Maximum likelihood estimates and confidence intervals of an M/M/R queue with heterogeneous servers

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O R I G I NA L A RT I C L E

Tsung-Yin Wang · Jau-Chuan Ke Kuo-Hsiung Wang · Siu-Chuen Ho

Maximum likelihood estimates

and confidence intervals of an M/M/R

queue with heterogeneous servers

Received: 15 March 2003 / Accepted: 15 July 2004 / Published online: 17 January 2006 © Springer-Verlag 2006

Abstract This paper studies maximum likelihood estimates as well as confidence intervals of an M/M/R queue with heterogeneous servers under steady-state condi-tions. We derive the maximum likelihood estimates of the mean arrival rate and the three unequal mean service rates for an M/M/3 queue with heterogeneous servers, and then extend the results to an M/M/R queue with heterogeneous servers. We also develop the confidence interval formula for the parameterρ, the probability of empty system P0, and the expected number of customers in the system E[N], of an M/M/R queue with heterogeneous servers.

Keywords Confidence interval · Heterogeneous servers · Maximum likelihood estimate· Queue

1 Introduction

In this paper, we study both point estimations and confidence intervals of an M/M/R queue with ordered heterogeneous servers under steady-state conditions. It is as-sumed that customers arrive following a Poisson process with rate λ and with service times according to an exponential distribution with R unequal mean ser-vice rates µi, (i = 1, 2, . . . R), where µ1 > µ2 > · · · > µR. We assume the

following: (i) Arriving customers at the servers form a single waiting line and are served in the order of their arrivals; (ii) Each server may serve only one customer T.-Y. Wang

Department of Industrial Engineering and Management, National Chiao Tung University, Hsinchu 300, Taiwan

J.-C. Ke

Department of statistics, National Taichung Institute of Technology Taichung 404, Taiwan K.-H. Wang (

B

)· S.-C. Ho

Department of Applied Mathematics, National Chung-Hsing University, Taichung, 402, Taiwan E-mail: khwang@amath.nchu.edu.tw

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at a time; (iii) If all servers are idle, the first customer in the waiting line goes to the fastest server; (iv) If part of the servers are idle, the first customer goes to the faster server; (v) If all servers are busy, the first customer waits until any one server becomes free; (vi) The arrival process and the service process are independent.

The statistical analysis of queueing systems are rarely found in the literature and the work of related systems in the past mainly concentrates on only one server or two servers. A landmark paper in parameter estimations for queueing models was first proposed by Clarke (1957), who derived maximum likelihood estimates for the arrival and service parameters of an M/M/1 queue. Lilliefors (1966) investigated the confidence intervals for the M/M/1, M/Ek/1 and M/M/2 queues. Basawa and

Prabhu (1981) examined moment estimates as well as maximum likelihood esti-mates for a G/G/1 queue. An illustration of statistical estimation technique applied to the queueing problems can be found in Rubin and Robson (1990). Jain (1991) obtained maximum likelihood estimates of the parameters for an M/Ek/1 queue.

Basawa et al. (1996) studied maximum likelihood estimates of the parameters in the single-server queue using waiting time data. Rodrigues and Leite (1998) used Bayesian analysis to investigate the confidence intervals of an M/M/1 queue. Max-imum likelihood estimates and confidence intervals in an M/M/2 queue with heter-ogeneous servers were derived by Dave and Shah (1980) and Jain and Templeton (1991), respectively. Abou-E1-Ata and Hariri (1995) developed point estimations and confidence intervals of an M/M/2/N queue with balking and heterogeneous servers. Recently, an overview of literature on the statistical analysis of several queueing systems was provided by Dshalalow (1997).

The main purpose of this paper is to derive maximum likelihood estimates and confidence intervals of an M/M/R queue with ordered heterogeneous servers. In section 2, we derive the maximum likelihood estimates of parameters for an M/M/3 queue with heterogeneous servers and consider two special cases. Similar procedure is used and extended to an M/M/R queue with heterogeneous servers and the results are presented in section 3. Two special cases are also considered. Finally, section 4 presents the confidence interval formula for the parameterρ, the probability of empty system P0, and the expected number of customers in the system E[N], of an M/M/R queue with heterogeneous servers.

