)n(w) + X(w) lKmfU n(w + z)pe-j“dz = -1, 0 5 w < K, (2.10) 
JO 
and n(w) = 0, w > K. As we show, lim,tK n(w) = n(K-) > 0 so that n(w) 
discontinuity at w = K. Equation (2.10) can be transformed as above into 
has a 
We obtain a boundary condition at w = 0 by evaluating (2.10) at w = 0, noting that 
r(0) = 0, and subtracting the result from the limit of (2.10) as w 1 0, thus yielding 
n’(O) = 0. By evaluating (2.10) as w t K, we obtain r(K)n’(K-) + X(K)n(K-) = 1. 
The solution of (2.11) that satisfies these two boundary conditions is 
n(w) = 
ep~e-~ s,” a(a)ds 
/ 
K ,,,(~-t),-rJ~~ a(s)da 
40) + 0 r(t) 
dt (2.12) 
-&= a(a)da s = ,~so’ a(sP* _ _ d e-fit 0 dt x(t) dtdz- [ 1 
From (2.12), we observe that n(K-) > 0. Thus, the mean first passage time n(w) is 
discontinuous at w = K. This is a consequence of the fact that the deficit level U(t) can 
jump across the boundary at w = K without actually hitting it. From (2.12), one can 
easily show that if o(w) < 1 for w 2 K, then n(w) is exponentially large in K. 
238 C. KNEXL et al. 
iii) Length of a Production Period. We now compute the mean residual length and mean 
length of a production period. We define the residual length of a production period, or 
the amount of time for the deficit level to reach zero, given w units of deficit level, by 
? = inf[t : U(t) = 0] (2.13) 
and the mean residual length of a production period by 
r(w) = E[?lU(O) = w]. 
The mean length of a production period T is then given by 
(2.14) 
s 
00 
T= r(w)pe-p”dw. (2.15) 
0 
We first compute r(w) which satisfies [7] 
27(w) = -1, r(0) = 0 (2.16) 
where 2 is the integrodifferential operator on the left side of (2.10) with the upper limit 
on the integral replaced by 00. The solution of (2.16) satisfying the condition of no 
exponential growth as w + oo is 
Using (2.17), we find that the mean length of a production period is 
Jo f-PI 
iv) Total Production. The residual total production 
/ 
7 
ii= U(t)dt 
0 
(2.18) 
is defined by 
(2.19) 
where ? is the residual length of a production period defined by (2.13). The mean residual 
total production u(w) is then 
u(w) = E[GlU(O) = to]. (2.20) 
The mean total production during a production period, P, is defined by 
P= 
SW 
u(w)peVP”dw . 
0 
The mean residual total production u(w) satisfies [7] 
L(w) = -w, 
so that 
(2.21) 
(2.22) 
A state dependent M/M/l queue 239 
The mean total production during a production period is constructed using (2.23) in 
(2.21) and repeated integration by parts. We obtain 
(2.24) 
v) Distribution of the Mazimum. We now compute the distribution of the maximum of 
U(t) during a production period which corresponds to the minimum of the inventory 
level Y(t) during a production period. We define the stopping times 
7* = inf[t : U(t) = 0 or U(t) 2 K] (2.25) 
and the probability that U(t) reaches or exceeds K before the end of the current prc+ 
duction period by 
q(w) = Pr{U(r*) 2 KIU(0) = w}. (2.26) 
The distribution of the maximum of U(t) during a production period, M(K), is then 
given by 
K 
M(K) = Pr{rr#J(t) > K} = 
,T / 
q(w)pe 
-W’Jdw + e-~K (2.27) 
where ? is the length of a production period. Tie conditional probability q(w) satisfies 
i%q(w) = 0; q(0) = 0; q(w) = 1, w 2 K. (2.28) 
The solution of (2.28) is 
q(w) = 
s; pa(z)e~ze-“.fos a(d)dd & 
s,” pcu(z)e~ze-~~o’ a(d)dd z + epKe-l'~oKu(d)dd ’ 
(2.29) 
from which we obtain 
K 
M(K) = e~Ke--PJoKa(s)da + 
I 
/..m(z)e% 
-ps,” a(a)ds dz 
0 I 
-’ . (2.30) 
ACKNOWLEDGEMENTS 
This research was supported in part by NSF Grants DMS-87-03110 and DMS-8620267, 
and grants from the U.S.-Israel Binational Science Foundation and the Israel Academy 
of Sciences. C. K. was partially supported by an N.S.F. Postdoctoral Fellowship. 
REFERENCES 
1. D.P. Gaver, Jr., Opera&g chamderisticd of a dimple production, inventory control moafel,, Oper. Res. 9 
(1961), 635-649. 
2. D.P. Heyman and M.J. Sobel, “Stochastic Models in Operations Research Vol.1,” McGraw-Hill, 
New York, 1982. 
3. J. Keilson and N.D. Mermin, The second order distribution of integrated shot noise, nans. I.R.E. 5 
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4. D.P. Gaver and R.G. Miller, Limiting distributions for dome dtonzge pmblemd, Studies in Applied Proba- 
bility and Management Science eds. K. J. Arrow, S. Karlin and H. Scarf (1962), Stanford University 
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5. J.W. Cohen, “The Single Server Queue,” North-Holland, New York, 1982. 
6. L. Kleinrock, “Queueing Systems I, II,” Wiley, New York, 1975, 1976. 
7. C. Knessl, B.J. Matkowsky, Z. Schuss and C. Tier, Adymptotic an&id of a dtate-dependent M/G/l 
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‘Dept. of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 
60680, 2Dept. of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, 
IL 60208, 3Dept. of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel 


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A state dependent M/M/1 queue: a production-inventory model

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