Monotonicity properties of the throughput of an open finite capacity queueing network

Monotonicity properties of the throughput of an open finite

capacity queueing network

Citation for published version (APA):

Adan, I. J. B. F., & Wal, van der, J. (1987). Monotonicity properties of the throughput of an open finite capacity queueing network. (Memorandum COSOR; Vol. 8712). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1987

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Memorandum COSOR 87-12

Monotonicity properties of the throughput of an open finite capacity.queueing network

by

1·0 Adan en J. van der Wal

Eindhoven, June 1987 The Netherlands

Monotonicity properties of the throughput of an open

finite capacity queueing network

by

Ivo Adan and Jan van der Wal

University of Technology, Eindhoven

ABSTRACT

Monotonicity properties of the throughput of an open finite capacity queueing network with general service and arrival times are established by using coupling and sample path arguments.

1. Introduction

Recently much attention has been devoted to the monotonicity properties of the throughput in a closed network. See e.g. Robertazzi and Lazar [1985]. Suri [1985]. Yao [1985]. Van der Wal [1985] and Shanthikumar and Yao [1985]. Most of the results deal with exponential or productform networks.

Monotonicity results for non-productform networks are established by Adan and Van der Wal [t987a] and [1987b]. Adan and Van der Wal [1987a] show for the case of a closed network with general services that the throughput is monotone in the number of servers and Adan and Van der Wal [1987b] establish monotonicity properties of the throughput of an open network with general arrivals and services. A motivation for studying monotonicity properties of the throughput can be found in the work of Van Dijk. He tries to give upper and lower bounds for the throughput of non-productform networks. see e.g. Van Dijk and Lamond [1986] and Van Dijk, Walrand and Tsoucas [1987].

In this paper we are concerned with an open finite capacity queueing network with general service and arrival times. For instance. such a network may be used to model a telephone exchange. where the total number of phone calls is restricted by the (finite) number of lines available. Another example is a production system. where jobs are circulated on a pallet. As soon as a job leaves the system. the pallet is available to the next arriving job. If all pallets are occupied. arriving jobs are not admitted and lost. Hence the total number of circulating jobs is restricted by the (finite) number of pallets available. We will show that the throughput increases if

we enlarge the capacity of the network. So, the number of phone calls per unit time increases if extra lines are available and the total number of processed jobs per unit time increases if extra pallets are available. The proof will be based on coupling and sample path arguments and it is organized as follows. In section 2 the model. notations and main theorem are given. The subsequent section presents a lemma needed for the proof of the main theorem. Section 4 contains the proof of the main theorem.

Using similar arguments. one can show that the throughput is monotone in the number of servers and in the service times. One would probably expect the throughput to be monotone in the arrival times as well. In general this is not true. This can be explained by the fact that a job is rejected if it arrives earlier. at a moment the system is still full. We give an example, which shows that the throughput can decrease if the arrivals are sped up. But in section 5 we prove that for the special case of a Poisson arrival stream the throughput is monotone in the arrival rate.

2. Model, notations and theorem

Let us consider an open queueing network with stations 1 •.... N with Markovian routing and independent generally distributed service times. Jobs arrive according to an arbitrary arrival process. The arrival times are independent of the service times. In each station there are

4

identical servers, i == 1 . ... . N. and the service discipline is FCFS. The network has a finite capacity. If the network is full. an arriving job leaves the network without receiving service. and else it joins queue i with probability Pj' In order to show that the throughput increases if the capacity increases, we compare a network having a capacity of K jobs with one having a capacity of K+l jobs. The network with a capacity of K jobs is denoted by system K and the other by system K+l. We couple the arrivals. services and transitions of both systems by taking the same arrival times. service times and transitions. This is justified by the assumption that the arrivals. services and transitions are independent of each other. FolloWing the same approach as Adan and Van der Wal [1987b] we will show that for any given realization of the (coupled) arrival times. service times and transitions for all t and for each station the throughput in sys-tem K+1. defined as the total number of service completions up to and including time t, is at least equal to the throughput in system K.

Let Ai' Xij'

Soj

and Sij be any given realization of arrival times. service times and transitions, i = 1 ....• N. j

=

1 . 2 ... That is, Aj denotes the time of the j-th arrival from the outside. Xjj denotes the service requirement by the j-th arriving job in queue 1.