2 M/M/3 queue with heterogeneous servers

In this section, our objective is to develop the maximum likelihood estimates of the mean arrival rateλ and the three unequal mean service rates µ1,µ2andµ3, whereµ1> µ2> µ3of the M/M/3 queue with heterogeneous servers.

In steady-state, the following notations are used.

P0≡ probability that there are no customers in the system,

Pn ≡ probability that there are n customers in the system,

where n= 1, 2, 3, . . . .

Steady-state equations for an M/M/3 queue with heterogeneous servers are given by:

λP0= µ1P1, (1)

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(λ + µ1+ µ2)P2= (µ1+ µ2+ µ3)P3+ λP1, (3)

(λ + µ1+ µ2+ µ3)Pn = (µ1+ µ2+ µ3)Pn+1+ λPn−1, n ≥ 3. (4)

Solving recursively, analytic solutions Pnare derived in the following:

Pn=      λ µ1P0, n= 1 λ2 µ1(µ1+µ2)P0, n= 2 λ3 µ1(µ1+µ2)(µ1+µ2+µ3)P0, n≥ 3 (5) where P0= 1+ λ µ1 + λ2 µ1(µ1+ µ2)(1 − ρ) ₋₁ , (6) and ρ = λ µ1+ µ2+ µ3.

Since the queue is in steady-state, soρ must be less than 1 or equivalently

λ < µ1+ µ2+ µ3.

2.1 Likelihood function and maximum likelihood estimates

At time t = 0, the queue has just started operation with m0customers present. Let

T denote a fixed sufficiently large interval of time during which the queue is being

observed. During T , we assume that there are Nanumber of arrivals to the queue

and Nd number of departures from the queue. Following Dave and Shah (1980),

we observe during T that:

Te≡ amount of time during which three servers are idle;

TB1≡ amount of time during which only the fastest server is busy;

TB2 ≡ amount of time during which both the fastest server and faster server

are busy;

TB3≡ amount of time during which three servers are busy;

Ne ≡ number of arrivals to an empty queue when three servers are idle

(transitions E0to E1);

NB1 ≡ number of arrivals to a partially busy queue when the fastest server is

busy (transitions E1to E2);

NB2≡ number of arrivals to a partially busy queue time when the fastest server

and faster server are busy (transitions E2to E3);

NB3 ≡ number of arrivals to a completely busy queue when three servers are

busy (transitions Ei to Ei+1, i ≥ 3);

ND1 ≡ number of departures from a partially busy queue when the fastest

server is busy (transitions E1to E0);

ND2 ≡ number of departures from a partially busy queue when the fastest

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ND3 ≡ number of departures from a completely busy queue when three servers

are busy (transitions Ei to Ei−1, i ≥ 3).

Obviously,

T = Te+ TB1+ TB2+ TB3, Na= Ne+ NB1+ NB2+ NB3, Nd = ND1+ ND2+ ND3.

Following Abou-E1-Ata and Hariri (1995), the corresponding likelihood func-tion can be broken down into the following five basic components:

(i) The probability that there are initial m0customers in the system can be ob-tained from (6) yielding Pm0 = λ

3

µ1(µ1+µ2)(µ1+µ2+µ3)ρ m0−3_P

0, m0≥ 3; (ii) The probability density function of Ne transitions (E0 to E1) occurring

during time Teis given by f1= λNee−λTe;

(iii) The probability density function of NB1 transitions (E1to E2) occurring and ND1 transitions (E1 to E0) occurring during time TB1 is given by

f2= λN_B1 e−λTB1µND1 1 e−µ1 T_B1 ;

(iv) The probability density function of NB2 transitions (E2to E3) occurring

and ND2 transitions (E2 to E1) occurring during time TB2 is given by f3= λN_B2_e−λT_B2_(µ 1+ µ2)ND2e−(µ1+µ2)TB2 ;

(v) The probability density function of NB3 transitions (Ei to Ei+1, i ≥ 3)

occurring and ND3transitions (Ei to Ei−1, i ≥ 3) occurring during time TB3 is given by f4 =

λN_B3_e−λT_B3_µN_D3_e−µT_B3_{, where} _{µ = µ}

1+

µ2+ µ3.

