Abstract
The paper deals with the sensitivity analysis of reliability and performance measures for a multi-server queueing system where customers at the head of the queue retries to occupy a server in exponentially distributed time. The servers can differ in service and reliability characteristics. We have proved the insensitivity of the mean number of customers in the system to the type of allocation policy for equal service rates and confirmed a weak sensitivity in a general case of unequal service rates. A further sensitivity analysis is conducted to investigate the effect of changes in system parameters on a reliability function, a distribution of the number of failures of a server and a maximum queue length during a life time.
Keywords
1 Introduction
Many multi-server queueing systems were investigated under the assumption of absolutely reliability of the servers. The experience of the last decade has shown the high potential to make the modern telecommunication systems more superior in performance and reliability by supplying them with a so-called hybrid or heterogeneous infrastructure. For example, the links of a data transmission channel can differ in reliability and performance characteristics. Such heterogeneous structure can be made very flexible to satisfy different constraints, e.g. packet delay, power consumption, link availability and so on. This paper deals with a multi-server retrial unreliable queueing system, where servers can have either different service rates or reliability attributes, which reflects possible heterogeneous nature of available data transmission facility. It is reasonable to assume that customers who are rejected to get a service immediately upon an arrival join a queue of repeated customers. A constant retrial discipline is assumed. In this case a customer at the head of the queue performs a repeated attempt to occupy a server. The usage of a retrial feature is motivated from one side due to practical interests and necessity to represent such systems more accurately as discussed in [1], but from other side due to the fact that obtained results can be generalized to the case of a truncated version of the classical retrial discipline [2], where a certain large number of customers can repeat independently the request for service, and to get the corresponding approximations for an ordinary queueing system by letting the retrial intensity be very large. Some results and literature overview dedicated to the multi-server retrial non-reliable queueing systems can be found e.g. in [4, 5, 7].
Sensitivity analysis of unreliable systems is often presented as a way of checking whether the reliability characteristics are sensitive to the type of life and repair time distributions, see e.g. [3, 8]. The proposed in this paper sensitivity analysis includes the following contributions:
-
(a)
We have proved that the mean number of customers in the system with identical service rates is insensitive to the type of an allocation policy. In case of different service rates we have shown a weak sensitivity which vanishes as unreliability of servers increases.
-
(b)
We have shown that the reliability function belongs to a class of the PH-type distributions. We have investigated a type of a discrete counterpart of the reliability function in form of a number of failures of a server during a life time of the system.
-
(c)
We have developed the equation to compute a distribution of the maximum queue length in a retrial system during its life time, which represents an alternative reliability descriptor for unreliable queueing systems.
In further sections we will use the notation \(\mathbf{e}_j\) for the column-vector with 1 in the j-th position beginning from 0-th and 0 elsewhere. There is no need to emphasize the dimension of these vector, since it will be clear from the context. The notation \(\delta _{i,j}\) will stand for a Kronecker-Delta function and |A| – the cardinality of a discrete set A.
2 Mathematical Model
We study a M / M / K queueing system with K unreliable servers and a constant retrial discipline (see Fig. 1). The arrival process is Poisson with rate \(\lambda \), the service rate of the j-th server is \(\mu _j\), \(j=1,\dots ,K\). The life time of server j is exponentially distributed with the rate \(\alpha _j\) and is independent of its state (idle or busy). At the moment of failure, if the server is in a busy state, the customer being served moves to another idle server or to the head of the queue. The repair time of the server j is exponentially distributed with the rate \(\beta _j\). The customers, which cannot be immediately served upon an arrival, form a queue of the retrial customers, where only a customer at the head of the queue can perform repeated attempts to get service in exponentially distributed time with the rate \(\tau \). The queueing model with heterogeneous servers always requires a control policy needed to allocate the customers between the servers according to some objective, e.g. minimization of the long-run average number of customers in the system. The system states at time t are described by a Markov process
where Q(t) stands for the number of retrial customers in the queue at time t and the vector \(\mathbf{D}(t)=(D_1(t),\dots ,D(t))\) specifies states of servers, where \(D_j(t)=0,1,2\) means that server j is idle, busy or failed at time t. The state space of \(\{X(t)\}_{t\ge 0}\) is defined by \(E=\{x=(q,\mathbf{d}):q\in \mathbb {N}_0,\mathbf{d}\in E_D\}\), where \(E_D=\{\mathbf{d}=(d_1,\dots ,d_K):d_j\in \{0,1,2\},\,j=1,\dots ,K\}\) is a state space of the process \(\{\mathbf{D}(t)\}_{t\ge 0}\). The optimization problem is formulated as a Markov decision process. The minimum long-run average number of customers in the system, given the initial state \(x\in E\), is
among all admissible control polices \(f:E\rightarrow A\), where \(L(t)=Q(t)+\sum _{j=1}^K\delta _{D_j(t),1}\). Using the method of uniformization [6] we get an optimal policy \(f^*\) together with \(g^*\) as solutions of the optimality equation \(Bv(x)=v(x)+g\), where B is a dynamic programming operator.
