1 queue

On the trade-off between queue congestion and server's reward in an M/M/I queue Guy Latouche Universit~ Libre de B,uxeiles, Boulevard du Triomphe, I050 Brussels, Belgium and University of Delaware, Newark, U.S.A. Received June 1978 Revised l~,~ovember1978 We consider an M/M/1 queue which is controlled by dynamically setting the customer's entrance fee to either b I or b 2 > b I and thereby setting the customer arrival rate to either X1 or ~2 < hl, respectively. With E[F] the expected fee collected per unit of time and PN the steady state probability that the system eontah~s more than N customers, we consider two criteria: (i) for some number E', minimize PN subject to F iF] I> 7, and (ii) for some number e, maximize EIFI subject to PN <~e. Under each criterion we consider two cases: (a) the admissible policies are single critical number switching policies and Co) a cost 3' I> 0 is incurred whenever the server switches between b I and b2, the admissible policies allow for hysteresis to appear in the arrival process. Optimal policies are computed for each criterion and each case.

O. Introduction

In the liteTature on control of the arrival process at a queuing station, the objective is usually to maximize the difference between fees colkcted and costs: it is assumed that the fees and costs can be measured and are expressed in the same units. 1hey are then merged into a single objective function (see e.g. [1,4,5,6,9 and 10]). In this paper we consider th; two objectives of maximizing the fees collected and minimizing queue congestion as distinct. When it is not possibi~: to eptimize these simultaneously the problem is to optimize one of them under the constraint that the other is maintained at a reasonable level. In [3], we consider similar models where the two objectives are merged into one single objective function. The analysis and results in [3] are both qualitatively and quantitatively different from those we present here. We assume that the server may dynamically set the entrance fee to either bl or b2 > bl; the arrivals term a Poisson process with parameter Xt or ~,2 < Xl respectively. The service times are independent, identically distributed, exponential 0a) random variables, independent of the arrival process. The waiting room is infinite, but the total number of customers in the system should preferably not exceed some value N, called the critical level. We consider two control mechanisms: (a) the admissible policies are single critical number switching policies, they are characterized by a number M: if there are fewer than M customers in the system, the entrance fee is b~ ~,~herwise it is b2; (b) a cost 7 1> 0 is incurred whenever the server switches between bl and b2 and the admissible policies allow for hysteresis to appear in the arrival process. These policies are characterized by two numbers, m ~ ~l M (m ~
The author is grateful to the referee for extremely helpful suggestions that greatly improved the presentation of the results. This research was sponsored by the Air Force Office of Scientific Research Air Force Systems Command USAF, under Grant No. AFOSR-77-3236. © North-Holland Publishing Company European Journal of Operational Research 4 (1980) 203- 214. 203

204

(7. Latouehe / Queue congestion and server'srewardin an M/M[1 queue

More complex controls have been applied to the service mechanism (Yadin and Naor [8]). To the best of our knowledge, only Scott [7] has considered a queue with hysteretic arrivals; we comment more on this paper in Section 4. We analyse the single critical number control in Sections 1 to 3 and the two-level control in Sections 4 to 6. Comparisons are presented in Section 7. Most of the proofs are not given: they are very lengthy while the methodology is sintple.

1. Control with a single critical number In this case the moael is simple to analyze and most of the results in Sections I to 3 are intuitively obvious. As stated in the introduction we assume b I < b2 and ~,1 >)-2Moreover, ifpi = Xt/~ i = 1, 2, we assume that P2 < 1, while Pl may be less than or greater than 1. It is obvious, therefore, that steady state conditions are satisfied ifM <*% or M = 0. and Pl < 1. Let lri~l) =P[i customers in the system in steady state I m], andPi(M) = Z~'=i+Ini(M). We shall refer to the Pi's as the tail probabilities. One can easily solve the system of equations for the probabilities ~ri(bO, i = 0, I, ... and hence one gets for P/I+I (1 -P2)-PMI(P!--P:z) 1 - P2 -- pM(pl -- P2) ,

P (M) =

(1 -

i-M+l PD. PlMP2

l - p2 - p

(pl -

i <~M,

i>~ M;

p2)

(la)

(lb)

'

for p I = 1, the corresponding formulae are

1+(M-i-

i
1X1-P2)

'

1 +/14(1 - P2)

ei(U)-

pi2-M+l I+M(1-p2)

i>~M.

