Cooperative Game Theoretic Analysis of Shared Services

Abstract

Shared services are increasingly popular among firms and are often modeled as multi-class queuing systems. Several priority scheduling rules are possible to schedule customers from different classes. These scheduling rules can be static, where a class has strict priority over the other class, or can be dynamic based on delay and certain weights for each class. An interesting and important question is how to fairly allocate the waiting cost for shared services.

In this paper, we address the above problem using the solution concepts of cooperative game theory. We first appropriately define worth functions for each player (class), each coalition, and the grand coalition for multi-class M/G/1 queue with non-preemptive priority. It turns out that the worth function of the grand coalition follows Kleinrock’s conservation law. We fully analyze the $2-$class game and obtain the fair waiting cost allocations from several cooperative games’ solution concepts viewpoints. These include Shapley value, the core, and nucleolus. We prove the $2-$class game is convex which implies that the core is non-empty and the Shapley value allocation belongs to the core. Cooperative game-theoretic solutions capture fairness. We characterize the closed-form expression for these scheduling policies as bringing out various fairness aspects amongst scheduling policies. We consider Delay dependent priority (DDP) rule to determine fair scheduling policies from the Shapley value and the core-based allocation. We present extensive numerical experiments by partitioning the stability region for 2-class queues in three sub-regions.

A Proofs

Proof of Proposition 1: It follows that for any convex game the core is non-empty and the nucleolus belongs to the core [15] (See page 424 and 455). As we already proved this is a Convex Game. so, the core is non-empty and the nucleolus belongs to the core. Also, Shapley value belongs to the core. From [15] (see page 416) the core, $\mathcal {C}(\mathcal {P}, v) = \{x = (x_1,\cdots ,x_n) \in \mathbb {R}^n: \sum \limits _{i=1}^n x_i = v(\{N\}); \sum \limits _{i \in C} x_i \ge v(\{C\}), \forall ~C\subseteq \mathcal {P}\}.$ For $2-$class game the core is $\mathcal {C}(\mathcal {P}, v) = \{x = (x_1, x_2) \in \mathbb {R}^2: \sum \limits _{i=1}^2 x_i = v(\{N\}); \sum \limits _{i \in C} x_i \ge v(\{C\}), \forall ~C\subseteq \mathcal {P}\}.$ For this $2-$class game by using Eqs. (1)-(3) we simplify write $x_1 \ge \dfrac{\rho _1 W_0}{(1-\rho _1)}, x_2 \ge \dfrac{\rho _2 W_0}{(1-\rho _2)}, x_1+x_2 \ge \dfrac{\rho W_0}{(1-\rho )}, x_2 = \dfrac{\rho W_0}{(1-\rho )}-x_1$. From these expressions we get Proposition 1.    $\blacksquare$

Proof of Proposition 2: It follows that there exists exactly one mapping $\phi : \mathbb {R}^{2^N-1} \rightarrow \mathbb {R}^N$ that satisfies all three Axioms (Symmetry, Linearity, and Carrier) [15] (see page 432). This mapping satisfies

where N is the total number of players. The above expression for $\phi _i(v)$ gives the expected contribution of player i to the worth of any coalition and is called Shapley value. For this $2-$class game all the possible coalitions are $(\{\phi \}),(\{1\}), (\{2\})$ and $(\{12\})$. From the above expression, the Shapley values of player 1 and player 2 are:

$$\phi _1(v) = \frac{1}{2}\left[ v(\{1\})+v(\{12\})-v(\{2\})\right] ,$$

$$\phi _2(v) = \frac{1}{2}\left[ (v(\{2\})+v(\{12\})-v(\{1\})\right] .$$

Now from Eqs. (1, 2, 3) and mean waiting time expressions we simplify the Shapley value of player 1 and player 2 as given in the proposition.    $\blacksquare$

Proof of Proposition 3: Using Eqs. (12) and (14) we can write $\hat{\phi _1} = \mathbb {E}(W_1^{\beta })$ and $\hat{\phi _2} = \mathbb {E}(W_2^{\beta })$. From Eqs. (15) and (16) mean waiting time for class 1 and class 2 under DDP are as follows:

$$\begin{aligned} \mathbb {E}[W_1^{DDP}(\beta )] = \frac{W_0 (1-\rho (1-\beta ))}{(1-\rho )(1-\rho _1(1-\beta ))}1_{\{\beta \le 1\}}+\frac{W_0}{(1-\rho )(1-\rho _2(1-\frac{1}{\beta }))}1_{\{\beta >1\}}, \end{aligned}$$

$$\begin{aligned} \mathbb {E}[W_2^{DDP}(\beta )] = \frac{W_0}{(1-\rho )(1-\rho _1(1-\beta ))}1_{\{\beta \le 1\}}+\frac{W_0 (1-\rho (1-\frac{1}{\beta }))}{(1-\rho )(1-\rho _2(1-\frac{1}{\beta }))}1_{\{\beta > 1\}}, \end{aligned}$$