Theorem 1
The dynamic programming operator B is defined as follows,
where \(T_0\), \(T_j\), \(T_{b,j,k}\), \(T_{r,j}\) and \(T_s\) – events operators, respectively, for a new arrival, service completion, failure occurrence in idle and busy state, repair completion of server j and retrial arrival,
where \(S_j^{\pm l}x=x\pm l\mathbf{e}_j\), \(S_jx:=S_j^1x\) and \(J_k(x)=\{j:d_j(x)=k\}\).
3 Optimality of the Allocation to an Arbitrary Server
In case of equal service rates, \(\mu _j=\mu \), \(j=1,\dots ,K\), the following statement can be proved:
Theorem 2
The value function \(v:E\rightarrow \mathbb {R}\) satisfies the condition
Proof
The proof is by induction on n in \(v_n\). Let us define \(v_0(x)\) for all states \(x\in E\). This function obviously satisfies the conditions (a)–(d). Now we assume (a)–(b) for the function \(v_n(x)\), \(x\in E\), and some \(n\in \mathbb {N}\). We have proved that \(v_{n+1}=Bv_n\) satisfies the proposed conditions as well. Due to the limit relation \(\lim B^nv_0(x)=v(x)\) we get the inequalities for v.
-
(a)
Since \(J_0(x)=J_0(S_0x)\), \(J_1(x)=J_1(S_0x)\), \(J_2(x)=J_2(S_0x)\) and \(|J_0(x)\cup J_2(x)|=|J_0(S_0x)\cup J_2(S_0x)|\), the inequality (a) can be rewritten as follows,
$$\begin{aligned}&v_{n+1}(x)-v_{n+1}(S_0x)=l(x)-l(S_0x)+\lambda [T_0v_n(x)-T_0v_n(S_0x)]&(I)\\&+\mu \sum _{j\in J_1(x)}[v_n(S_j^{-1}x)-v_n(S_j^{-1}S_0x)]&(II)\\&+\sum _{j\in J_0(x)}\alpha _j[v_n(S_j^2x)-v_n(S_j^2S_0x)]&(III)\\&+\sum _{j\in J_1(x)}\alpha _j[T_0v_n(S_jx)-T_0v_n(S_jS_0x)]&(IV)\\&+\sum _{j\in J_2(x)}\beta _j[v_n(S_j^{-2}x)-v_n(S_j^{-2}S_0x)]&(V)\\&+(1-\delta _{q(x),0})\tau T_0v_n(S_0^{-1}x)-\tau T_0v_n(x)+\delta _{q(x),0}\tau v_n(x)&(VI)\\&+\Big (|J_0(x)\cup J_2(x)|\mu +\sum _{j\in J_2(x)}\alpha _j+\sum _{j\in J_0(x)\cup J_1(x)}\beta _j\Big )&(VII)\\&\times [v_n(x)-v_n(S_0x)]\le 0. \end{aligned}$$The first term (I) is less then 0, since \(l(x)-l(S_0x)=-1< 0\) and
$$\begin{aligned} T_0v_n(x)-T_0v_n(S_0x)\le T_0v_n(x)-v_n(S_0S_jx)\le v_n(S_jx)-v_n(S_0S_jx)\le 0, \end{aligned}$$where \(j\in J_0(S_0x)\cup \{0\}\), by virtue of (a) in state \(S_jx\) and inequality (c). The terms (II), (III),(V) and (VII) are non-positive due to assumption (a) in states \(S_j^{-1}x\), \(S_j^2x\), \(S_j^{-2}x\) and x. For the term (IV) we get,