'

I.emma 1. For all freed i, Pi(M) is an increasing function o f M. Moreover, PM_I(M) is a decreasing function o f M. Proof. The first assertion is obvious; ifM increases, the number of states for which the arrival rate is kl (>he) increases and so does the probability of having more than i customers. The second follows from the fact that PM_I(M) may be written as ((1 - p2)/((l

-

Pl) PlM) -- (Pl -- P2)/(1 - Pl)) -1

and that (1 - Pl) pM is a decreasing function of M.

Now, let r(M) -- E[fee collected per unit of time in steady state I M']. Then r(M) = ~,2b2 + (~tbl - ~,2b2Xl

-

PM-I(M))

•

Lemma 2. r(M) is an increasing function of M if ~tb I > ~2b2; it is constant if hlbl = ~2b2 and decreasing if ~ b l < ~2b2. ProoL The proof is obvious. Theorem 1. I f ~ l b l <<.~2b2, it is optimal to set M equal to O.

(2)

G. Latouche / Queue congestion and server's reward in an M/M~1 queue

205

Ihroof. This is an immediate consequence of Lemmas I and 2. Remark. One can reach simultaneously the two objectives: maximize r(M)and minimize PN(M) only if hlbl < ~2b2. We shall assume in the next two sections th~.t ~qbl > ~k2b2.

2. Constraint on the expected fee

The problem is to find that M, which minimizes the probability PN(M) under the following constraint: the expected fee r(M) must be at least equal to some predetermined value F. It follows from Lemmas 1 and 2 that MI = min {M I r(M) >f T}. Writing r(**)= limM.~**r(M) one gets easily from (1) and (2) that 0,

ifp I ~< 1,

Pl-1 , Pl - P2

ifp I > 1,

lim PM_I(M) =

hence ~,lbl,

ifp 1 ~< 1,

X2b2 P I - I +Xlbl l-p_._.__~2< ~ q b l , Pl - P 2 Pl - P 2

ifpl>l.

However, ifM is equal to 0% the expected fee over any interval of time is ~lbl times the length of that interval, even ifpi > 1. Therefore we shall consider that the expected fee is equal to ~lb I for all Xl, when M is set equal t o oo.

Theorem 2.

ff

Xlb I
there is no solution;

r(°*) <~F < ~lbl,

MI = oo;

~,2b2 <~" < r(**), ~2 > 0 ,

M l = Ix I ];

0

M 1=


= 0,

F < X2b2

max(N, [xl ]);

MI =0;

where

-log( 1 + y)/log p l,

if p I :/: 1,

(F - ~t2b2)/((~lb I - FX1 - P2)),

if Pl = 1,

X1 =

y = ((r - X262)O

- Pl))/(0,1b,

- r)(l

- P2)),

and Ix ] denotes the smallest integer greater than or equal to x.

Observe that if 7,2 = 0, the level M takes on a special meaning: it is the maximum number of customers the server allows in the system. Ifx I as defined in Theorem 2 is less than (N - ]L),then the solution is not unique, since P/v(M) = 0 and r(M) i> F for all M = ~xl ], Ix1 ] + 1, ..., N. However, since the critical level has been set to N and not a smaller value,Ml should be N since r(N) > r(N - 1) > .... Therefore, if X2 = 0, th~n Ml = max(N, Ix i 1)"