Soj

denotes the station where the j-th admitted job enters the system and Sij denotes the station where the j-th departing job from station i will jump to. The Sij may equal zero. In that case the j-th departing job leaves the network. We also allow the Aj's to be equal. So jobs may arrive in batches at the network. Observe that the j-th admitted job need not be the j-th arriving job. For conveni-ence we assume At

>

0 and both systems to be empty on t == O.

In the sequel we need the following notations.

Aij the time of the j-th arrival in queue i.

Dij the departure time of the job that arrived as j-th job in queue i. In the multi server case this need not be the time of the j-th departure.

3

-Ai(t) the total number of arrivals from the in- and outside up to and including time t in queue i.

Ao(t) the total number of jobs admitted in the system up to and including time t.

Aoi(t) the total number of jobs who entered the system from the outside in station i up to and including time t.

Di( t) the total number of departures up to and including time t in queue i.

the total number of admitted jobs who left the system up to and including time t.

These variables will have a superscript K or K+l to indicate whether they correspond to the K or K + 1 system.

Let ef

<

e2

c

< ...

be the time instants in system C upon which one or more ser-vices are completed or one or more jobs arrive from the outside. Then define the sequence to. tl . '" of time instants upon which an event occurs in at least one of the two systems by

to:= 0

tn := min { min { ejK I elK> tn-I} , min { ~K+l I ~K+l

>

tn-I} }. n ~ 1. We make the following assumption.

Assumption

0) A_{j ....} co if j.... co

n

(ii)

1:

Xij "" co (n .... co) for all i j= 1

(iii) Xij

>

0 for all i and j

The first condition guarantees that in every interval [ 0 . t ] only a finite number of jobs arrives from the outside at station i. and hence only a finite number will be admitted. i

=

1 , .... N. The first and second condition guarantee tn .... co for n .... co (see Appendix of Adan and Van der Wal [1987b]). The third condition Xij

>

0 guarantees that a job can complete only one service at a time and hence make at most one transition at a time.

We will prove that for any given realization of the Aj's. Xij·s. and Sij'S for all t and each station the total number of service completions and arrivals up to and includ-ing time t in system K+l is at least equal to the one in system K. Further we will prove that for all t the number of jobs who are admitted in system K+1 and the number of admitted jobs who left system K+l up to and including time t is at least equal to the number in system K.

Theorem

For all t ~ 0 and all i

=

0 . 1 . 2 ... N

and

Since the functions

Dr

_{are constant on the intervals [tm . tm+} 1) and

tn - t 00 (n - t 00) it suffices to prove the theorem for the time instants to . tl ...

3. Preliminaries

The proof of the theorem will be based on the following lemma. stating that if arrivals come earlier then so do departures.

Lemma 1 If

A _ij K+l t:. _~ A _ij K f _{or 1 -}. - 1 2 _{. . .... n}

then

D_ij K+l t:. _~ DK _ij f _{or J -}. - 1 2 _{• ••••• n.}

In order to prove the lemma we need more notation. Define for C

=

K. K+l

Tif

the time a server in station i becomes available to the j-th arriving job

Then we have

and for

Tif

follows

(4)

o

ifj~ ~

the time j-Li jobs of the first j-l arriving jobs in station i have departed if j

>

Li

5

-Let a (at ... an ) be a vector.

Then Ua denotes the nondecreasing reordering of a:

with aj ~ aj ~ ... ~ aj , and ( it • i

_{z •...•}

in ) a permutation of ( 1 • 2 •... , n ).

1 2 n

For two vectors a .. ( at .

az ' ...•

an ) and b"" ( bi ' b2 ••••• bn ) we write

a ~ b if (Ua)j ~ (Ub)i . i = 1 ... n. One may easily verify the following result.

Lemma

2

If aj ~ bi for i - 1 . 2 ... n, then a ~ b. This gives us Corollary If D K+1 ij ~ "'<;;: DK ij f or k -- 1 •.. , • n then T K_in+l"'<;;: +1 ~ TK _in+t· Proof

Note that, if n+ 1

>

Lj • then

is the time upon which n+ 1-Lj jobs of the first n arriving ones have departed

sta-tion i in system C. C .. K, K+1.

Hence the corollary follows immediately from (4) and lemma 2. 0 Now we give the proof of the first lemma.

Proof of lemma 1 By induction.

n-1.

D K+l -il - AK+l it

+

X ..,., it ~ AK it

+

X -il - DK il .

Assume the lemma holds for n

=

m. Then follows from the corollary

TK₊1 _{~ TK}

im+l -.-: im+l.

and hence. by (3)

DK₊1 ~ DK

im+l -.-: im+l·

which proves the lemma for n

=

m+l.