$$\begin{aligned} (IV)\le \sum _{j\in J_1(x)}\alpha _j[v_n(S_kS_jx)-v_n(S_kS_jS_0x)]\le 0, \end{aligned}$$where \(k\in J_0(S_jS_0x)\cup \{0\}\), due to the inequality (a) in state \(S_kS_jx\). For the term (VI) in case \(q(x)=0\) we obtain,
$$\begin{aligned} (VI)=\tau [v_n(x)-T_0v_n(x)]\le \tau [v_n(x)-v_n(S_jx)]\le 0 \end{aligned}$$for any \(j\in J_0(x)\cup \{0\}\) due to assumptions (a) and (b). In case \(q(x)>0\),
$$\begin{aligned} (VI)=\tau [T_0v_n(S_0^{-1}x)-T_0v(x)]\le \tau [v_n(S_jS_0^{-1}x)-v(S_jx)]\le 0, \end{aligned}$$for any \(j\in J_0(x)\cup \{0\}\) according to assumption (a) in state \(S_jS_0^{-1}x\).
-
(b)
For the next inequality it holds that
$$\begin{aligned}&v_{n+1}(x)-v_{n+1}(S_kx)=l(x)-l(S_kx)+\lambda [T_0v_n(x)-T_0v_n(S_kx)]&(I)\\&+\mu \sum _{j\in J_1(x)}[v_n(S_j^{-1}x)-v_n(S_j^{-1}S_kx)]&(II)\\&+\sum _{j\in J_0(S_kx)}\alpha _j[v_n(S_j^2x)-v_n(S_j^2S_kx)]&(III)\\&+\sum _{j\in J_1(x)}\alpha _j[T_0v_n(S_jx)-T_0v_n(S_jS_kx)]&(IV)\\&+\alpha _k[v_n(S_k^2x)-T_0v_n(S_k^2x)]&(V)\\&+\sum _{j\in J_2(x)}\beta _j[v_n(S_j^{-2}x)-v_n(S_j^{-2}S_kx)]&(VI)\\&+\Big (|J_0(x)\cup J_2(x)|\mu +\sum _{j\in J_2(x)}\alpha _j+\sum _{j\in J_0(x)\cup J_1(x)}\beta _j+\tau \Big )&(VII)\\&\times [v_n(x)-v_n(S_kx)]\le 0, \end{aligned}$$which can be obviously proved in the same way as an inequality (a).
-
(c)
The next iequality yields,
$$\begin{aligned}&v_{n+1}(S_kx)-v_{n+1}(S_lx)=\lambda [T_0v_n(S_kx)-T_0v_n(S_lx)]&(I)\\&+\mu \sum _{j\in J_1(x)}[v_n(S_j^{-1}S_kx)-v_n(S_j^{-1}S_lx)]&(II)\\&+\sum _{j\in J_0(S_kS_lx)}\alpha _j[v_n(S_j^2S_kx)-v_n(S_j^2S_lx)]&(III)\\&+\sum _{j\in J_1(x)}\alpha _j[T_0v_n(S_jS_kx)-T_0v_n(S_jS_lx)]&(IV)\\&+\alpha _l[v_n(S_l^2S_kx)-T_0v_n(S_l^2x)]+\alpha _k[T_0v_n(S_k^2x)-v_n(S_k^2S_lx)]&(V)\\&+\sum _{j\in J_2(x)}\beta _j[v_n(S_j^{-2}S_kx)-v_n(S_j^{-2}S_lx)]&(VI)\\&+(1-\delta _{q(x),0})\tau [T_0v_n(S_0^{-1}S_kx)-T_0v_n(S_0^{-1}S_lx)]&(VII)\\&+\Big ((|J_0(x)\cup J_2(x)|+1)\mu +\sum _{j\in J_2(x)}\alpha _j+\sum _{j\in J_0(x)\cup J_1(x)}\beta _j+\delta _{q(x),0}\tau \Big )&(VIII)\\&\times [v_n(S_kx)-v_n(S_lx)]=0. \end{aligned}$$The terms (I)–(IV), (VI) and (VIII) are obviously equal to 0 due to assumption (c) in corresponding shifted states. For the term (V) we have,
$$\begin{aligned} (V)=\alpha _l[v_n(S_l^2S_kx)-v_n(S_kS_l^2x)]+\alpha _k[v_n(S_lS_k^2x)-v_n(S_k^2S_lx)]=0 \end{aligned}$$and
$$\begin{aligned} (VII)=(1-\delta _{q(x),0})\tau [v_n(S_lS_0^{-1}S_kx)-v_n(S_kS_0^{-1}S_lx)]=0. \end{aligned}$$according to (c) and (d).