Since the random variables m0, TBi , NBi and NDi (i = 1, 2, 3) are mutually independent, the likelihood function is given by

L1(λ, µ1, µ2, µ3) = λm0+Nae−λTµND3−m0+3µ N_D1 1 ×(µ1+ µ2)ND2e−µ1TB1−(µ1+µ2)TB2−µTB3 × P0 µ1(µ1+ µ2)µ . (7)

Since the queue is in steady-state, the probability Pm0can be neglected. Taking

the logarithm of (7), it implies that

lnL1= lnL1(λ, µ1, µ2, µ3) = Nalnλ − λT + ND1lnµ1+ ND2ln(µ1+ µ2) +ND3lnµ − µ1TB1− (µ1+ µ2)TB2− µTB3.

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Differentiating (8) with respect to the parametersλ, µ1, µ2 andµ3, respec-tively, we finally obtain

∂lnL1 ∂λ |λ=ˆλ= Na ˆλ − T = 0, (9) ∂lnL1 ∂µ1 |µi= ˆµi = ND1 ˆµ1 + ND2 ˆµ1+ ˆµ2 + ND3 ˆµ − (TB1+ TB2+ TB3) = 0, (10) ∂lnL1 ∂µ2 |µi= ˆµi = ND2 ˆµ1+ ˆµ2 + ND3 ˆµ − (TB2+ TB3) = 0, (11) ∂lnL1 ∂µ3 |µi= ˆµi = ND3 ˆµ − TB3= 0. (12) From (9), we have ˆλ = Na T . (13)

Subtracting (11) from (10), we get

ˆµ1=

ND1 TB1

. (14)

Subtracting (12) from (11) and using (14), we get

ˆµ2= ND2 TB2 − ND1 TB1 . (15) We obtain from (12) ˆµ = ˆµ1+ ˆµ2+ ˆµ3= ND3 TB3. (16) It implies from (14)–(16) ˆµ3= ND3 TB3 − ND2 TB2 . (17)

Thus, the maximum likelihood estimates ofλ , µ1,µ2andµ3are given in (13), (14), (15), and (17), respectively.

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2.2 Special cases

We consider the following two special cases:

Case 1:

Letµi− µi₊₁= δi, (i = 1, 2), we have

µ1= µ2+ δ1= µ3+ δ1+ δ2, (18)

µ2= µ3+ δ2. (19)

Substituting (18)–(19) into (8) and using = TB1+2TB2+3TB3,1= δ1+δ2 and2= δ1+ 2δ2, yields the following log-likelihood function:

lnL2=lnL2(λ, µ3) = Nalnλ−λT +ND1ln(µ3+1)+ND2ln(2µ3+2) + +ND3ln(3µ3+2) − µ3TB1−2µ3TB2−3µ3TB3 (20)

Using a derivation analogous to that of the previous section, we get the estimates ofλ and µ3as follows: ˆλ = Na T , (21) and ˆµ3 3 + ₆₁_{+ 5}₂ 6 − Nd ˆµ2 3 +512+ 22 6 − 52ND1+ 2(31+ 2)ND2+ 3(21+ 2)ND3 6 ˆµ3 +122 6 − 2 2ND1+ 212ND2+ 312ND3 6 = 0. (22)

It should be noted that the positive real value of ˆµ3should be taken in order to give the maximum log-likelihood function, and then ˆµ1and ˆµ2can be obtained from the expressions (18)–(19).

Ifδ1= δ2= 0, the estimates of µ1,µ2andµ3are given by

ˆµ1= ˆµ2= ˆµ3= Nd TB1+ 2TB2+ 3TB3. (23) Ifδ1= 0 and δ2= 0, we have ˆµ3 3+ ₈_δ₂ 3 − Nd ˆµ2 3+ δ2 ₇_δ₂ 3 − 2Nd + ND1+ ND2 3 ˆµ3+ δ2 2 ₂_δ₂ 3 − Nd + ND1+ ND2 3 = 0. (24)

The positive real value of ˆµ3should be taken in order to give the maximum log-likelihood function, and then ˆµ1and ˆµ2can be obtained from the expressions (18)–(19).