Remark

o

Note that the proof of lemma 1 also holds if Li

=

00. That is. station i is infinite

server.

4. Proof of the theorem

As noted before it suffices to prove (1) for the instants

to .

tl ... This will be done by induction.

For

to

(= 0) inequalities (1) and (2) trivially hold. since DiK+1(O)

=

DiK(O)

=

0 and AiK+1(O)

=

AiK(O)

=

0 for all i.

Assume (1) and (2) hold for

to .

tl ... t_m• Since A8t) is constant on [ tk-l • tk )

thus

and from lemma 1

Further. for all j for which

it follows from X_ij

>

0 that

-7

which proves (1) for tm+l and i = 1 .. _ .. N.

For the total number of admitted jobs who left the network up to and including time tm+l we have. with 80 . j) = 1 if i = j and 0 otherwise.

that is. (1) holds for tm+l and i = O. Now. let us prove that (2) holds for tm+l and i = O. We assume that if one or more jobs leave the network just at the moment a job arrives at the network. they leave just before the arrival and thus they are not counted as being in the system at the arrival instant. So

is _{the number of jobs in system C just before tm+l' C ... K . K+l. Let N(tm+1)} denote the number of arrivals at tm+l. so

00

N(tm+1)

=

1:.

8(Aj • tm+1)'

j=l

Further the number of jobs admitted in the system K+1 at time tm+l is mine K

+

1 - _AJ'+l(tm)

+

DOK+l(tm+l) • N(tm+l»)

~ mine K - _AoK+l(tm)

+

_{D!'(tm+l) • N(tm+l) )}

= mine

K -

_AJ'(tm)

+

_{D!'(tm+l) - [ AoK+1(tm) - AoK(tm) ] . N{tm+l} )}

~ mine K - AoK(t_m)

+

_{DOK(tm+1) • N(tm+1) ) -} [ AoK+1(tm} - AoK(tm)

1.

where the latter inequality follows from

mine a - b • c ) ~ mine a . C ) - b if b ~ O.

Hence

(8) AoK_+1(tm+l)

=

AoK₊1

(tm)

+

mine K

+

1 - _AoK+1(tm)

+

DOK+l(tm+l) • N(tm+l) ) ~ AoK(tm)

+

mine

K -

AoK(tm}

+

D!'(tm+l) . N(tm+l) )

=

AoK(tm+l). so (2) holds for tm+1 and i ... O.

Finally we prove that (2) holds for tm+l and i "" 1 ... N. From (6) and (8) fol-lows

1=1 j=l

that is, inequality (2) holds for tm+l and i = 1 , ... , N. This completes the proof of the theorem.

Remarks

o

- We proved that for all t the total number of jobs admitted in system K+l up to and including time t is at least equal to the number in system K. This does not imply that for all t the number of jobs being in system K+l is at least equal to the number in system K. We give an example. Consider a network consisting of two single server stations (see figure 1). In station 1 a job receives a service time 1 and in station 2 a service time 9. At t

=

0, 20, 40, ... a batch of two jobs arrives at the network and at t == 10,30.50, ... one job arrives at the network. The assignment of jobs to a station is alternating. the first admitted job is assigned to station 1. the second to station 2, the third to station 1, etc. The job leaves the network as soon as the service is completed. If the network has a capacity of two jobs. it is empty at t = 15, 55. 95, ... whereas if it has a capacity of one job it is not empty at those time instants (see figure 2).

figure 1: Model

n~

t

10

C=2

_~

II

I~

9 10 29 39 '+a

n~

h

I

C = 1

Ip

I

IP

0 19 20 30 '+0

" t d~notl!'S thl!' numb~1'" of jobS

in the- SySt~m

C d~not~S th~ capac i t\:l of tht'

ne-twol"'k

figure 2: Situation in the network

10

S9 t "'3110

I

S9 t +

- We assumed that the routing of the jobs within the network followed a Markov chain and the next station where a job would jump to. was drawn after the service completion of a job. If a station is single server it does not make any difference if

-

9-the drawing takes place before 9-the servicing is started or after 9-the service comple-tion. but if a station is multi server it does. since then the j-th arriving job need not be the j-th departing job. The theorem still holds if the drawing takes place before the servicing of a job is started. In this case the Sij denotes the station where the j-th arriving job will jump to. The theorem can be obtained using similar argu-ments as before. As a matter of fact. in the proof only the arguments in order to obtain inequalities (7) and (9) are different. We have to replace (7) and (8) by

N

L

BCSlj.