-
(d)
The next inequality yields,
$$\begin{aligned}&v_{n+1}(S_kx)-v_{n+1}(S_0x)=\lambda [T_0v_n(S_kx)-T_0v_n(S_0x)]&(I)\\&+\mu \sum _{j\in J_1(x)}[v_n(S_j^{-1}S_kx)-v_n(S_j^{-1}S_0x)]&(II)\\&+\sum _{j\in J_0(S_kx)}\alpha _j[v_n(S_j^2S_kx)-v_n(S_j^2S_0x)]&(III)\\&+\sum _{j\in J_1(x)}\alpha _j[T_0v_n(S_jS_kx)-T_0v_n(S_jS_0x)]&(IV)\\&+\alpha _k[T_0v_n(S_k^2x)-v_n(S_k^2S_0x)]&(V)\\&+\sum _{j\in J_2(x)}\beta _j[v_n(S_j^{-2}S_kx)-v_n(S_j^{-2}S_0x)]&(VI)\\&+(1-\delta _{q(S_kx),0})\tau T_0v_n(S_0^{-1}S_kx)-\tau T_0v_n(x)+\delta _{q(S_kx),0}]\tau v_n(S_kx)&(VII) \end{aligned}$$$$\begin{aligned}&+\Big ((|J_0(x)\cup J_2(x)|+1)\mu +\sum _{j\in J_2(x)}\alpha _j+\sum _{j\in J_0(x)\cup J_1(x)}\beta _j\Big )&(VIII)\\&\times [v_n(S_kx)-v_n(S_0x)]\le 0. \end{aligned}$$
4 Stationary Performance and Reliability Measures
We partition the set of states E into the subsets of states E(q) according to the number of customers in the orbit q, where \(E(q)=\{(q,\mathbf{d}):\mathbf{d}\in E_D\}\), \(q\in \mathbb {N}_0\). To enumerate the vector system states, the \(K+1\)-dimensional state space E is converted to a one-dimensional equivalent state space \(\mathbb {N}_0\) by the function \(h:E\rightarrow \mathbb {N}_0\),
Obviously, for the fixed control policy the process (1) is of a QBD-type with a tri-diagonal block infinitesimal matrix \(Q=[\lambda _{x,y}]_{x,y\in E}\),
consisting of the blocks of dimension \(|E_D|\times |E_D|\). The blocks \(Q_{11}\) and \(Q_{12}\) includes the transitions between the states x and y within a certain subgroup of states, i.e. \(q(x)=q(y)\). In this case for \(x\not =y\) (\(h(x)\not =h(y)\)) and subgroup \(q=0\) we have
The diagonal elements of this block are equal to \( \lambda _{{x,x}}=-\sum \limits _{x\in E_D\atop y\not =x} \lambda _{x,y}. \) The block \(Q_{12}\) at the main diagonal is defined for the subgroup \(q>0\) as
The transitions to the upper subgroup \(q\,+\,1\) takes place in state x with \(|J_0(x)|=0\) and are described by the block \(Q_{01}\) with elements
The transitions to the lower subgroup \(q-1\), \(q\ge 1\), occurs by retrials in states x with \(|J_0(x)|>0\) and are summarized in the block \(Q_{21}\),
The stationary state probability vector \({\varvec{\pi }}=({\varvec{\pi }}_0,{\varvec{\pi }}_1,{\varvec{\pi }}_2,\dots )\), where the sub-vectors \({\varvec{\pi }}_q=(\pi _{(q,\mathbf{d})}:\mathbf{d}\in E_D)\), \(q\in \mathbb {N}_0\), under ergodicity condition is a unique solution to \({\varvec{\pi }}\varLambda =\mathbf{0}\) and \({\varvec{\pi }}\mathbf{e}=1\). The stationary state probability vector exists if and only if
where \(\mathbf{p}\) is the invariant probability of the matrix \(Q=Q_{10}+Q_{12}+Q_{21}\) and satisfies the system \(\mathbf{p}Q=\mathbf{0}\) and \(\mathbf{p}\mathbf{e}=1\). We omit here a closed form of the inequality, since it can not be represented as a compact formula. It is well known from the theory of matrix-analytic solutions that the elements of the vector \({\varvec{\pi }}\) have a property,
The matrix R is the unique non-negative solution with spectral radius less than one of the equation
It is well known that this matrix can be calculated by recursively by successive substitution. The probability of the boundary states \({\varvec{\pi }}_0\) and \({\varvec{\pi }}_1\) by solving the equations for the corresponding states together with a normalization condition,
Some stationary performance and reliability measures are listed below:
-
1.