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Ifδ1= 0 and δ2= 0, we get ˆµ3 3+ ₁₁_δ₁ 6 − Nd ˆµ2 3+ δ1 δ1− 3Nd 2 + 4ND1+ ND2 6 ˆµ3+ δ2 1 δ1 6 − Nd 2+ 2ND1+ ND2 6 = 0. (25)

The positive real value of ˆµ3should be taken in order to give the maximum log-likelihood function, and then ˆµ1and ˆµ2can be obtained from the expressions (18)–(19). Ifδ1= δ2= δ = 0, we obtain ˆµ3 3 + ₉_δ 2 − Nd ˆµ2 3+ δ ₁₃_δ 2 − 7Nd 2 + 2ND1+ ND2 2 ˆµ3 +δ2₃_{δ −} 3Nd + 3ND1+ 2ND2 2 = 0. (26)

The positive real value of ˆµ3should be taken in order to give the maximum log-likelihood function, and then ˆµ1and ˆµ2can be obtained from the expressions (18)–(19). Case 2: Let µi+1 µi = θi < 1, (i = 1, 2), we have µ2= θ1µ1, µ3= θ2µ2= θ1θ2µ1.

Using a derivation similar to that of Case 1, the estimates ofµ1,µ2andµ3are given by ˆµ1= Nd TB1+ (1 + θ1)TB2+ (1 + θ1+ θ1θ2)TB3, (27) ˆµ2= θ1 Nd TB1+ (1 + θ1)TB2+ (1 + θ1+ θ1θ2)TB3, (28) ˆµ3= θ1θ2 Nd TB1+ (1 + θ1)TB2+ (1 + θ1+ θ1θ2)TB3. (29) Ifθ1= θ2= 1, we have ˆµi = Nd TB1+ 2TB2+ 3TB3, i = 1, 2, 3. (30) Ifθ1= θ2= θ < 1, we obtain ˆµi = θ i−1_N d TB1+ (1 + θ)TB2+ (1 + θ + θ2)TB3 , i = 1, 2, 3. (31)

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3 M/M/R queue with heterogeneous servers

In this section, we will derive the maximum likelihood estimates of the mean arrival rateλ and the R unequal mean service rates µi, (i = 1, 2, . . . R), where µ1> µ2> · · · > µRof the M/M/R queue with heterogeneous servers.

As in the M/M/3 queue, the steady-state equations for an M/M/R queue with heterogeneous servers are as follows:

λP0= µ1P1, (32) (λ + n j=1 µj)Pn= n+1 j=1 µjPn+1+ λPn−1, 1 ≤ n ≤ R − 1 (33) (λ + µ)Pn= µPn+1+ λPn−1, n ≥ R (34) whereµ =R_j₌₁µj.

Solving recursively for this set of linear equations, we have

Pn=    λn n k=1kj=1µjP0, 1≤ n ≤ R − 1 λR R k=1kj=1µjρ n−R_P 0, n ≥ R (35) and P0= 1+ R−1 n=1 λn n k=1 k j=1µj + λR (1 − ρ)R k=1 k j=1µj −1 . (36)

Let E[N] denote the expected number of customers in the system. From (34), we finally obtain E[N]= ∞ n=1 nPn= _R₋₁ n=1 nλn n k=1 k j=1µj +R λR k=1 k j=1µj R− Rρ + ρ (1 − ρ)2 P0, (37) whereρ = λ/R_j₌₁µj.

3.1 Likelihood function and maximum likelihood estimates

At time t = 0, the queue has just started operation with m0customers present. Let

T denote a fixed sufficiently large interval of time during which the queue is being

observed. During T , we assume that there are Nanumber of arrivals to the queue and Ndnumber of departures from the queue. During T , we observe the following:

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TB1≡ amount of time during which only the fastest server is busy;

TBi ≡ amount of time during which i faster servers are busy where i = 2, 3, . . . , R − 1;

TBR≡ amount of time during which all servers are busy;

Ne ≡ number of arrivals to an empty queue when all servers are idle (transitions E0to E1);

NB1≡ number of arrivals to a partially busy queue when only the fastest server

is busy (transitions E1to E2);

NBi ≡ number of arrivals to a partially busy queue time when i servers are busy (transitions Ei to Ei+1) where i = 2, 3, . . . , R − 1;

NBR ≡ number of arrivals to a completely busy queue when all servers are busy (transitions Ei to Ei+1, i ≥ R);

ND1 ≡ number of departures from a partially busy queue when only the fastest

server is busy (transitions E1to E0);

NDi ≡ number of departures from a partially busy queue when i faster servers are busy (transitions Ei to Ei−1) where i = 2, 3, . . . , R − 1;

NDR ≡ number of departures from a completely busy queue when all servers are busy (transitions Ei to Ei−1, i ≥ R).