0) ~

L

and N (11) AiK(tm+l)

=

L L

B(Slj. j)

+

~~(tm+l) 1=1 j D~~tm+l N ~

L

which follow from implication (5). stating that Dif+l ~ Dif for all j for which Dif

~

tm+l'

- The throughput of an open finite capacity network increases if the servicing is being sped up. One can show this by comparing the sample paths of a Cslow) net-work with the ones of a netnet-work having stochastically faster service times. The coupling of the service times of both systems is not as trivial as before (i.e. taking them the same). In this case the service times in the slow system have to be cou-pled with the corresponding ones in the fast system such that the slow service times are with probability 1 at least equal to the fast service times. In order to show that for all t and any given realization of the arrival times. service times and transitions the throughput in the fast system is at least equal to the one in the slow system. one can use similar arguments as before.

- It is easy to verify that the throughput of an open finite capacity network increases if an extra server is added.

- One would probably expect that the throughput increases if the arrival times are being sped up. However. in general this is not true. We give an example. Consider a network consisting of one single server station. The network has a capacity of one job. All service times are equal to 5. In the slow system jobs arrive at t "" O. 6. 12. 18 .... and in the fast system at t ., O. 4. 8, 12 ... The throughput. defined by the average number of service completions per unit time. is

116

job per unit time in the slow system and 1/8 per unit time in the fast system. In this example we con-sidered a deterministic arrival process. In the following section we will show that the throughput is monotone in the arrival rate if jobs arrive according to a Poisson process.

s.

A Poisson arrival stream

We will show that in case jobs arrive according to a Poisson arrival stream the throughput increases if the arrival rate is being sped up. For this purpose. let us consider again an open finite capacity network with multi server stations 1 , ... , N and independent generally distributed service times. The network has a capacity of K jobs. Jobs arrive according to a Poisson arrival stream with rate A. An admitted job joins the queue at station i with probability Pi'

Since jobs arrive according to a Poisson arrival stream. the interarrival times are independent and exponentially distributed with mean A-I. Moreover. it follows from the strong Markov property that from the moment a departure unblocks the system the time till the next job arrives is exponentially distributed with mean A -1. So we can do as if the Poisson arrival stream is switched off as soon as an

arrival blocks the system and is turned on as soon as a departure unblocks the sys-tem. It follows from these observations that the open queueing network can be transformed into a closed queueing network as follows (see figure 3). Replace the outside of the open network by a single server station. station 0 say. with indepen-dent exponential distributed service times with mean A-I. Jobs who leave the net-work. now queue up at station 0 and a job jumps from station 0 to station i with probability Pi' This closed network with K circulating jobs is equivalent with the open network with a capacity of K jobs.

opt'o S~Stem cloSed SyStem

network

Stot ion 0

a r r i v a l r o t e workload in S t a t i o n 0

figure 3: Transformmion to a closed network

It follows from the results of Adan and Van der Wal [1987b] that the throughp'Qt of a closed network is monotone in the number of jobs. the service times and the number of servers. Hence. we have for the open finite capacity network that the throughput is monotone in the capacity (the number of jobs in the closed net-work). the arrival rate (the service times of station 0 in the closed netnet-work). the service times and the number of servers.

Remark

- Consider an open finite capacity network. where jobs arrive according to a Pois-son arrival process with an arrival rate depending on the total number of jobs in the network and the capacity of the network. The arrival rate is (K-k)A. where k denotes the number of jobs in the network and K the capacity of the network. Such an open system can also be transformed into a closed system. In this case one has to replace the outside of the network by a multi server station with K identical

-11-servers and independent exponentially distributed service times with mean

A

-1. Hence. the throughput is monotone in the rate A, the capacity of the network. the service times and the number of servers.

6. Conclusions

In the preceding we studied an open queueing network with independent general service times and Markovian routing. The arrival process was arbitrary and the capacity of the network :finite. We proved the intuitively obvious result that it is rewarding to enlarge the capacity. speed up the service times or add an extra server. If the jobs arrive according to a Poisson process it is also rewarding to speed up the arrival rate. In case of an arbitrary arrival process this need not be true.

We note that the restrictions to Li.d. service times and Markovian routing can be

relaxed. For a discussion. see Adan and Van der Wal [1987a] and [1987b].

Remarks

- Van Dijk. Walrand and Tsoucas {1987] proved the throughput of a multi server queue with a :finite number of servers and waiting facilities (a special case of an open :finite capacity network) to be monotone in the number of servers and waiting places. Their proof was also based on sample path and coupling arguments.