206

G. Latouche / Queue congestion and server'sreward in an M/M/I queue

3. Constraint on the tail probability Suppose now, in contrast with the preceding section, that it is decided that the probability of having more than N customers in the system has to be less than some given value e. The problem is then to find M2 = max {M i Ply(M) < e}. Then, since r(M) is an increasing function of M, r(M2) i> r(M) for all M such that PA,(.M)~< e. M2 is determined very easily. Two cases must be distinguished: if PN(N) < e, it follows from Lemma 1 that M2 is greater than or equal t o n and is determined using eq. (la); ifPA,(N) > e, M2 < N and eq. (lb) has to be used. Note that e,~(oo) = limM~® Pzv(M) = rain(l, pN+l) and that e~v(0) = p2N+I should be smaller than or equal to e in order to have a solution. We can thus formulate

Theorem 3.

if

e < p~+l,

there is no solution;

p~+l ~< e < min(1, p ~ + l ) ,

M2 =

min(!, p~+l) ~< e <~ 1,

/]/2 =oo;

Lx2J;

where x2 < N is" the unique nonnegative root o f the equation ( , 0 1 / P 2 ) x = e(1 -- P2 - - P ~ ( P l

- - P 2 ) ) / ( p 2 N * I ( 1 -- P l ) ) ,

ifPN2+1 <~e <~PN(N)

(3)

and x2 = log{(l - p2)(p~ +l - e)/((91 - p 2 X I - e))}/log

pl >1-1V, ifPN(N) ~< e <

min(l, p~+1)

and Lx J denotes the greatest integer less than or equal to x. Remark. In the strict sense, eq. (3) may have two nonnegative roots if e = p2A'+I" x = 0 and some positive root, say x*. It can be seen easily that in this case, x* < 1 and therefore there is no ambiguity on the value of M2: both L0 J and Lx* J are equal to 0. There is no explicit form for M2 if M2 ~2b2. I f X l b l < ~,2b2, it can be shown that the optimal rule is to impose the fee b2 always. Let us assume furthermore that T I> 0 where 3' is the cost incurred by the server each time there is a change of fee and let us assume also that M - 1 >~m~>0.

(4)

This last inequality means that the fee is certainly b i when the system is empty and the only purpose of this assumption is to give a more uniform presentation of the results. The determination of the state probabilities and the expected reward is not as straightforward as for the preceding type of control. Scott [7] derives the probabilities when the maximum queue size is finite. We determine these quantities - when the maximum queue size is infinite - by using a different technique. Let r(m, M) = E[fee collected per unit of time in steady state I (m, M)], and Pi(m, M) = P[more than i customers in the system in steady state i (m. M)].

G. Latouche / Queue congestion and server's reward in an M/M/1 queue

207

Theorem 4. The expressions for r(m, 34) and Pi(m, M) are given by plM-I( 1 -- pl)2(2T( 1 - P2) + ( M - m ) ( b l P l - b2P2))

r(m, M) = Xlbt - Xl (1 - p2X1 -- pM-m) + (M - m) pM(p 2 -- plX1 - p l ) ' and

(a) O <<.i <. m

Pi(m, M) =

(I -- 0 2 ) 0 (I --

-- pM-m)PiI+I + (114-

p2X1

--

pMI-m)

+

m) pM(p2 -

( M - m) plM(p2

Pl)(I -- Pl)

-- pl)(l -- Pl)

(b)

m -

1 <~i<~M

e~(,n,~O = p~l , X(I-o2)2(I-PMI-i)+(M-i)PMI-i-I(p2-OlXI-PlXI-B2)+oM-i-I(,°2-oi2-m+IXI-oI) I -

02

2

(I - 02)0 -o~u-m) + ( M - m) oiM(02 -- p~XI - p~)

(c) pM1p~-M+l

M - i
P~(m,~I - P2

(I - pl)2(l

- [f -m)

(1 - 02)(1 - p M - m ) + (M - m) pM~2 -- pIX1 - or)"