It is clear that T = Te+ R j₌₁ TBj, Na= Ne+ R j=1 NBj, Nd = R j=1 NDj.

Since the queue is in steady-state, the probability Pm0can be neglected. As in

the M/M/3 queue, the corresponding likelihood function can be broken down into the following three basic components:

(i) The probability density function of Ne transitions (E0 to E1) occurring during time Teis given byλNee−λTe;

(ii) The probability density function of NBi transitions (Ei to Ei+1, 1≤ i ≤ R−1) occurring and NDitransitions (Eito Ei−1, 1≤ i ≤ R−1) occurring during time TBiis given by

λN_Bi e−λTBii j=1µj N_Di e− i j=1µj T_Bi ;

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(iii) The probability density function of NBR transitions (Ei to Ei+1, i ≥ R) occurring and NDRtransitions (Eito Ei−1, i ≥ R) occurring during time TBRis given by λN_BR e−λTBR µN_DR e−µTBR .

Using a derivation similar to that of the previous section, the log-likelihood function is given by lnL3= lnL3(λ, µ1, µ2, . . . , µR) = Nalnλ − λT + R−1 k=1 NDkln k j=1 µj +NDRlnµ − R−1 k=1 k j=1 µjTBk − µTBR. (38) Using (38) and after some algebraic manipulations, we obtain the maximum likelihood estimates ofλ and µi (i = 1, 2, . . . , R)

ˆλ = Na T , (39) ˆµ1= ND1 TB1 , (40) and ˆµi = NDi TBi − NDi−1 TBi−1 , for 2 ≤ i ≤ R. (41) From (40)–(41), we get ˆµ = R i=1 ˆµi = NDR TBR . (42)

Thus, we get the maximum likelihood estimate ofρ

ˆρ = NaTBR NDRT . (43) 3.2 Special cases Let µi+1 µi = θi, (i = 1, 2, . . . , R − 1), we have µi+1= µ1 i k=1 θk, i = 1, 2, . . . , R − 1.

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Using the procedure in section 2.2, we get the estimates ofµi ˆµi= Ndik−1=1θk R−1 k₌₁ 1+k_j₌₂k_l₌₁−1θl TBk + 1+R_j₌₂_lj₌₁−1θl TBR , 1 ≤ i ≤ R (44) where theb_n_=anotation indicates the term is 0 when a > b and theb_n_=anotation indicates the term is 1 when a> b.

Two special cases are considered in the following:

Case 1: θi = 1, (i = 1, 2, . . . , R − 1), (44) can be simplified to ˆµ1= ˆµ2= . . . = ˆµR = Nd R k₌₁kTBk . (45) Case 2: θi = θ < 1, (i = 1, 2, . . . , R − 1), (44) can be simplified to ˆµi = (1 − θ)θ i−1_N d R k=1(1 − θk)TBk , 1 ≤ i ≤ R. (46)

4 Confidence interval formula forρ, P0and E[N]

In this section, we will develop the confidence interval formula for ρ, P0 and

E[N] of an M/M/R queue with heterogeneous servers. To achieve our aim, we first

establish the following results.

For a simple birth–death process to an M/M/R queueing system with hetero-geneous servers, it follows from Appendix that

E[N] = −ρ∂lnP0

∂ρ ≥ 0, (47)

and the variance

Var[N] = ρ∂ E[N]

∂ρ ≥ 0. (48)

Next, applying the approach by Lilliefors (1966), we have the(1 − α) × 100% lower and upper confidence limits L_ρand U_ρ ofρ as follows;

L_ρ = ˆρF_1−α/2(2Na, 2Nd), (49)

and

U_ρ = ˆρF_α/2(2Na, 2Nd), (50)

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One observes from (47) that P0is a monotonic decreasing function ofρ. Hence, the(1 − α) × 100% lower and upper confidence limits, LP0and UP0of P0can be obtained through (36) and (49)–(50). That is,