Availability of the system
$$\begin{aligned} \bar{A}=1-\sum _{x\in E}\mathbb {P}[\cap _{j=1}^K(d_j(x)=2)]=1-{\varvec{\pi }}_1G\mathbf{e}_{|E_D|-1}. \end{aligned}$$ -
2.
Availability of server k
$$\begin{aligned} \bar{A}_k=1-\sum _{x\in E}\mathbb {P}[d_k(x)=2]=1-{\varvec{\pi }}_1G\sum _{x\in E_D}\mathbf{e}_{u(x)}\delta _{d_k(x),2}. \end{aligned}$$ -
3.
Utilization of server k
$$\begin{aligned} \bar{U}_k=\sum _{x\in E}\mathbb {P}[d_k(x)=1]={\varvec{\pi }}_1G\sum _{x\in E_D}\mathbf{e}_{u(x)}\delta _{d_k(x),1}. \end{aligned}$$ -
4.
Mean number of busy servers \(\bar{C}=\sum _{k=1}^K\bar{U}_k\).
-
5.
Mean number of customers in the orbit \(\bar{Q}={\varvec{\pi }}_1(I-R)^{-2}\mathbf{e}\).
-
6.
Mean number of customers in the system \(\bar{N}=\bar{C}+\bar{Q}\).
5 Reliability Measures on the Life Cycle
Denote by T the random value of the time to the first complete failure of the system which occurs in one of the states of the set \(E_1=\{x_1=(q,2,\dots ,2):q\in \mathbb {N}_0\}\subset E\) with \(|J_2(x_1)|=K\) and enumerated by \(u(x_1)=3^K(q(x_1)+1)-1\), \(q(x_1)\in \mathbb {N}_0\), given that initial state is \(x_0=(0,\dots ,0)\in E\) with \(u(x_0)=0\). Define reliability function as \(R(t)=\mathbb {P}[T>t]\). Since the busy and idle state of the server has no influence on its failures, the reliability function is independent on arrivals and service completions. The time T is equivalent to a first passage time of the auxiliary Markov process \(\{\hat{X}(t)\}_{t\ge 0}=\{\hat{D}_1(t),\dots ,\hat{D}_K(t)\}_{t\ge 0}\) with state space \(\hat{E}=\{(d_1,\dots ,d_K):d_j\in \{0,1\}\}\), where \(d_j=0\) means that server j is operational and \(d_j=1\) - server j is failed. The K-dimensional state space \(\hat{E}\) is converted to a one-dimensional space \(\mathbb {N}_0\) by
The process \(\{\hat{X}(t)\}_{t\ge 0}\) starts in state \(x_0\in \hat{E}\) with \(\hat{u}(x_0)=0\) and is absorbed in a single state \(x_1=(1,\dots ,1)\in \hat{E}\), where \(\hat{u}(x_1)=2^K-1\). The infinitesimal matrix \(\hat{\varLambda }=\begin{pmatrix}Q_0 &{} Q_1\\ \mathbf{0} &{} \mathbf{0}\end{pmatrix}\) consists of matrix block \(Q_0\) of dimension \((|\hat{E}|-1)\times (|\hat{E}|-1)\) with transition intensities within the class of transient states and of the vector block \(Q_1\) of dimension \(|\hat{E}|-1\) with transition intensities to the absorbing state \(x_1\). The matrix \(Q_0\) has the following transition intensities for any x with \(\hat{u}(x)\not =2^K-2^{k-1}-1\),
for the vector \(Q_1\) we have
and \(\hat{\lambda }_{x,x}=-\sum _{y\not =x} \hat{\lambda }_{x,y}\).