Now by comparing $\hat{\phi _1}$ and $\hat{\phi _2}$ with the mean waiting time of class 1 and class 2 under DDP for $\beta \le 1$ we get the following:

$$\begin{aligned} \frac{W_0 [\rho \rho _2+2(1-\rho )]}{2(1-\rho _1)(1-\rho _2)(1-\rho )} = \frac{W_0 (1-\rho (1-\beta ^{Shapley}))}{(1-\rho )(1-\rho _1(1-\beta ^{Shapley}))} \end{aligned}$$

  and  

$$\begin{aligned} \frac{W_0 [\rho \rho _1+2(1-\rho )]}{2(1-\rho _1)(1-\rho _2)(1-\rho )} = \frac{W_0}{(1-\rho )(1-\rho _1(1-\beta ^{Shapley}))} \end{aligned}$$

Similarly, for $\beta > 1$ we get the following:

$$\begin{aligned} \frac{W_0 [\rho \rho _2+2(1-\rho )]}{2(1-\rho _1)(1-\rho _2)(1-\rho )} = \frac{W_0}{(1-\rho )(1-\rho _2(1-\frac{1}{\beta ^{Shapley}}))} \end{aligned}$$

  and  

$$\begin{aligned} \frac{W_0 [\rho \rho _1+2(1-\rho )]}{2(1-\rho _1)(1-\rho _2)(1-\rho )} = \frac{W_0 (1-\rho (1-\frac{1}{\beta ^{Shapley}}))}{(1-\rho )(1-\rho _2(1-\frac{1}{\beta ^{Shapley}}))} \end{aligned}$$

Compiling all the results we get the outcome mentioned in Proposition 3.    $\blacksquare$

Proof of Proposition 4: From Eq. (15) and (16) mean waiting time for class 1 and class 2 under DDP are as follows:

$$\begin{aligned} \mathbb {E}[W_1^{DDP}(\beta )] = \frac{W_0 (1-\rho (1-\beta ))}{(1-\rho )(1-\rho _1(1-\beta ))}1_{\{\beta \le 1\}}+\frac{W_0}{(1-\rho )(1-\rho _2(1-\frac{1}{\beta }))}1_{\{\beta >1\}}, \end{aligned}$$

$$\begin{aligned} \mathbb {E}[W_2^{DDP}(\beta )] = \frac{W_0}{(1-\rho )(1-\rho _1(1-\beta ))}1_{\{\beta \le 1\}}+\frac{W_0 (1-\rho (1-\frac{1}{\beta }))}{(1-\rho )(1-\rho _2(1-\frac{1}{\beta }))}1_{\{\beta > 1\}}, \end{aligned}$$

For the 2-class game The Core expression we got, $x_1 \ge \dfrac{\rho _1 W_0}{(1-\rho _1)}, x_2 \ge \dfrac{\rho _2 W_0}{(1-\rho _2)}, x_1+x_2 \ge \dfrac{\rho W_0}{(1-\rho )}, x_2 = \dfrac{\rho W_0}{(1-\rho )}-x_1$. Now by comparing the mean waiting time for class 1 under DDP with Core allocation of class 1 $(x_1)$ and mean waiting time for class 2 under DDP with Core allocation of class 2 $(x_2)$ we get the following:

$$\frac{W_0}{(1-\rho )(1-\rho _1(1-\beta ^{Core}))} \ge \frac{W_0}{(1-\rho _1)}, \beta ^{Core} \le 1$$

$$\frac{W_0}{(1-\rho )(1-\rho _2(1-\frac{1}{\beta ^{Core}}))} \ge \frac{W_0}{(1-\rho _2)}, \beta ^{Core} > 1$$

After simplifying the above inequalities we get the ranges of $\beta$ as mentioned in this proposition.    $\blacksquare$

Proof of Proposition 5: Let’s assume Nucleolus is Shapley value ($\phi _1,\phi _2$). Now excess of coalition 1, $e(1,\phi _1) = v(1)-\phi _1$ and excess of coalition 2, $e(2,x) = v(2)-\phi _2$. Excess of the grand coalition, $e(12, \phi ) = 0$. We found $e(1,\phi _1)=e(2,\phi _2)=\dfrac{-\rho _1\rho _2W_0(2-\rho )}{2(1-\rho _1)(1-\rho _2)(1-\rho )}$. Now suppose there are any other allocations $(x_1^*,x_2^*)$, that can further reduce $e(1,\phi _1)$ and $e(2,\phi _2)$. It means $v(\{1\})-x_1^*<v(\{1\})-\phi _1$ and $v(\{2\})-x_2^*<v(\{2\})-\phi _2$. This implies $x_1^*>\phi _1$ and $x_2^*>\phi _2$ which cannot be true as $\phi _1 +\phi _2 = \dfrac{\rho W_0}{1-\rho }$. Thus Shapley value is Nucleolus.