Roof. Let us denote by rm, M the expected total reward during a busy cycle, given m and M. To determine rm,M, we decompose a busy cycle as follows. First, there is an idle period, at the end of which a customer enters the queue and pays b 1. Then a random walk begins, with absorbing boundary at 0 and M, parameter ~,1, to each step to the right is associated a reward bl. If absorption is at O, the busy cycle terminates. Otherwise, a cost V is incurred and a new random walk begins, with absorbing boundary at m, parameter ~,2, reward b2 for each step to the right. Upon absorption at m, a cost T is incurred and a new random walk of the first type is initiated, etc. Thus, one gets after simple algebraic manipulations

rm.M=b,[i 1 -p,

(M-m)oM(P'~)

]

(l - ~ X i - ~ ; ~ - ~

j -

pMI-I(I-PI)[2T(I-o2)+(M-m)(bIPI-b2P2)] (1 - p ~ ) ( 1 _ p ~ - m )

. . . .

•

(5)

Let COm,Mbe the expected length of a busy cycle: it is the idle period (mean 1/~1) plus the amount of work brought by the customers during the busy cycle. To get this amount of work, one just replaces in (5) 7 by O, b l and b2 by l[/~. This leads to 1 ~ 1 ¢Om'M=~ll--Pl

( M - m ) pM(p I - p2)] (1 - ~22)0 - - P f - - ~ ] "

From (5) and (6), it is easy to get the expression for r(m, M). The expressions for Pi(m, M) are proved in much the same way. The following three lemmas characterize the main aspects of the behaviour of the tail probability and the expected fee as functions of the parameters m and M. I.emma 3. For all i, Pi(m, M) is an increasing function of m and M. Lemma 4. For any given m, r(m, M) is an increasingfunction of M.

(6)

G. Latouch¢/ Queuecongestionand server°zrewardin an M/M/l queue

205

Lemma 5 For any given M, r(m, M) is a unimodal function of m and we denote by mM the corre~onding maxi= mizing va'~ue.Moreover, mM+l >1mM for all M ~ 1.

5. Constraint on the expected fee

The problem is to find (ml, MI) such that

r(ml, Ml)>~V

and

P~mI, M I ) < P ~ m , M )

foraH (m, bOsuch thatr(m,M)>~'.

Let ~2 be the set {(m, M') ! M = 1, 2, ...;m = 0, 1, ...,M - 1} of all pairs satisfying (4) and let ~2t = {(m, b0 [ (m, M) E [2, r(m,/tO I> 7}. Obviously, (ml, M I ) E I2t. I f ~ l is empty, the problem has no solution; ff~2 t = [l, it follows from Lemma 3 that (mr, Mr) = (0, I) and ~he fee should be be whenever there is a customer in the system; otherwise, we proceed in two steps to determine (mr, M1). Firstly, by using repeatedly the properties of r(m, M), we prove that ~2t has the form indicated on Fig. 1" the dashed region corresponds to ~ - ~2t; the boundary of ~2t is made up of two parts: for increasing m, it is first non-increasing and second it is non-decreasing until it coincides with the line m =M - 1. We have marked by an x the 'comers' in the first part of the boundary. Secondly, we use Lemma 3 to prove the following: Theorem $. (mI, MI)E {(mX, MiX), i = l,...,lX};

where

MX < M~xl

and

m x > m~+l

for all i,

M~: = rain {M I r(mM, M) >i ~},

for a given m = m x,

r(m~, M) >~V iff M >~Mf ,

for a given M =M f ,

r(m, MiX) < F for all m < m~.

Remark. The (mx, M~'s are those elements of [21 marked by an x on Fig. 1. We do not give their precise definition as it is very cumbersome. There is no explicit expression for (m l, M1), one should compare all (m:[,Mf)'s to determine it. This can easily be done numerically. Note that I x is at most equal to [xt ] (defined in Theorem 2) and is,

M

°l

j

m

Fig. 1. ~ t i~ the set of admissiblesolutions;(m I, MI) is to be found among the points marked by a x.

G. Latouche/ Queuecongestionand server'srewardin an M/M/I queue

209

d

(Hl,m I) 50

f

30

•

II

• • • • • • • •

. . • • . .