LP0 = P0|ρ=Uρ, (51)

and

UP0 = P0|ρ=Lρ. (52)

From (48), we observe that E[N] is a monotonic increasing function of ρ. Thus, the(1 − α) × 100% lower and upper confidence limits, LE[N]and UE[N]of E[N], can be obtained through (37) and (49)–(50). That is,

LE[N]= E[N]|ρ=L_ρ, (53)

and

UE[N]= E[N]|ρ=Uρ. (54)

5 Conclusions

In this paper, we have developed the maximum likelihood estimates for the arrival and service parameters of the M/M/3 queue and M/M/R queue with heterogeneous servers, respectively. We also have demonstrated that both results for the maxi-mum likelihood estimates of the parameters are the functions of the observations only which are consistent with the results of Dave and Shah (1980). The estimates ofλ and µi (i = 1, 2, . . . , R) can be easily computed, due to the fact that these

observations can easily be made for an M/M/R queue with heterogeneous servers. Next, we have derived the confidence interval formula forρ, P0and E[N] of an M/M/R queue with heterogeneous servers.

Appendix

Derivations of (47) and (48)

Taking the logarithm of(36) and differentiating it with respect to λ, we finally get

∂lnP0 ∂λ = − R−1 n=1 nλn−1 n k=1 k j=1µj +R λR−1 k=1 k j=1µj · R− Rρ + ρ (1 − ρ)2 P0. (A–1) Multiplying (A–1) by−λ and using (37), we obtain

E[N] = −λ∂lnP0

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Since ∂lnP0

∂ρ = ∂lnP∂λ0/∂ρ∂λ, we have ∂lnP∂ρ0 = µ∂lnP∂λ0. We obviously get

ρ∂lnP0

∂ρ = λ ∂lnP0

∂λ . (A–3)

It follows from (A–2) and (A–3) that

E[N] = −ρ∂lnP0

∂ρ ≥ 0, (A–4)

which is the result given in (47). Since ∂ln P0

∂λ = ∂ P∂λ0∂ln P∂ P00 =

∂ P0

∂λ P01, we have ∂ P 0

∂λ = P0∂ln P_∂λ0. It yields from (A–2) that ∂ P0 ∂λ = − P0 λ E[N]. (A–5) Let Q= R−1 n=1 nλn n k=1 k j=1µj +R λR k=1 k j=1µj  R − Rρ + ρ (1 − ρ)2 . Thus Q= −λ P0 ∂ln P0 ∂λ = − λ P₀2 ∂ P0 ∂λ. (A–6)

From (A–5) and (A–6), we obtain E[N] = Q P0. Differentiating E[N] with respect toλ yields ∂ E[N] ∂λ = P0∂ Q ∂λ + Q ∂ P0 ∂λ, (A–7) where ∂ Q ∂λ = R−1 n=1 n2λn−1 n k=1 k j=1µj +R λR−1 k=1 k j=1µj _R2 1− ρ+ (2R + 1)ρ (1 − ρ)2 + 2ρ2 (1 − ρ)3 .

After doing some algebraic manipulations in (35), we obtain

E[N2]= ∞ n=1 n2Pn = R−1 n=1 n2λn n k=1 k j=1µj +RλRρ−R k=1 k j=1µj ∞ n=R n2ρn P0 = R−1 n=1 n2λn n k=1 k j=1µj +R λR k=1 k j=1µj  R2 1− ρ+ (2R+1)ρ (1−ρ)2 + 2ρ2 (1−ρ)3 P0 =λP0∂ Q ∂λ. (A–8)

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Since ∂ E[N]_∂ρ = ∂ E[N]_∂λ ∂λ_∂ρ, we have ∂ E[N]_∂ρ = µ∂ E[N]_∂λ . From (A–7), we get

ρ∂ E[N]_∂ρ = λ∂ E[N]_∂λ = λP0∂ Q

∂λ + λQ ∂ P0

∂λ. (A–9)

It follows from (A–5) and (A–8) that

ρ∂ E[N]

∂ρ = E[N2] − (E[N])2= V ar[N] ≥ 0, (A–10)

which is the result given in (48).

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