Denote by \(T_x\) the first passage time to the complete failure state of the system \(x_1\) given that the initial state is \(x\in \hat{E}\), i.e. \(T=T_{x_0}\), and by \(\tilde{r}_x(s)=\mathbb {E}[e^{-s T_x}]\), \(Re[s]>0\), the corresponding Laplace-Stieltjes transform (LST). The column-vector \(\tilde{\mathbf{r}}(s)=(\tilde{r}_x(s),\,x\in \hat{E}\setminus \{x_1\})\) comprises the conditional LSTs enumerated by (9). The employment of the first step analysis yields
Theorem 3
The time to the first failure T has a PH-type distribution with representation \((\mathbf{e}_0,Q_0)\),
Proof
The employment of the first step analysis yields the following system for all \(x\in \hat{E}\setminus \{x_1\}\), i.e. for all x with \(|J_0(x)|>0\),
where \(\tilde{r}_{x_1}(s)=1\) for the absorbing state \(x_1\). By expressing the last equations in matrix form we obtain the expression
for the LST of the PH-type distribution, which together with \(\tilde{R}(s)=\frac{1}{s}(1-\mathbf{e}_0\tilde{\mathbf{r}}(s))\) completes the proof.
Corollary 1
In particular case of \(K=2\) the proposed in (10) reliability measures are of the form,
Now we derive another reliability descriptor, namely a distribution of the number of failures (repairs) of server j during the life time of the system. Denote by \(N_j\) the number of failures of server j left up to absorption time T, \(\psi ^{(j)}_x(n)=\mathbb {P}[N_j=n|X(0)=x]\) – the probability density function (PDF), \(\tilde{\psi }^{(j)}_x(z)=\sum _{n=1}^{\infty }\psi _x^{(j)}(n)z^n\), \(|z|\le 1\) – the probability generating function (PGF), \(\tilde{\varvec{\psi }}^{(j)}(z)=(\tilde{\psi }_x^{(j)}(z),\,x\in \hat{E}\setminus \{x_1\})\) and \(\tilde{\psi }^{(j)}(z)=\tilde{\psi }_{x_0}(z)\). The study of this descriptor complements the reliability analysis providing a type of a discrete counterpart of the length of T.
Theorem 4
The PGF \(\tilde{\psi }^{(k)}(z)\) satisfies the following expression,
where \(\hat{u}(x)=\sum _{l=1\atop l\not =j}^Kd_l(x)2^{l-1}\), \(\hat{u}(x)\not =2^K-2^{j-1}-1\) and \(\hat{u}(y)=\hat{u}(x)+2^{j-1}\), \(x\in \hat{E}\setminus \{x_1\}\).
Proof
Once again we may use the first step analysis to get the system reflecting the dynamic of the PGFs \(\tilde{\psi }_x^{(k)}(z)\),
where \(\tilde{\psi }_{x_1}(z)=1\) for the absorbing state \(x_1\). The resulting system has the same form as for LSTs in case \(s=0\) and the failure rates \(\alpha _j\) are substituted by \(z\alpha _j\). The corresponding result can be expressed in form (11).
Noting that \(\mathbb {E}[N_j(N_j-1)\dots (N_j-n+1)]=\frac{d^n}{dz^n}\tilde{\psi }^{(j)}(z)\Big |_{z=1}\) we get the recursive formula for computing arbitrary factorial moment,
where \(\hat{u}(x)\) and \(\hat{u}(y)\) are the same as for the function \(\tilde{\psi }^{(j)}(z)\).