. . . . . . . . . . . . . .

10

M

m]

20

~=1., ~1=1.2, 1~2=.I, b=l, B=4, y=.0625, ~=.8, r(~)=.809 Fig. 2. Variations

of M 1 and m I as functions

of N.

Table 1 Systems with null 3' and hysteretic solutions ?~2

0.05

0.1

0.2

0.3

mI

2

2

3

3

tz= 1.0,~. 1 = 0.9,F"

M1

4

4

4

5

3, = 0 , b 2 = 0 . 4 / ~ . 2, N --- 0 , 1 . . . . .

= 0.8,b

I = 1.0,

10

in general, smaller; we have thus an upper bound on the number of pairs that will have to be determined and compared. To analyze the solution as a function of the parameters, one has to resort to computations. As an example, we have examined how (ml, MI) depends on N (see Fig. 2). For the single critical number policies, M 1 is independent of N. For control with hysteresis, it appears that (ml, M1) varies in general only slightly with N, if at all; however, if • is very close to the limM_***r(m, M) and ~l is much larger than ~2, the variations are greater, as can be seen in Fig. 2. Those cases do not seem typical. One can observe that if 3' = 0, m I is not necessarily equal to (M I - 1) (Table 1). In other words, even if the server suffers no cost each time the fee is changed, it is optimal in some cases to introduce hysteresis in the arrival process•

6• Constraint on the tail probability In this case, (m2, M2) is defined as follows:

P~(m2,1142) < e,

and

r(m2, M2) >ir(m, M)

for all (m, M) such that P~(m, hO <<.e.

Let I22 = {(m, 21//) [ (m, M) E ~2 and PN(m, M) < e}. If [22 is empty the problem has no solution. If ~2 = [2 it

210

G. Latouche / Queue congestion and server's reward in an M/M/! queue

4. 4.

m

Fig. 3. 122 is the set of admissible solutions, (m 2, M 2) is to be found among the points marked by a +.

follows from Lemma 4 that M 2 = ** and therefore m 2 can take any value: the fee is b 1 always. Otherwise we use 1.emma 3 to prove that I22 has the form indicated on Fig. 3: the dashed region corresponds to I2 - I22; the boundary of I22 forms a non-increasing line for increasing m. We have marked by a + some of the points in ~22. We then use the properties ofr(m, M) to prove the following: Theorem 6.

(m2, M2)

{(mL

i - 1 .... ,I +'.~;

where

M~
and

Mi~ = max {M I

"

m+>m++ 1

foralli,

< e},

ml = m

+,

M1

foragivenm=m +,

PN(m~,M)
for a given M =M +,

PN(m, M+) < e iff m < m~.

Moreover, if (m~, M~ • 1)~ I22, then (m2, M2) ~ (m~, MD. Remarks. (1) Again we do not define formally the (m +, M~)'s, merely mentioning that they correspond to the elements of I22 marked by a + on Fig. 3. (2) Ohserve that (m2, M2) lies on the boundary of I22 and (m~, M1+) cannot be optimal if it lies inside ~22. x2 (defined in Theorem 3) provides an upper bound on the number of pairs that have to be determined and compared, for I + < ( L x 2 j + 1). We have analyzed numerically (m2, 342) as a function of 3, and it appears (Fig. 4) that 342 is a (non strictly) increasing function of 3' while m2 is decreasing. This shows that, in contrast with the single critical number policies (see Section 3), the optimal solution depends on the values assigned to cost and fees. Observe that if 3' - 0, m2 is not necessarily equal to (M2 - 1) (Fig. 4 and 6).