- We considered an open network with a global capacity constraint and proved the throughput to be monotone in the (global) capacity. We give an example of an open network with a local capacity constraint. for which the monotonicity of the throughput in the (local) capacity still holds. Consider an open queueing network consisting of stations 1 ....• N. Station 1 is the only station with a local :finite buffer. All jobs arrive at station 1. An arriving job is lost if the buffer is full. Once a job has completed service at station 1. it is further serviced by the other stations (see :figure 4).

---,

I

I

I 1

-11

b u f f e r

~I:>

I

l

1 _ _ _ _ _ _ _ _ _ _ J open f i n te

.

COpOClt~ sub-network

---,

I

I

Ils,o,;ons

_L

1.2, ••• ,N

r

1 1 1 _ _ _ _ _ _ _ _ J open i n f i n i t e copoc t~ sub-network

figure 4: A loa:zl buffer rrwdel

The throughput increases if the size of the local buffer at station 1 is enlarged. In order to prove so. observe that the network can be uncoupled in two complemen-tary sub-networks. one consisting of station 1 and the other consisting of the remaining stations 2 , ...• N. The sub-network consisting of station 1. is an open

finite capacity network and independent of the complementary sub-network. Hence the throughput of station 1 increases if the capacity (of the buffer) increases. The other sub-network is an open (infinite capacity) network and since it is fed by the departing jobs from station 1. its arrival process is sped up and therefore its throughput increases if the buffer of station 1 is enlarged. Similarly one can prove the throughput of such a network to be monotone in the service times. the number of servers and the arrival rate (in case of a Poisson arrival stream).

- Consider an open finite capacity network with a central buffer (see figure 5). Whenever the network is full. arriving jobs are not admitted in the network. but placed in the central buffer. which has an infinite capacity. If a departure unblocks the system and the central buffer is not empty at the departure instant. another job

is immediately released from the buffer into the network.

f i n i t t >

--~--l:

c t> n t r o b u f f t> r

~

<: a pac ; t

~

' - - - ' n t> t.", 0 r k

figure 5: A central buffer model

For such a network the main theorem. stating that the throughput is monotone in the capacity. still holds. It can be proved by using similar arguments as before. In fact. only the arguments in order to prove inequality (2) for tm+1 and i

=

0 are different. For this we need more notation. Let B(tm+l) denote the total number of arrivals from the outside up to and including time 1m+1' Then follows for the total number of jobs admitted in the network up to and including time tm+l

AOK+l(tm+l)

=

mine K

+

1

+

_{DJ'+l(tm+l) • B(tm+1) )}

~ mine K

+

DJ'(tm+l) . B(tm+l) )

=

AJ'(tm+l). which proves inequality (2) for tm+l and i

=

O.

13-References

- Adan. I.J.B.F. and J. van der Wal (1987a). Monotonicity of the throughput of it

closed queueing network in the number of jobs. Eindhoven University of Technology. Department of Mathematics and Computing Science, Memoran-dum COSOR 87-03.

- Adan. I.J.B.F. and J. van der Wal (1987b), Monotonicity of the throughput of a open queueing network in the interarrival and service times. Eindhoven University of Technology. Department of Mathematics and Computing Sci-ence. Memorandum COSOR 87-05.

- Dijk. N.M. van and B.F. Lamond (1986), Bounds for the call congestion of finite single-server exponential tandem queues, to appear in Operations Research. - Dijk. N.M. van. J. Walrand and P. Tsoucas (1987), Simple bounds for the call

congestion of finite multi-server delay systems, Twente University of Tech-nology, Department of Applied Mathematics, Memorandum 606.

- Robertazzi. T.O. and A.A. Lazar (1985). On the modeling and optimal flow con-trol of the Jacksonian network. Performance Evaluation 5, 29-43.

- Shanthikumar. 1.0. and D.O. Yao (1985), Stochastic monotonicity of the queue lengths in closed queueing networks, to appear in Operations Research.

- Suri. R. (1985). A concept of monotonicity and its characterization for closed queueing networks, Operations Research 33, 606-624.

- Wal. J. van der (1985). Monotonicity of the throughput of a closed exponential queueing network in the number of jobs. Eindhoven University of Technol-ogy, Department of Mathematics and Computing Science. Memorandum COSOR 85-21.

- Yao, D.O. (1985), Some properties of the throughput function of closed networks of queues. Operations Research Letters 3. 313-317.