Corollary 2
The PDF \(\psi ^{(j)}(n)\), \(n\ge 1\), \(j=1,2\) for \(K=2\) is of the form,
We study next the distribution of the maximum number of customers in the orbit \(Q_{max}\) reached by the QBD process during a life time. Denote by \(\varXi _x(n)=\mathbb {P}[Q_{max}\le n|x]\) the probability that starting in \(x\in E\) the QBD process reaches the state from the set \(\tilde{E}_1=\{x_1=(q,2,\dots ,2):q\in \mathbb {N}_0\cap [0,n]\}\) avoiding the states with an orbit size \(n+1\). Therefore, for any fixed n we study an auxiliary process \(\{\tilde{X}(t)\}_{t\ge 0}\) with a state space \(\tilde{E}=E(q),\,q\in \mathbb {N}_0\cap [0,n]\), initial state \(x\in E\) and absorbing states \(\tilde{E}_1\cup E(n+1)\). The employment of the first step analysis leads to the following result.
Theorem 5
The CDF \(\xi (n)\), \(n\ge 0\), satisfies the following expression,
\({\varvec{\alpha }}=\sum _{j=1}^K\alpha _j(\mathbf{e}_{3^K-2\cdot 3^{j-1}-1}+\mathbf{e}_{3^K-3^{j-1}-1})\) and Q are the block matrices of the infinitesimal matrix of the corresponding QBD process \(\{X(t)\}_{t\ge 0}\).
6 Numerical Results
Consider the system M / M / 2 and fix the system parameters at values:
-
Case 1: \(\alpha _1=0.1,\,\alpha _2=0.2,\,\beta _1=0.8,\,\beta _2=0.1\); Figs. 2(a), and 4(a);
-
Case 2: \(\alpha _1=0.01,\,\alpha _2=0.01,\,\beta _1=8,\,\beta _2=8\); Figs. 2(b), and 4(b).
Cases 1 and 2 specify the system supplied by unreliable and reliable servers. The following allocation policies are studied: Fastest Free Server (FFS), Random Server Selection (RSS) and Optimal Threshold Policy (OTP). For comparison analysis we evaluate also the characteristics homogeneous systems with equal service rates operating under FFS and RSS.
The sensitivity analysis of functions g, \(\psi ^{(j)}(n)\) and \(\varXi (n)\) is summarized in Figs. 2, 3 and 4. The first five curves of g at the right hand side of Fig. 2 correspond to the retrial queueing system with \(\tau =2\), while five other curves can be treated as approximations for the ordinary queues. We confirm insensitivity of g to changes of allocation policy in system with identical service rates. For unequal service rates we observe sensitivity of g to policy changes in ordinary queues with reliable servers but it vanishes as \(\tau \) decreases and servers become more unreliable. Figure 3 depicts the PDFs \(\psi ^{(j)}(n)\) for \(j=1\) (a) and \(j=2\) (b) versus reliability attributes. We notice that for more reliable servers the tail of the distribution becomes heavier. The effect of allocation policies to the maximum queue length is shown in Fig. 4. The jump at the point \(n=0\) equals to the probability of a complete failure which occurs earlier as a customer comes to an empty system. We observe that the maximum queue length takes the highest values in ordinary queueing system with reliable servers.
7 Conclusions
In this paper, we have provided explicit expressions for system performance and reliability measures of the multi-server unreliable retrial queueing system with constant retrial policy and threshold-based allocation mechanism. Some interesting conclusions about insensitivity of the optimal control policy to the changing of reliability characteristics of servers as well as about system reliability were performed in the paper.
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The publication was prepared with the support of the “RUDN University Program 5-100”, RFBR according to the research project No. 16-37-60072 mol.a.dk.
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Efrosinin, D. (2017). Sensitivity Analysis of Reliability and Performability Measures for a Multi-server Queueing System with Constant Retrial Rate. In: Rykov, V., Singpurwalla, N., Zubkov, A. (eds) Analytical and Computational Methods in Probability Theory. ACMPT 2017. Lecture Notes in Computer Science(), vol 10684. Springer, Cham. https://doi.org/10.1007/978-3-319-71504-9_23
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