7. Comparisons between the two types of policies To conclude, we mention numerical comparisons between the two types of policies when the cost 3, is positive: we can determine as in Sections 1 to 3 optimal single critical number policies. Figs. 5 and 6 present 4 examples corresponding to 2 values for N (N -- 5 ~ d N = 10) :,.zd 2 values for 3' (3' = 0.5

G. L~touche / Queue congestion und server's reward in an JW/M/1 queue

211

( M2 •m2 )i M2

.0

~2 -( Uffil

, AI=~

~

~ =.h

NffilO

bffil

, E = .25

Bf2

(M 2 ,m 2

M~

l

,

t .0 p=l

•

,

a .0 ~I =-

5

~2

i 0.0

,

~2=.2

8.0

,

N=

5

,

bffil.

LO .0 Bffi2

,

e= . 0 1

(M2,m2/

----i

I

i

]

~2

Y .1..ffi .

,

~'I = " 5

,

X "" ffi " 1

,

N=5

'

b=l

"

'

B=I

F i g . 4 . V a r i a t i o n s o f M 2 a n d m 2 as f u n c t i o n s o i T-

" 5

,

E=

.01

G. Latouche /Queue ¢ongegion and server'sreward in an M/M/I queue

212

--"l

",------x.

(4)

.5

.4:6

.?

~8 ~I=.9, ~2=.3, ~=I., b=1., 8=2., N=5

.9

;

PN

1

.9

.8

I

.7

I, L

.6

.?

.8

.9

~1=.9, ~,2=.3, u=l., b=l., B=2., N=IO

r

Fig. 5. Comparisons of PN(M1) and PN(m 1, MI) for various £'.

- which is small compared to bl and b2 - and 7 = 5 - which is large compared to bl and bz). Curves 1 and 2 correspond to 7 -- 0.5, curves 3 and 4 to 7 = 5; curves I and 3 correspond to control with hysteresis, curves 2 and 4 to control with a single critical number. Obviously, controls with a single critical number yield less good results, since they are special cases of controls with hysteresis. As can be seen in the figures the difference can i:~ substantial. Also it appears that the optimal

G. Latouche / Queue congestion and server's reward in an M/M/1 queue

213

rI

I.

(1)

o

.~

~i-.9,

/

./

:3

~"

.3

e

E

X2-.3, p=l.,'b=l., B=2., N=5

iI i /

/

.5

,,/

(4) .3 0

.I

.2

Fig. 6. Comparisons of r(m2) and r(m 2, M2) for various e.

control with hysteresis is efficient, except under severe circumstances, such as small value for N, large cost 3' and quite large F (or small ¢).

Bibliography [ 1 ] N. Edelson and D. Hildebrand, Congestion tolls for Poisson queueing processes, Econometrica 43 (1975) 81-92. [2] G. Lato.uche, ModUles de contr6~ optimal d'une file d'attente avec droit d'entr6e variable, Ph.D. Dissertation, Facult6 des Sciences, Universit6 Libre de ~ruxelles,'Brussols (1976)•

214

G. Latouche / Queue congestion and server'sreward in an M/M/1 queue

[3] G. Latouche, Optimal pricing policy for a M]M/1 queue (a) without and (b) with hysteresis in the arrival process, in: Proc. Second European Congress Operations Research, Stockholm 1976 (North-Holland, Amsterdam, 1976) 251-258. [4] D. Low, Optimal dyr~anic operating policies for an M/M/s queue with variable arrival rate, IBM Los Angeles Scientific Center, G320-2654 (1971). [5] D. Low, Optimal dynamic pricing policies for an M/M/s queue, Operations Res. 22 (1974) 545-561. [6] P. Naor, The regulation of queue size by Levying tolls, Econometrica 37 (1969) 15-24. [7] M. Scott, A queueing process with some discrimination, Management Sci. 16 (1969) 227-233. [8] M. Yadin and P. Naor, On queueing systems with variable service capacities, Naval Res. Logist. Quart. 14 (1967) 43-53. [9] U. Yechiali, On optimal balking rules and toll charges in the GI/M/I queueing process, Operations Res. 19 (1971) 349-370. [10] U. Yechiali, Customer's optimal joining rules for the GI/M/s queue, Management Sci. 18 (1972) 434-443.